# The complete Pick property for pairs of kernels and Shimorin’s factorization

SCOTT MCCULLOUGH  DEPARTMENT OF MATHEMATICS, UNIVERSITY OF FLORIDA,
GAINESVILLE, FL<EMAIL_ADDRESS>and  GEORGIOS TSIKALAS  DEPARTMENT OF MATHEMATICS
AND STATISTICS, WASHINGTON UNIVERSITY IN ST. LOUIS, ST. LOUIS, MO
<EMAIL_ADDRESS>
###### Abstract.

Let $(\mathcal{H}_{k},\mathcal{H}_{\ell})$ be a pair of Hilbert function
spaces with kernels $k,\ell$. In a 2005 paper, Shimorin showed that a certain
factorization condition on $(k,\ell)$ yields a commutant lifting theorem for
multipliers $\mathcal{H}_{k}\to\mathcal{H}_{\ell}$, thus unifying and
extending previous results due to Ball-Trent-Vinnikov and Volberg-Treil. Our
main result is a strong converse to Shimorin’s theorem for a large class of
holomorphic pairs $(k,\ell),$ which leads to a full characterization of the
complete Pick property for such pairs. We also present a short alternative
proof of sufficiency for Shimorin’s condition. Finally, we establish necessary
conditions for abstract pairs $(k,\ell)$ to satisfy the complete Pick
property, further generalizing Shimorin’s work with proofs that are new even
in the single-kernel case $k=\ell.$ Our approach differs from Shimorin’s in
that we do not work with the Nevanlinna-Pick problem directly; instead, we are
able to extract vital information for $(k,\ell)$ through Carathéodory-Fejér
interpolation.

###### Key words and phrases:

pairs of reproducing kernels, complete Pick property, complete Carathéodory
property, interpolation

###### 2010 Mathematics Subject Classification:

46E22

Tsikalas partially supported by National Science Foundation Grant DMS 2054199.
Also funded by the Onassis Foundation - Scholarship ID: F ZR 061-1/2023-2024.

##  1\. Introduction

Given $n$ points $z_{1},\dots,z_{n}$ in the unit disc $\mathbb{D}=\\{|z|<1\\}$
in the complex plane and $n$ complex numbers $w_{1},\dots,w_{n},$ when does
there exist a holomorphic function $\phi:\mathbb{D}\to\overline{\mathbb{D}}$
such that

$\phi(z_{i})=w_{i}\hskip 2.84544pt\text{ for }1\leq i\leq n?$

Pick’s theorem [37] tells us that such a $\phi$ exists if and only if the
$n\times n$ self-adjoint matrix

$\bigg{[}\frac{1-w_{i}\overline{w_{j}}}{1-z_{i}\overline{z_{j}}}\bigg{]}$
(1.1)

is positive semi-definite, henceforth simply positive or PsD. This result has
had a major, and continuing, impact on both function theory and operator
theory. The operator theory approach to interpolation, pioneered by Sarason
[39], rests on interpreting Pick’s Theorem in terms of a partially defined
multiplier.

This operator theoretic multiplier view of Sarason was dramatically expanded
upon in unpublished work of Agler by viewing Pick’s theorem as a special
property of multipliers of reproducing kernel Hilbert spaces. A reproducing
kernel Hilbert space is a complete Pick space if it hosts a matrix-valued
analogue of Pick’s theorem (see Definition 1.1 below). Examples include the
classical Hardy space $H^{2}$, standard weighted Dirichlet spaces on the unit
disc, the Sobolev space $W^{2}_{1}$ on the unit interval and the Drury–Arveson
space $H^{2}_{d}$ on the unit ball of $\mathbb{C}^{d}.$ A complete
characterization of complete Pick spaces was achieved through work of the
first author [34, 33] and Quiggin [38], while Agler and McCarthy [2] proved
that $H^{2}_{d}$ is universal among such spaces (see subsection 2.2). Since
their inception, complete Pick spaces have proven to be a natural venue for a
number of function-theoretic operator theory topics such as Carleson measures
[10], interpolating sequences [32, 5, 17], invariant subspaces [35],
factorizations [3, 4], weak product spaces [27, 7], the column-row property
[24] and operator models [18, 13]. The book [2] is a standard reference (see
also the surveys [23, 40]).

Here we will study interpolation for multipliers between spaces. Accordingly,
let $\mathcal{H}_{k},\mathcal{H}_{\ell}$ denote reproducing kernel Hilbert
spaces on the non-empty set $X,$ with kernels $k$ and $\ell,$ respectively.
Thus, $k,\ell:X\times X\to\mathbb{C}$ are positive semi-definite functions.
Given a positive integer $N$, let $\mathbb{M}_{N}$ denote the set of all
$N\times N$ matrices with complex entries. A function
$\Phi:X\to\mathbb{M}_{N}$ is a multiplier
$\mathcal{H}_{k}\otimes\mathbb{C}^{N}\to\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}$
if $f\in\mathcal{H}_{k}\otimes\mathbb{C}^{N}$ implies $\Phi
f\in\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}.$ In this case, the principle of
uniform boundedness implies $\Phi$ determines a bounded linear operator
$M_{\Phi}$ from $\mathcal{H}_{k}\otimes\mathbb{C}^{N}$ to
$\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}$ given by $M_{\Phi}f=\Phi f$ and we
let
$\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathbb{C}^{N},\mathcal{H}_{\ell}\otimes\mathbb{C}^{N})$
denote the set of these multipliers. In the case that $k=\ell$ and $N=1,$ we
write $\operatorname{Mult}(\mathcal{H}_{k}).$ The norm of a multiplier refers
to the norm of the operator $M_{\Phi}.$

For $w\in X,$ letting $k_{w}\in\mathcal{H}_{k}$ denote the function
$k_{w}(z)=k(z,w),$ a routine computation shows if
$\Phi\in\operatorname{Mult}\big{(}\mathcal{H}_{k}\otimes\mathbb{C}^{N},\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}\big{)}$
and $h\in\mathbb{C}^{N},$ then $M_{\Phi}^{*}k_{w}\otimes
h=k_{w}\otimes\Phi(w)^{*}h.$ It is well-known, and not difficult to verify,
that a function $\Phi:X\to\mathbb{M}_{N}$ is a multiplier of norm at most one
if and only if

$X\times X\ni(z,w)\mapsto\ell(z,w)I_{N\times N}-k(z,w)\Phi(z)\Phi(w)^{*}$

is a (positive) kernel. Thus, given an $n,$ points $z_{1},\dots,z_{n}\in X$
and $n$ matrices $W_{1},\dots,W_{n}\in\mathbb{M}_{N}$, a necessary condition
for the existence of a multiplier
$\Phi\in\operatorname{Mult}\big{(}\mathcal{H}_{k}\otimes\mathbb{C}^{N},\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}\big{)}$
of norm at most one that satisfies $\Phi(z_{i})=W_{i},$ for all $1\leq i\leq
n,$ is that the block matrix

$\begin{bmatrix}\ell(z_{i},z_{j})I_{N\times
N}-k(z_{i},z_{j})W_{i}W^{*}_{j}\end{bmatrix}_{i,j=1}^{n}$ (1.2)

is positive. Returning to Pick’s Theorem, a bounded analytic function
$\varphi:\mathbb{D}\to\mathbb{C}$ determines a multiplier of Hardy-Hilbert
space $H^{2}(\mathbb{D}).$ The space $H^{2}(\mathbb{D})$ is a reproducing
kernel Hilbert space whose kernel is Szegő’s kernel
$\mathfrak{s}(z,u)=(1-z\overline{u})^{-1}$ for $z,u\in\mathbb{D};$ that is, if
$f\in H^{2}(\mathbb{D})$ and $u\in\mathbb{D},$ then,

$\langle f,\mathfrak{s}_{u}\rangle=f(u).$

The matrix-valued version of Pick’s Theorem says, in the case
$k=\ell=\mathfrak{s},$ that the necessary condition above is also sufficient.

######  Definition 1.1.

A pair $(k,\ell)$ of kernels on $X$ is a complete Pick pair if, for every
positive integer $n,$ for every choice of points $z_{1},\dots,z_{n}\in X,$ for
every positive integer $N$ and for every choice of matrices
$W_{1},\dots,W_{n}\in\mathbb{M}_{N}$ for which the matrix in (1.2) is
positive, there exists a multiplier
$\Phi\in\operatorname{Mult}\big{(}\mathcal{H}_{k}\otimes\mathbb{C}^{N},\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}\big{)}$
of norm at most one that satisfies,

$\displaystyle\Phi(z_{i})=W_{i},$

for $i=1,2,\dots,n.$ ∎

In the special case $k=\ell$, the kernel $k$ is known in the literature as a
complete Pick kernel. As the prototypical example, Szegő’s kernel is a
complete Pick kernel. Since they play an important role in this paper, a
further discussion of complete Pick kernels appears in Subsection 2.2 below.
The expressions $(k,\ell)$ is a complete Pick pair,
$(\mathcal{H}_{k},\mathcal{H}_{\ell})$ is a complete Pick pair, $(k,\ell)$ has
the complete Pick property will be used interchangeably in the sequel. Also,
for brevity, we will often use the terms CP pair and CP property instead.

It is expected that many deep properties of complete Pick kernels have natural
extensions in the setting of complete Pick pairs. In fact, such extensions
have already been worked out for reproducing kernels with a complete Pick
factor (see e.g. [3, 19, 43, 20] and [24, Section 3.8]). These are kernels
$\ell$ with the property that there exist a complete Pick kernel $s$ and a
kernel $g$ such that

$\ell=s\cdot g,$

giving rise to the complete Pick pair $(s,\ell)$ (see [5, Proposition 4.4]).
Another class of examples is implicitly contained in work of Volberg and Treil
[42], who proved a commutant lifting theorem for certain pairs of function
spaces on the unit disc. A consequence of their results is that if
$\mathcal{H}_{k}$ and $\mathcal{H}_{\ell}$ are (normalized) Hilbert spaces of
holomorphic functions on $\mathbb{D}$ such that the shift $S:z\mapsto zf(z)$
is contractive on $\mathcal{H}_{\ell}$ and expansive on $\mathcal{H}_{k},$
then $(\mathcal{H}_{k},\mathcal{H}_{\ell})$ is a complete Pick pair. We note
that the spaces $\mathcal{H}_{k}$ in this setting can be viewed as generalized
de Branges-Rovnyak spaces; see [31, 26, 6] for recent activity concerning such
spaces.

A substantial contribution to the theory of complete Pick pairs can be found
in the 2005 paper [41] of Shimorin, who proved a commutant lifting theorem for
pairs of spaces that unifies and extends analogous results found in [12] and
[42]. In particular, Shimorin was able to show that a certain factorization
property of $k$ and $\ell$ is sufficient for $(k,\ell)$ to be a complete Pick
pair.

######  Definition 1.2.

Assume $k,\ell$ are kernels on $X.$ A kernel $s$ on $X$ will be called a
strong Shimorin certificate for $(k,\ell)$ if it is a complete Pick kernel and
also there exist kernels $h,g$ on $X$ with

$k=(1-h)s\hskip 8.5359pt\text{ and }\hskip 8.5359pt\ell=sg.$ (1.3)

######  Theorem 1.3 ([41], Theorem 1.3).

Suppose $(k,\ell)$ is a pair of kernels on the set $X.$ If there exists a
strong Shimorin certificate for $(k,\ell),$ then $(k,\ell)$ is a complete Pick
pair.

Setting $X=\mathbb{D}$ and $s=\mathfrak{s}$ in (1.3) and choosing
$\mathcal{H}_{\ell}$ to be a holomorphic space allows one to recover the
Volberg-Treil class of pairs of kernels described above, while setting
$h\equiv 0$ leads to pairs $(s,\ell)$ where $s$ is a complete Pick factor of
$\ell.$ Shimorin actually worked with operator-valued pairs $(k,\ell)$. His
proof of Theorem 1.3 rested on recasting the Pick interpolation problem in
this setting as a matrix completion problem, which then allows for the use of
Parrott’s Lemma, a strategy previously employed in [38] and [1], as well as in
unpublished work of Agler, in the single-kernel setting. We also point out
that in order to prove the full commutant lifting theorem for
$(\mathcal{H}_{k},\mathcal{H}_{\ell})$, Shimorin imposed an additional
regularity condition on $k$ (see condition “(R)” in the statement of [41,
Theorem 1.1]), which, roughly, ensures that $\text{Mult}(\mathcal{H}_{k})$
shares sufficiently many elements with $\text{Mult}(\mathcal{H}_{s})$. An
alternative approach to the commutant lifting theorem in the important special
case that $k=s$ is the Drury-Arveson kernel can be found in [20] (if $k=s$,
then Shimorin’s condition (R) is automatically satisfied; see the discussion
in the second half of [41, p. 139]).

In Section 4, we will give a new, short proof of Theorem 1.3 that utilizes a
version of Leech’s Theorem valid in the context of a complete Pick kernel.
Further, in Section 3, we will consider the notion of a Shimorin certificate
for a pair $(k,\ell)$ (see Definition 3.1). Our definition is motivated by
Shimorin’s proof of Theorem 1.3 and, in particular, the consideration of the
precise conditions that have to be satisfied in order to extend a multiplier
$\mathcal{H}_{k}\to\mathcal{H}_{\ell}$ on a set of points to one more point.
Every pair that possesses a strong Shimorin certificate also has a Shimorin
certificate, though the converse does not always hold (see Proposition 3.6).
Still, it turns out that the existence of a Shimorin certificate for
$(k,\ell)$ continues to be sufficient for the complete Pick property to hold,
which leads us to a generalization of Theorem 1.3 (see Theorem 4.5).

###  1.1. Diagonal complete Pick pairs

Before we state our first main result, we establish some notation and
terminology. A kernel $f(z,w)$ on a domain111Following the usual convention,
domain means open, connected and non-empty.
$\Omega\subseteq\mathbb{C}^{\mathsf{g}}$ is a holomorphic kernel if
$f:\Omega\times\Omega\to\mathbb{C}$ is a kernel (a PsD function) that is
holomorphic in $z$ and conjugate holomorphic in $w.$ Let $\mathbb{N}$ denote
the set of all non-negative integers and fix an integer ${\mathsf{g}}\geq 1$.
Given $a\in\mathbb{N}^{\mathsf{g}}$ and $z\in\mathbb{C}^{\mathsf{g}},$ let

$a=(a_{1},\dots,a_{{\mathsf{g}}}),\hskip
11.38092pt|a|=|a_{1}|+\dots+|a_{{\mathsf{g}}}|,\hskip
11.38092ptz^{a}=z^{a_{1}}_{1}\cdots z_{{\mathsf{g}}}^{a_{{\mathsf{g}}}}.$

Also, let $\mathbf{0}=(0,0,\dots,0)\in\mathbb{N}^{\mathsf{g}}.$ As a
definition, assuming $\mathbf{0}\in\Omega,$ the holomorphic kernel $f$ is a
diagonal holomorphic kernel if there exists a sequence of positive numbers
$\\{f_{a}\\}_{a\in\mathbb{N}^{\mathsf{g}}}$ such that the power series
expansion of $f$ at $\mathbf{0}$ takes the form

$f(z,w)=\sum_{a\in\mathbb{N}^{\mathsf{g}}}f_{a}z^{a}\overline{w}^{a},\hskip
8.5359ptz,w\in\Omega.$ (1.4)

Finally, a diagonal holomorphic $k$ will be termed normalized if
$k_{\mathbf{0}}=1$ in (1.4). A further discussion of such kernels appears in
subsection 2.3.

Our main result is a strong converse to Shimorin’s Theorem 1.3 for diagonal
holomorphic kernels.

######  Theorem 1.4.

A pair of diagonal holomorphic kernels $(k,\ell)$ on a domain
$\mathbf{0}\in\Omega$ is a complete Pick pair if and only if it possesses a
strong Shimorin certificate.

In the context of Theorem 1.4, more is true. Not only does a complete Pick
pair of diagonal holomorphic kernels have a strong Shimorin certificate, but
it has a distinguished certificate $s$ that depends only upon $k.$ This kernel
$s$ is, in a sense, minimum among all diagonal holomorphic certificates for
$(k,\ell).$ Moreover, there are restrictions placed on the domain $\Omega$ in
terms of the domains of convergence of the power series for $k$ and $\ell$ as
discussed below. These domain restrictions are illustrated by examples
involving Bergman-type kernels in Section 6 (Examples 6.6, 6.8) and Section 8.

Before stating Theorem 1.6 below, two additional ingredients require an
introduction. The first is a canonical domain for holomorphic diagonal
kernels. For a diagonal holomorphic kernel $f$ on a domain
$\mathbf{0}\in\Omega\subseteq\mathbb{C}^{\mathsf{g}}$ with series expansion at
the origin as in equation (1.4), let $\Omega_{f}$ denote the domain of
convergence (see [30, Definition 2.3.11]) of the series $\sum_{a}f_{a}x^{2a}.$
In particular, $\Omega_{f}$ is a domain222It is a logarithmically convex
complete Reinhardt domain and is thus, in particular, star-like with respect
to the origin. and

$\Omega_{f}=\operatorname{interior}\mathcal{C}_{f}=\cup_{r>0}\\{x:\sum_{a}f_{a}|y|^{2a}<\infty,\
\ \text{ for all }\|x-y\|<r\\},$ (1.5)

where

$\mathcal{C}_{f}=\\{x\in\mathbb{C}^{\mathsf{g}}:\sup f_{a}|x|^{2a}<\infty\\}.$

In this case, $\Omega\subseteq\Omega_{f}$ and $f$ extends to be a diagonal
holomorphic kernel on $\Omega_{f}$; see Proposition 2.8.

The second ingredient is the notion of a master certificate.

######  Definition 1.5.

Let $k$ be a normalized diagonal holomorphic kernel on
$\Omega\subseteq\mathbb{C}^{\mathsf{g}},$ so that

$k(z,w)=1+\sum_{a\in\mathbb{N}^{\mathsf{g}}\setminus\\{0\\}}k_{a}z^{a}\overline{w}^{a}.$

Define $\vartheta_{\mathbf{0}}=0$ and $\vartheta_{e_{j}}=k_{e_{j}}$ for all
$1\leq j\leq{\mathsf{g}}$. Assuming $M\geq 1$ and $\vartheta_{b}$ has been
defined for all $|b|\leq M$, let, for $|b|=M+1$,

$\vartheta_{b}=\max\Bigg{\\{}0,k_{b}-\sum_{\begin{subarray}{c}w+u=b,\\\
w,u\neq\mathbf{0}\end{subarray}}\vartheta_{w}k_{u}\Bigg{\\}}.$

The master certificate associated with $k$ is the formal power series in
${\mathsf{g}}$ complex variables defined as333If not for the max below, we
would have $\vartheta=1-\frac{1}{k}.$ Moreover, in the case that $k$ is a
complete Pick kernel, $\vartheta=1-\frac{1}{k}.$

$\vartheta(x)=\sum_{b\in\mathbb{N}^{\mathsf{g}}}\vartheta_{b}x^{b}.$

Set

$\Omega_{\vartheta}^{1}=\\{x\in\Omega_{\vartheta}:\sum_{a\neq\mathbf{0}}\vartheta_{a}|x|^{2a}<1\\}.$
(1.6)

We can now state a refined version of Theorem 1.4. For convenience, we will
work with normalized kernels.

######  Theorem 1.6.

Let $(k,\ell)$ be a pair of normalized diagonal holomorphic kernels on a
domain $\mathbf{0}\in\Omega\subseteq\mathbb{C}^{\mathsf{g}}$ and let
$\vartheta$ denote the master certificate from Definition 1.5. If $(k,\ell)$
is a complete Pick pair, then

$\Omega\subseteq\Omega_{\ell}\subseteq\Omega_{\vartheta}^{1}\subseteq\Omega_{k}.$

Moreover, the following are equivalent:

  1. (i)

$(k,\ell)$ is a complete Pick pair on $\Omega;$

  2. (ii)

$(k,\ell)$ has a Shimorin certificate on $\Omega;$

  3. (iii)

$(k,\ell)$ has a strong Shimorin certificate on $\Omega;$

  4. (iv)

$s(z,w)=\frac{1}{1-\vartheta(z\overline{w})}$ is a strong Shimorin certificate
for $(k,\ell)$ on $\Omega;$

  5. (v)

there exists a positive kernel $g$ on $\Omega$ such that

$\ell(z,w)=\frac{g(z,w)}{1-\vartheta(z\overline{w})}$

on $\Omega.$

Theorem 1.6 is expanded upon as Theorem 6.4 and proved in Section 6.

###  1.2. Carathéodory interpolation for diagonal pairs

Our proof of Theorem 6.4, and thus Theorem 1.6, does not proceed directly from
the complete Pick property as Shimorin’s proof in [41, Theorem 1.3] did.
Instead we obtain our necessary and sufficient conditions through
consideration of the Carathéodory interpolation problem for pairs of
(normalized) diagonal kernels.

Recall that the classical Carathéodory interpolation problem for $H^{\infty}$
(also termed the Carathéodory-Fejér problem in the literature), originally
introduced and studied by Carathéodory [16, 15], can be phrased as follows:
Given $c_{0},c_{1},\dots,c_{n}\in\mathbb{C},$ when do there exist complex
numbers $c_{n+1},c_{n+2},\dots$ so that
$\phi(z)=\sum_{i=0}^{\infty}c_{i}z^{i}$ is analytic and satisfies
$\sup_{z\in\mathbb{D}}|\phi(z)|\leq 1$? This problem has since been studied in
many different settings (see [39, p. 186] for additional references, also [9,
44, 21, 8, 28, 22, 11]).

Our interest lies in a matrix-valued version of Carathéodory interpolation
that applies to pairs of kernels. Assume $(k,\ell)$ is a pair of (normalized)
diagonal holomorphic kernels on a domain
$\Omega\subseteq\mathbb{C}^{\mathsf{g}}$ and let
$F\subseteq\mathbb{N}^{\mathsf{g}}$ be a co-invariant set of indices (see
Definition 2.13). Given a positive integer $J$ and a collection of matrix
coefficients $\\{c_{a}:a\in F\\}\subseteq\mathbb{M}_{J}$, when does there
exist a collection $\\{c_{a}:a\in\mathbb{N}^{\mathsf{g}}\setminus
F\\}\subseteq\mathbb{M}_{J}$ such that the function

$\Phi(z)=\sum_{a\in\mathbb{N}^{\mathsf{g}}}z^{a}\otimes c_{a}$ (1.7)

is a contractive multiplier from $\mathcal{H}_{k}\otimes\mathbb{C}^{J}$ to
$\mathcal{H}_{\ell}\otimes\mathbb{C}^{J}$? A necessary condition for the
existence of such a multiplier is that the block upper-triangular matrix $C$
indexed by $F\times F$ with block $J\times J$ entries

$C_{a,b}=\begin{cases}c_{b-a}^{*}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise,}\end{cases}$ (1.8)

is a contraction, where $b\geq a$ means $b_{j}\geq a_{j}$ for
$a=(a_{1},\dots,a_{\mathsf{g}})$ and similarly for $b.$ If, for any $J\geq 1$
and any choice of co-invariant $F\subseteq\mathbb{N}^{\mathsf{g}}$ and
matrices $\\{c_{a}:a\in F\\}\subseteq\mathbb{M}_{J}$, this necessary condition
is also sufficient for the existence of a contractive multiplier $\Phi$ as in
(1.7), we say $(k,\ell)$ is a complete Carathéodory pair. In the special case
$k=\ell,$ the kernel $k$ is known as a complete Carathéodory kernel. For a
more extensive discussion of this definition and further background on the
classical setting, see subsection 2.4. As with the complete Pick property, the
expressions $(k,\ell)$ is a complete Carathéodory pair,
$(\mathcal{H}_{k},\mathcal{H}_{\ell})$ is a complete Carathéodory pair,
$(k,\ell)$ has the complete Carathéodory property will be used interchangeably
in the sequel. We will often replace them with the shorter versions CC pair
and CC property.

In [33], the first author characterized those kernels that satisfy an abstract
version of the complete Carathéodory property formulated in terms of a
backwards shift operator; see [33, Theorem 7.2]. This characterization was
further extended in [25], where it was shown that a single holomorphic kernel
$k$ (satisfying a few very mild regularity assumptions) is CC if and only if
it is CP; see [25, Corollary 3.3]. The fact that this equivalence continues to
hold for pairs of kernels will be one of the main components of our proof
strategy for Theorem 1.6. First, we show directly that any CP pair of diagonal
holomorphic kernels is also a CC pair (Theorem 5.19) and are thus led to the
following question. Does every diagonal CC pair $(k,\ell)$ possess a strong
Shimorin certificate? We show that the answer is yes for diagonal kernels and,
in fact, in that case we can always choose the $s$ from Theorem 1.6(iv) as our
certificate (Corollaries 5.11 and 6.3). The key observation here is that the
inequalities obtained by looking at the positivity condition

“$s=(1-\vartheta)^{-1}$ is a factor of $\ell$”

coefficient-wise can all be encoded by appropriate choices of contractive
block matrices $C$ as in (1.8). We also point out that the factorization
$k=(1-h)s$ follows immediately from the definition of $\vartheta$; see
Proposition 5.9. After obtaining that $s=(1-\vartheta)^{-1}$ is a strong
Shimorin certificate for the CC pair $(k,\ell),$ an application of Theorem 1.3
shows that $(k,\ell)$ is CP, concluding our argument. In this indirect fashion
we obtain the following corollary.

######  Theorem 1.7.

A pair of diagonal holomorphic kernels $(k,\ell)$ is a complete Pick pair if
and only if it is a complete Carathéodory pair.

###  1.3. The Bergman kernel as an example

Recall that the conclusion of Theorem 1.6 can be reformulated as follows: for
every CP pair of diagonal holomorphic kernels $(k,\ell)$, there exists a
minimum strong Shimorin certificate that only depends on $k$. Unfortunately,
this strong canonicity result does not survive if we drop the assumption that
$\ell$ is diagonal.

######  Theorem 1.8.

Let $\mathfrak{b}$ denote the Bergman kernel on the unit disc
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}.$ There exists an $0<r<1$, and a one-
parameter family of CP kernels $\\{s_{\lambda}\\}_{\lambda\in\Lambda}$ with
the following properties:

  1. (i)

For every $\lambda\in\Lambda,$ there exists a kernel $h_{\lambda}$ on
$r\mathbb{D}$ such that

$\mathfrak{b}=(1-h_{\lambda})s_{\lambda};$

  2. (ii)

There does not exist a CP kernel $s$ on the disc $r\mathbb{D}$ such that
$\mathfrak{b}=(1-h)s$ for some kernel $h$ on $r\mathbb{D}$ and with the
property that, for every $\lambda\in\Lambda,$ there exists a kernel
$g_{\lambda}$ on $r\mathbb{D}$ such that

$s_{\lambda}=sg_{\lambda}.$ (1.9)

In particular, $(\mathfrak{b},s_{\lambda})$ is a family of complete Pick pairs
over $r\,\mathbb{D},$ but there does not exist a kernel $s$ that serves
simultaneously as a certificate for each pair.

Theorem 1.8 further implies that, given a CP pair $(\mathfrak{b},\ell)$, there
will not, in general, exist a strong Shimorin certificate that is a factor of
every other strong Shimorin certificate for $(\mathfrak{b},\ell),$ even if
$\ell$ is taken to be diagonal; see Corollary 8.4. Thus, the certificate $s$
from Theorem 1.6(iv) will, in general, only be minimum among diagonal
holomorphic strong Shimorin certificates.

###  1.4. Necessary conditions

Despite the negative results of subsection 1.3, we do provide new necessary
conditions for a pair of abstract kernels to be a CP pair including an
extension of the following theorem of Shimorin.

######  Theorem 1.9 ([41], Theorem 1.4).

If $(k,\ell)$ is a complete Pick pair of kernels on $X,$ then there exists a
kernel $g$ on $X$ such that $\ell=kg.$

Given a kernel $k$ on a set $X$ and a non-empty subset $Y\subseteq X$, let
$k^{Y}$ denote the reproducing kernel associated with the subspace

$\\{f\in\mathcal{H}_{k}:f(w)=0\text{ for all }w\in Y\\}$

of $\mathcal{H}_{k}.$ If $Y=\\{w\\}$ is a singleton, we write $k^{w}$ in place
of $k^{\\{w\\}}$. Recall (see also subsection 2.2) that $k$ is a CP kernel if
and only if, for every $w\in X,$ there exists a kernel $g_{w}$ on $X$ such
that

$k^{w}=kg_{w}.$ (1.10)

Theorem 1.11 below generalizes this result to pairs of kernels and can be seen
as an extension of [41, Theorem 1.4] quoted above as Theorem 1.9. Its proof
proceeds directly from the complete Pick property.

######  Theorem 1.10.

Let $(k,\ell)$ be a CP pair of kernels on a set $X.$ If $Y\subseteq X$ is
finite, then $(k,\ell^{Y})$ is also a CP pair and, consequently, there exists
a kernel $g_{Y}$ on $X$ such that

$\ell^{Y}=kg_{Y}.$ (1.11)

In the special case of a complete Pick kernel, where $\ell=k,$ the argument
used in the proof of Theorem 1.10 (see Theorem 7.3), while following the same
general approach as other proofs (besides those mentioned already another
proof can be found in [29]), yields a new and somewhat simpler proof of the
necessity of (1.10) for the single kernel case.

As it turns out, conditions of the form (1.11) are far from sufficient for a
pair $(k,\ell)$ to be CP, even in the diagonal holomorphic case. In fact, we
will see that even replacing $\ell^{Y}$ by the kernel of any multiplier-
invariant subspace of $\mathcal{H}_{\ell}$ will, in general, not be enough to
guarantee sufficiency; see Proposition 7.4 and the discussion preceding it.

Finally, in Theorem 9.1, the necessary and sufficient conditions of Theorem
1.6 are formulated without reference to the certificate
$s(z,w)=(1-\vartheta(z\overline{w}))^{-1},$ which we know may not be available
in the non-diagonal case. Even though these conditions are still specific to
diagonal holomorphic kernels, we hope that they may provide a more practical
framework for a future derivation of new necessary conditions for non-diagonal
pairs to be CP.

###  1.5. Reader’s guide

The paper is structured as follows: in Section 2, we review some basic facts
concerning reproducing kernels and the complete Carathéodory and Pick
properties. In Section 3, we introduce the notion of a Shimorin certificate
for a pair $(k,\ell)$ and explore some basic consequences of its existence.
Our main results are Propositions 3.6 and 3.13, which illuminate the relation
between Shimorin and strong Shimorin certificates in different settings. Next,
in Section 4, we give our alternative proof of Theorem 1.3, which also yields
a parametrization of all contractive multipliers
$\mathcal{H}_{k}\to\mathcal{H}_{\ell}$; see Corollary 4.1. In addition, we
generalize Theorem 1.3 by showing that the existence of Shimorin certificates
suffices for the CP property; see Theorem 4.5. Section 5 is exclusively
devoted to the characterization of the CC property for pairs of diagonal
holomorphic kernels. First, we reformulate the CC property in terms of one-
point extensions (Proposition 5.2), and then show that the existence of a
strong Shimorin certificates is both necessary (Theorem 5.10) and sufficient
(Theorem 5.6) for a pair to have the CC property. We also present a direct
proof that the CP property implies the CC property in this setting; see
Theorem 5.19. In Section 6, we gather up our results so far to prove Theorem
6.4, a refined version of Theorem 1.6. We also compute the master certificates
associated with certain Bergman-type kernels; see Examples 6.6 and 6.8.
Further, in Section 7, we give necessary conditions for general (not
necessarily holomorphic) pairs to be CP, our main results being Theorem 1.10
and Proposition 7.4. In Section 8, we compute several non-diagonal
certificates for CP pairs $(\mathfrak{b},\ell)$, which are then used to
demonstrate how Theorem 1.6 can fail if $\ell$ is not taken to be diagonal
(Theorem 1.8 and Corollary 8.4). However, we also show that the domain
$\Omega_{\vartheta}=\frac{1}{\sqrt{2}}\mathbb{D}$ of the master certificate
$\vartheta$ associated with $\mathfrak{b}$ continues to be, in a certain
sense, maximal for CP pairs $(\mathfrak{b},\ell)$ even if $\ell$ is not
diagonal. Section 9 contains the previously mentioned reformulation of Theorem
1.6 and a brief discussion interpreting the new necessary conditions it
contains; see Remarks 9.2-9.4. We close the paper with some open questions,
all contained in Section 10.

##  2\. Preliminaries

Background and preliminary results needed in the remainder of the paper are
collected here. Vector-valued reproducing kernel Hilbert spaces associated to
scalar kernel is the subject of Subsection 2.1; complete Pick kernels are
exposited in Subsection r2.2; more details on diagonal holomorphic kernels and
their domains are contained in Subsection 2.3; and initial results on
Carathéodory interpolation for pairs of kernesl are provided in Subsection
2.4.

###  2.1. Vector-valued kernels

Let $X$ be a non-empty set. A function

$k:X\times X\to\mathbb{C}$

is a positive kernel or just a kernel, denoted $k\succeq 0,$ if it is positive
semi-definite in the sense that the $n\times n$ complex matrix
$[k(z_{i},z_{j})]$ is positive semi-definite for every choice of points
$z_{1},\dots,z_{n}\in X.$ Here we make the standing assumption that our
kernels never vanish along the diagonal. The associated reproducing kernel
Hilbert space will be denoted by $\mathcal{H}_{k}$ or $\mathcal{H}_{k}(X)$ to
emphasize the domain, while the algebra of all functions $\phi:X\to\mathbb{C}$
that multiply $\mathcal{H}_{k}$ into itself will be denoted by
$\operatorname{Mult}(\mathcal{H}_{k})$.

We will often have to work with vector-valued versions of these objects. Given
a Hilbert space $\mathcal{E}$, we regard elements of
$\mathcal{H}_{k}\otimes\mathcal{E}$ as $\mathcal{E}$-valued functions on $X.$
It is not hard to see that the reproducing kernel of
$\mathcal{H}_{k}\otimes\mathcal{E}$ is equal to $k\otimes I_{\mathcal{E}}$,
where $I_{\mathcal{E}}$ is the identity operator on $\mathcal{E}$. If
$\mathcal{F}$ is another Hilbert space and $\ell$ is another kernel on $X,$ we
write
$\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathcal{E},\mathcal{H}_{\ell}\otimes\mathcal{F})$
to denote the space of all $B(\mathcal{E},\mathcal{F})$-valued functions on
$X$ that multiply $\mathcal{H}_{k}\otimes\mathcal{E}$ into
$\mathcal{H}_{\ell}\otimes\mathcal{F}$. A multiplier
$\Phi\in\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathcal{E},\mathcal{H}_{\ell}\otimes\mathcal{F})$
is contractive if the corresponding multiplication operator
$M_{\Phi}:\mathcal{H}_{k}\otimes\mathcal{E}\to\mathcal{H}_{\ell}\otimes\mathcal{F}$
is contractive. The following characterization of contractive multipliers is
standard (see e.g. [36, Theorem 6.28]).

######  Lemma 2.1.

Let $\mathcal{H}_{k},\mathcal{H}_{\ell}$ be reproducing kernel Hilbert spaces
on $X.$ Fix Hilbert spaces $\mathcal{E},\mathcal{F}.$ A function $\Phi:X\to
B(\mathcal{E},\mathcal{F})$ is a contractive multiplier in
$\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathcal{E},\mathcal{H}_{\ell}\otimes\mathcal{F})$
if and only if

$\ell(z,w)I_{\mathcal{F}}-k(z,w)\Phi(z)\Phi(w)^{*}\succeq 0\hskip
5.69046pt\text{ on }X\times X$

if and only if the operator $T$ densely defined by

$T(\ell_{w}\otimes v)=k_{w}\otimes\Phi(w)^{*}v$

extends to be a contractive operator from
$\mathcal{H}_{\ell}\otimes\mathcal{F}$ to $\mathcal{H}_{k}\otimes\mathcal{E}$,
in which case $T=M^{*}_{\Phi.}$

The following result will be used repeatedly in the sequel, usually without
special mention. Its proof follows from the definition of positivity.

######  Lemma 2.2.

Let $\mathcal{H}_{k}$ be a reproducing kernel Hilbert space on $X$. If
$\delta:X\to\mathbb{C}$ does not vanish, then the function

$\widetilde{k}(z,w):=\delta(z)\overline{\delta(w)}k(z,w)$

is a positive kernel on $X.$ Moreover, given any Hilbert spaces
$\mathcal{E},\mathcal{F},$ we have

$\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathcal{E},\mathcal{H}_{k}\otimes\mathcal{F})=\operatorname{Mult}(\mathcal{H}_{\widetilde{k}}\otimes\mathcal{E},\mathcal{H}_{\widetilde{k}}\otimes\mathcal{F})$

isometrically.

We also record the following well-known maximum modulus principle for
multipliers. For the readers convenience we provide a proof.

######  Lemma 2.3 (Maximum Modulus for multipliers).

Let $\mathcal{H}_{k}$ be a reproducing kernel Hilbert space on $X$ such that
$k$ is non-vanishing. Given a Hilbert space $\mathcal{E}$, assume the row-
valued multiplier
$\Phi\in\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathcal{E},\mathcal{H}_{k})$
is contractive. If there exists $z\in X$ such that $\|\Phi(z)\|=1,$ then
$\Phi$ is constant. In particular, if $B$ is a positive kernel on $X$ and
$(I-B(z,w))k(z,w,)$ is a (positive) kernel, then either $|B(z,w)|<1$ for all
$z,w\in X$ of $B(z,w)=1$ for all $z,w\in X.$

###### Proof.

Observe, for $u\in X,$ that the operator
$\Phi(u)^{*}:\mathbb{C}\to\mathcal{E}$ is naturally identified with an element
of $\mathcal{E}$ and if $v$ is also in $X,$ then

$\langle
k_{u}\otimes\Phi(u)^{*},k_{v}\otimes\Phi(v)^{*}\rangle=\langle\Phi(u)^{*},\Phi(v)^{*}\rangle\,k(u,v).$

Since $M_{\Phi}^{*}$ is a contraction, the function $F:X\times X\to\mathbb{C}$
defined by

$F(u,v)=(1-\Phi(v)\Phi(u)^{*})k(u,v)=(1-\langle\Phi(u)^{*},\Phi(v)^{*}\rangle)k(u,v)$

is a kernel by Lemma 2.1. By assumption $F(z,z)=0.$ By positivity, $F(z,w)=0$
for all $w\in X.$ Thus $\langle\Phi(z)^{*},\Phi(w)^{*}\rangle=1.$ Since also
$\|\Phi(z)^{*}\|=1\geq\|\Phi(w)^{*}\|,$ we conclude $\Phi(w)=\Phi(z).$

To prove the last statement, by standard reproducing kernel Hilbert space
considerations, there exists an auxiliarly Hilbert space $\mathscr{E}$ and a
function $\Phi:X\to\mathscr{E}$ such that $B(z,w)=\Phi(z)\Phi(w)^{*}.$ By
assumption $(I-\Phi(z)\Phi(w)^{*})k(z,w)\succeq 0$ and thus $\Phi$ is a
contractive multiplier by Lemma 2.1. Hence the last part of the lemma follows
from the first part. ∎

Finally, we will also make repeated use of a simple formula for the
reproducing kernels of zero-based invariant subspaces.

######  Lemma 2.4.

Let $\mathcal{H}_{k}$ be a reproducing kernel Hilbert space on $X$ and fix a
point $z\in X.$ The reproducing kernel $k^{z}$ of the subspace
$\\{f\in\mathcal{H}_{k}:f(z)=0\\}$ of $\mathcal{H}_{k}$ is given by

$k^{z}(w,v)=k(w,v)-\frac{k(w,z)k(z,v)}{k(z,z)}.$

###  2.2. The complete Pick property

A reproducing kernel Hilbert space $\mathcal{H}_{k}$ on a set $X$ is said to
be irreducible if $X$ cannot be partitioned into two non-empty disjoint sets
$X_{1},X_{2}$ so that $k(z_{1},z_{2})=0$ for all $z_{1}\in X_{1}$ and
$z_{2}\in X_{2}.$ The kernel of an irreducible complete Pick space satisfies
$k(z,w)\neq 0$ for all $z,w\in X,$ see [2, Lemma 1.1]. By the Agler-McCarthy
universality theorem [2, Theorem 3.1], the space $\mathcal{H}_{k}$ is an
irreducible complete Pick space if and only if there exist a function
$\delta:X\to\mathbb{C}\setminus\\{0\\}$, an integer $1\leq d\leq\infty$ and a
function $b:X\to\mathbb{B}_{d}$, where $\mathbb{B}_{d}$ denotes the open unit
ball of $\mathbb{C}^{d}$, so that

$k(z,w)=\frac{\delta(z)\overline{\delta(w)}}{1-\langle
b(z),b(w)\rangle},\hskip 5.69046pt\text{ for all }z,w\in X.$ (2.1)

Equivalently (see [2]), $\mathcal{H}_{k}$ is an irreducible complete Pick
space if and only if $k$ is non-vanishing and there exists $z\in X$ such that

$\frac{k^{z}}{k}\succeq 0.$ (2.2)

We note that, in view of Lemma 2.2, one may rescale any irreducible complete
Pick kernel $k$ so that $\delta\equiv 1$ in (2.1). Also, given any $z_{0}\in
X$, one may choose

$\delta(z)=\frac{\sqrt{k(z_{0},z_{0})}}{k(z,z_{0})},$

so that $k(z,z_{0})=1$ for all $z\in X.$

The study of general complete Pick kernels can be reduced to the study of
irreducible ones.

######  Lemma 2.5 ([1, Theorem 1.1]).

Every complete Pick space can be decomposed as an orthogonal direct sum of
irreducible complete Pick spaces. Explicitly, if $s$ is a complete Pick kernel
on a nonempty set $X,$ then there is a unique partition $X=\cup X_{i}$ such
that for each $i$ the function $s|_{X_{i}\times X_{i}}$ is non-vanishing and
for each $i\neq j$ the function $s|_{X_{i}\times X_{j}}$ is identically $0.$

Complete Pick kernels actually satisfy a stronger positivity condition than
(2.2). Given two kernels $\ell,k$ on a set $X,$ we will say that $k$ is a
factor of $\ell$ if there exists a kernel $g$ on $X$ such that $\ell=gk$.

######  Lemma 2.6 ([19, Lemma 2.2]).

Let $\ell$ be a kernel with an irreducible complete Pick factor $s.$ If $M$ is
a $\operatorname{Mult}(\mathcal{H}_{\ell})$-invariant subspace of
$\mathcal{H}_{\ell},$ and $\ell_{M}$ is the reproducing kernel for $M,$ then

$\frac{\ell_{M}}{s}\succeq 0.$

###  2.3. Diagonal holomorphic kernels

Given $z,w\in\mathbb{C}^{\mathsf{g}},$ let

$z\overline{w}=(z_{1}\overline{w_{1}},z_{2}\overline{w_{2}},\cdots,z_{\mathsf{g}}\overline{w_{\mathsf{g}}}).$

Recall, from the introduction, that a kernel $f$ on a domain
$\Omega\subseteq\mathbb{C}^{\mathsf{g}}$ containing $\mathbf{0}$ in its
interior is a normalized diagonal holomorphic kernel if it’s power series
expansion at $\mathbf{0}$ has the form

$f(z,w)=\sum_{a\in\mathbb{N}^{\mathsf{g}}}f_{a}(z\overline{w})^{a}$

with $f_{\mathbf{0}}=1$ and $f_{a}>0$ for all $a.$

The diagonal holomorphic kernel $f$ has its canonical domain $\Omega_{f}$ as
described in equation (1.5) and thus it gives rise to the reproducing kernel
Hilbert spaces $\mathcal{H}_{f}(\Omega)$ and $\mathcal{H}_{f}(\Omega_{f}).$
Among other things, Lemma 2.7 below says these two spaces, as well as their
spaces of multipliers, are canonically isometrically isomorphic. Indeed,
throughout this paper, we will often have to work with restrictions of
holomorphic kernels to open subsets of their domain of convergence.
Fortunately, no important information is lost by doing so.

######  Lemma 2.7.

Let $k$ be a holomorphic kernel on a domain
$\Delta\subseteq\mathbb{C}^{\mathsf{g}}.$ If $\Omega\subseteq\Delta$ is a non-
trivial open subset, then the restriction mapping

$\iota:f\mapsto f|_{\Omega}$

maps $\mathcal{H}_{k}(\Delta)$ unitarily onto $\mathcal{H}_{k}(\Omega).$
Moreover, $\iota$ is an isometric isomorphism from
$\operatorname{Mult}(\mathcal{H}_{k}(\Delta))$ onto
$\operatorname{Mult}(\mathcal{H}_{k}(\Omega)).$

###### Proof.

This result follows from combining the Identity Principle with basic facts
regarding restrictions of reproducing kernels; see [36, Section 5.4]. ∎

The following proposition says more is true for a diagonal holomorphic kernel
$f.$ Namely, the domain $\Omega_{f}$ is the maximum domain (containing
$\mathbf{0}$) for $f.$

######  Proposition 2.8.

Suppose $f$ is a holomorphic diagonal kernel and $\widetilde{f}$ is a
holomorphic kernel on a domain $\Omega\subseteq\mathbb{C}^{\mathsf{g}}.$ Thus,
it is assumed that $\mathbf{0}\in\Omega$ is open and connected and contains
$\mathbf{0}$ and moreover that $\widetilde{f}:\Omega\times\Omega\to\mathbb{C}$
is analytic in the first coordinate and conjugate analytic in the second. If
$f(z,w)=\widetilde{f}(z,w)$ for $z,w\in\Omega\cap\Omega_{f},$ then
$\Omega\subseteq\Omega_{f}.$

The proof will use the following variant of a theorem of Pringsheim. See for
instance [14, p. 15]. For the record, we provide a proof, borrowing heavily
from the arguments from [14] in both the proof of Lemma 2.9 and Proposition
2.8 below. Recall $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ denotes the (open)
unit disc in the complex plane.

######  Lemma 2.9.

Suppose

$f(\zeta)=\sum_{a}f_{a}\zeta^{a}$

is a power series in ${\mathsf{g}}$ complex variables and $f_{a}\geq 0$ for
all $a.$ If the domain of convergence of $f$ contains the polydisc
$\mathbb{D}^{\mathsf{g}}$ and if there exists a neighborhood $N$ of
$e=(1,1,\dots,1)$ and a holomorphic function $h:N\to\mathbb{C}$ such that
$f(\zeta)=h(\zeta)$ for $\zeta\in N\cap\mathbb{D}^{\mathsf{g}},$ then there is
a $t>1$ such that $\sum f_{a}(te)^{a}$ converges.

In the proofs of Lemma 2.9 and Proposition 2.8, we let $N_{\eta}(z)$ denote
the usual Euclidean $\eta$-neighborhood of a point
$z\in\mathbb{C}^{\mathsf{g}},$ where $\eta>0$. As is customary,
$a!=a_{1}!\,a_{2}!\,\cdots\,a_{\mathsf{g}}!$ for
$a=(a_{1},\dots,a_{\mathsf{g}})\in\mathbb{N}^{\mathsf{g}}.$

###### Proof.

We follow the applicable part of There is an $1>\epsilon>0$ such that
$N_{3\epsilon}((1-\frac{\epsilon}{\sqrt{{\mathsf{g}}}})e)\subseteq N.$ For
notational convenience, let $u=\frac{\epsilon}{\sqrt{{\mathsf{g}}}}.$ Since
$h$ is holomorphic on $N_{3\epsilon}((1-u)e),$ it has a power series expansion
centered at $(1-u)e$ that converges in this neighborhood and in particular at
$(1+u)e.$ Letting $h_{a}$ denote the coefficients of this power series, it
follows that

$h((1+u))e)=\sum_{a}h_{a}\,(2ue)^{a}$

and this series converges. On the other hand,

$h_{a}=\frac{1}{a!}f^{(a)}((1-u)e)=\sum_{b\geq
a}\frac{b!}{a!\,(b-a)!}f_{b}((1-u)e)^{b-a}=\sum_{b\geq
a}\binom{b}{a}\,f_{b}((1-u)e)^{b-a},$

where $f^{(a)}$ is the corresponding partial derivative of the function $f.$
Hence, the series

$\sum_{a}\bigg{[}\sum_{b\geq
a}\binom{b}{a}\,f_{b}((1-u)e)^{b-a}\bigg{]}(2ue)^{a}$

converges. Since everything in sight is non-negative, this sum converges after
rearranging; e.g., using the binomial theorem,

$\sum_{b}\sum_{a\leq
b}\binom{b}{a}\,f_{b}((1-u)e)^{b-a}\,(2ue)^{a}=\sum_{b}f_{b}((1+u)e)^{b}$

converges. Hence the conclusion of the lemma holds with $t=1+u.$ ∎

Another ingredient in the proof of Proposition 2.8 is Abel’s Lemma. For a
reference, see [30, Proposition 2.3.4].

######  Lemma 2.10 (Abel’s Lemma).

Given a power series $\sum_{a}c_{a}x^{a},$ if
$0\neq\mathbf{y}\in\mathbb{C}^{\mathsf{g}}$ and there is a $C>0$ such that
$|c_{a}y^{a}|\leq C$ for all $a,$ then the series $\sum c_{a}x^{a}$ converges
absolutely and uniformly on compact subsets of
$\\{(\zeta_{1}y_{1},\dots,\zeta_{\mathsf{g}}y_{\mathsf{g}}):\zeta_{j}\in\mathbb{D}\\}.$

###### Proof of Proposition 2.8.

Arguing by contradiction, suppose $\Omega\not\subseteq\Omega_{f}.$ In this
case, by connectedness of $\Omega$ and since
$\mathbf{0}\in\Omega\cap\Omega_{f},$ there exists a point
$p\in\Omega\setminus\Omega_{f}$ that lies in the boundary of
$\Omega_{f}.$444If no such $p$ exists, then
$\Omega\setminus\Omega_{f}=\Omega\,\cap\,[\operatorname{b}\Omega_{f}\cup\operatorname{e}\Omega_{f}]=[\Omega\cap\operatorname{e}\Omega_{f}],$
where $\operatorname{b}\Omega_{f}$ and $\operatorname{e}\Omega_{f}$ are the
boundary and exterior of $\Omega_{f}.$

Since $\mathbf{0}$ is in the interior of $\Omega_{f}$ (by a standing
assumption), $p\neq\mathbf{0}.$ However, it is still possible that some of the
coordinates of $p$ are $0.$ By renaming, we may assume
$p=(p_{1},\dots,p_{a},0,\dots,0)\in\mathbb{C}^{\mathsf{g}}$ where
$p_{1},\dots,p_{a}\neq 0.$ Let $\eta=\min\\{p_{1},\dots,p_{a}\\}>0.$ In what
follows we use the max norm (metric) $\|x\|_{\infty}=\max\\{|x_{j}|:j\\}$ on
$\mathbb{C}^{\mathsf{g}}.$ Since $\Omega$ is open, there exists an
$\epsilon<\frac{\eta}{2}$ such that
$N^{\infty}_{\epsilon}(p)=\\{x:\|x-p\|_{\infty}<\epsilon\\}\subseteq\Omega.$
It follows that there is a point $q=(q_{1},\dots,q_{\mathsf{g}})\in
N^{\infty}_{\epsilon}(p)\cap\Omega_{f}$ such that
$q_{a+1},\dots,q_{\mathsf{g}}$ are all non-zero. Note, at the same time,
$|q_{j}|\geq\frac{\eta}{2}$ for $1\leq j\leq a$, since
$|q_{j}-p_{j}|<\frac{\eta}{2}$ and $|p_{j}|\geq\eta$ for $1\leq j\leq a.$
Consider the point $z=(p_{1},\dots,p_{a},q_{a+1},\dots,q_{\mathsf{g}}).$ Note
$z\in\Omega,$ since $\|z-p\|_{\infty}=\max\\{|q_{j}|:a+1\leq
j\leq{\mathsf{g}}\\}\leq\|q-p\|_{\infty}<\epsilon.$ By construction all the
coordinates $z_{1},\dots,z_{\mathsf{g}}$ of $z$ are non-zero. Arguing by
contradiction, suppose $z\in\Omega_{f}.$ In this case there is a $t>1$ such
that $tz\in\Omega_{f}.$ By Lemma 2.10 applied to the power series $\sum
f_{a}x^{2a},$ the set $\Omega_{f}$ contains the polydisc
$D=\\{t(\zeta_{1}z_{1},\dots,\zeta_{\mathsf{g}}z_{\mathsf{g}}):\zeta_{j}\in\mathbb{D}\\}.$
Since none of the coordinates of $z$ are $0,$ this polydisc is an open set. In
particular, choosing $\zeta_{j}=\frac{1}{t}$ for $j=1,\dots,a$ and $0$
otherwise, we find $p\in D\subseteq\Omega_{f},$ which is a contradiction as
$\Omega_{f}$ is open and $p$ is in its boundary.

At this point we have $q\in N_{\epsilon}(p)\cap\Omega_{f}$ and $z\in
N_{\epsilon}(p)\setminus\Omega_{f}.$ Hence $\\{tq+(1-t)z:0\leq t\leq
1\\}\subseteq\Omega$ and there is a $0<t\leq 1$ such that $y=tq+(1-t)z$ is in
the boundary of $\Omega_{f}.$ Note that the coordinates of $y$ are given by
$y_{j}=tq_{j}+(1-t)p_{j}$ for $1\leq j\leq a$ and $y_{j}=tq_{j}\neq 0$ for
$j>a.$ Suppose $y_{j}=0$ for some $j\leq a.$ Thus, $q_{j}=-\frac{1-t}{t}p_{j}$
and therefore,

$\frac{\eta}{2}\geq\epsilon>|p_{j}-q_{j}|=\frac{|p_{j}|}{t}\geq\eta,$

a contradiction. Hence all the coordinates of $y_{j}$ are non-zero.

Another application of Lemma 2.10 to $\sum f_{a}x^{2a}$ says that the polydisc
$D=\\{(\zeta_{1}y_{1},\dots,\
\zeta_{\mathsf{g}}y_{\mathsf{g}}):\zeta_{j}\in\mathbb{D}\\}$ lies in
$\Omega_{f}.$ Let $g$ denote the power series, for
$\zeta\in\mathbb{C}^{\mathsf{g}},$

$g(\zeta)=\sum_{a\in\mathbb{N}^{\mathsf{g}}}f_{a}|y_{a}|^{2a}\zeta^{2a}.$

In particular, as power series,

$g(\zeta)=f(\zeta y,\overline{\zeta}y)=\sum
f_{a}((\zeta\,y)_{a})^{a}\overline{((\overline{\zeta}\,y)_{z})^{a}}.$

It follows that $\mathbb{D}^{\mathsf{g}}$ is contained in the domain of
convergence of $g$ and moreover, as functions, $g(\zeta)=f(\zeta
y,\overline{\zeta}y)$ for $\zeta\in\mathbb{D}^{\mathsf{g}}.$ Define $h$ by
$h(\zeta)=\widetilde{f}(\zeta y,\overline{\zeta}y).$ Since $y\in\Omega,$ the
function $h$ is defined in a neighborhod $N$ of $e=(1,\dots,1)$. Moreover,
$g(\zeta)=h(\zeta)$ for $\zeta\in\mathbb{D}^{\mathsf{g}}\cap N$ and $h$ is
analytic. By Lemma 2.9, it follows that there exists a $t>1$ such that $\sum
g_{a}(te)^{a}$ converges. Thus $\sum_{a}f_{a}|ty|^{2a}$ converges. By Lemma
2.10, $\Omega_{f}$ contains the polydisc
$\\{(\zeta_{1}ty_{1},\dots,\zeta_{\mathsf{g}}ty_{\mathsf{g}}):\zeta_{j}\in\mathbb{D}\\}$
and, by choosing $\zeta_{j}=\frac{1}{t},$ we have arrived at the contradiction
$y\in\Omega_{f}.$ ∎

The following lemma is used in the proof of Proposition 6.2.

######  Lemma 2.11.

Let $f=\sum\limits_{|a|>0}f_{a}z^{a}\overline{w}^{a}.$ If $f_{a}\geq 0$ for
all $|a|>0$ and $\Omega_{f}\neq\emptyset,$ then

$\Omega_{f}^{1}=\\{x\in\Omega_{f}:\sum f_{a}|x|^{2a}<1\\}$

is non-empty, open and connected; that is, a domain in
$\mathbb{C}^{\mathsf{g}}.$

###### Proof.

Let $g(\zeta)=\sum_{|a|>0}f_{a}\zeta^{2a}.$ The domain of convergence of the
power series $g$ is, by definition, $\Omega_{f}.$ By assumption, $\Omega_{f}$
contains a neighborhood of the origin. The domain of convergence of a power
series is open and star-like with respect to the origin. Since $g$ defines a
holomorphic function on $\Omega_{f},$ the set $\Omega_{f}^{1}$ is open and
contains $\mathbf{0}.$ Moreover, $\Omega_{f}^{1}$ is the intersection of two
sets that are star-like with respect to $\mathbf{0},$ namely $\Omega_{f}$
([30, Proposition 2.3.15]) and $\\{x\in\mathbb{C}^{\mathsf{g}}:\sum
f_{a}|x|^{2a}<1\\}.$ Hence, $\Omega_{f}^{1}$ is star-like with respect to
$\mathbf{0}$ too. In particular, it is connected and thus a domain. ∎

Informed by Proposition 2.8, we conclude this subsection with the following
observation. A normalized holomorphic diagonal kernel $f$ and its
corresponding Hilbert space $\mathcal{H}_{f}$ can also be understood as
follows. A point $z\in\Omega_{f}$ gives rise to a vector
$E_{z}=(\sqrt{f_{a}}\overline{z}^{a})$ in the Hilbert space
$\ell^{2}(\mathbb{N}^{\mathsf{g}}).$ Thus we obtain a mapping
$E:\Omega_{f}\to\ell^{2}(\mathbb{N}^{\mathsf{g}})$ and, if $z,w\in\Omega_{f},$
then

$\langle
E_{w},E_{z}\rangle=\sum_{a\in\mathbb{N}^{\mathsf{g}}}f_{a}z^{a}\overline{w^{a}}=f(z,w).$
(2.3)

From standard facts about reproducing kernels, the monomials $z^{a}$ form an
orthogonal basis for $\mathcal{H}_{f}$ and

$\langle z^{a},z^{a}\rangle=\|z^{a}\|^{2}=\frac{1}{k_{a}}.$

###  2.4. The complete Carathéodory property

As we saw in Section 1, the classical Carathéodory problem asks, given
$c_{0},\dots,c_{n}\in\mathbb{C},$ does there exist an analytic function
$g:\mathbb{D}\to\mathbb{C}$ such that the function

$f=\sum_{j=0}^{n}\overline{c}_{j}z^{j}+z^{n+1}g$

satisfies $|f(z)|\leq 1$ for all $z\in\mathbb{D}?$ From an operator-theoretic
viewpoint, there is a natural necessary condition that, when combined with
Parrott’s Lemma, can be seen to be sufficient. A function
$f:\mathbb{D}\to\mathbb{D}$ determines an operator $M_{f}:H^{2}(\mathbb{D})\to
H^{2}(\mathbb{D})$ defined by $M_{f}h=fh$ for $h\in H^{2}(\mathbb{D})$ and
$\|M_{f}\|\leq 1.$ The subspace $M_{n}$ of $H^{2}(\mathbb{D})$ spanned by the
orthonormal set $\\{1,z,\dots,z^{n}\\}$ is invariant for $M_{f}^{*}$ and
moreover, the matrix representation of $M_{f}^{*}$ with respect to this basis
is

$C_{n}=\begin{pmatrix}c_{0}&c_{1}&c_{2}&\dots&c_{n-1}&c_{n}\\\
0&c_{0}&c_{1}&\dots&c_{n-2}&c_{n-1}\\\ 0&0&c_{1}&\dots&c_{n-3}&c_{n-2}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\dots&0&c_{0}\end{pmatrix}.$

Thus, $\|C_{n}\|\leq 1$ is a necessary condition for a solution to the
Caratheodory problem. Sufficiency follows from the following version of the
Parrott Lemma (see, for instance, [2, Lemma B1]).

######  Lemma 2.12.

Suppose $p,q$ and $m,n$ are positive integers such that $p+q=m+n,$

  1. (a)

$A\in M_{p,m};$

  2. (b)

$C\in M_{q,m};$

  3. (c)

$D\in M_{q,n};$

such that

$\big{\|}\begin{pmatrix}A\\\ C\end{pmatrix}\big{\|},\ \
\big{\|}\begin{pmatrix}C&D\end{pmatrix}\big{\|}\leq 1,$

then there exists a matrix $B\in M_{p,n}$ such that

$\big{\|}\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\big{\|}\leq 1.$

With $c_{n+1}$ to be determined, consider the following partition of the
matrix $C_{n+1},$

$C_{n+1}=\left(\begin{array}[]{cccccc|c}c_{0}&c_{1}&c_{2}&\dots&c_{n-1}&c_{n}&c_{n+1}\\\
\hline\cr 0&c_{0}&c_{1}&\dots&c_{n-2}&c_{n-1}&c_{n}\\\
0&0&c_{1}&\dots&c_{n-3}&c_{n-2}&c_{n-1}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&\dots&0&c_{0}&c_{1}\\\ 0&0&0&\dots 0&0&0&c_{0}\end{array}\right).$

Since $\|C_{n}\|\leq 1,$

$\big{\|}\begin{pmatrix}C_{n}\\\ 0\end{pmatrix}\big{\|},\ \
\big{\|}\begin{pmatrix}0&C_{n}\end{pmatrix}\big{\|}\,\leq 1.$

Thus, by Parrott’s Lemma there is a choice of $c_{n+1}$ such that
$\|C_{n+1}\|\leq 1.$ An induction argument now produces $c_{m}$ such that
$\|C_{m}\|\leq 1,$ for all $m$, which in turn implies, with
$f=\sum_{j=0}^{m}\overline{c}_{j}z^{j},$ that the restriction of $M_{f}^{*}$
to the subspaces $M_{m}$ is a contraction. Thus, we obtain an
$f=\sum_{j=0}^{\infty}\overline{c_{j}}z^{j}$ such that $M_{f}^{*}$ defines an
operator on $H^{2}(\mathbb{D})$ of norm at most one. It is also not hard to
see that
$M_{f}^{*}\mathfrak{s}_{\lambda}=\overline{f(\lambda)}\,\mathfrak{s}_{\lambda},$
where $\mathfrak{s}$ is Szego’s kernel and $\lambda\in\mathbb{D}.$ Hence
$\|f(\lambda)\|\leq 1$ for all $\lambda\in\mathbb{D}$, as desired. Note that
the argument goes through unchanged if the $c_{j}$’s are replaced by $J\times
J$ matrices for any positive integer $J.$

We highlight the fact that this argument proceeds in two parts. There is the
matrix completion problem: Starting with the contraction $C_{n},$ find $c_{m}$
for $m>n$ such that $C_{m}$ is a contraction for each $m.$ And there is the
function-theoretic operator theory interpretation: the assumption that each
$C_{m}$ is a contraction implies that the power series
$\sum\overline{c}_{j}z^{j}$ defines a function $f:\mathbb{D}\to\mathbb{D}.$
Equivalently, $f:\mathbb{D}\to\mathbb{C}$ and $M_{f}^{*}:H^{2}(\mathbb{D})\to
H^{2}(\mathbb{D})$ is a contraction.

In the remainder of this section, we consider the complete Carathéodory
property for a pair of kernels $(k,\ell),$ where $k$ and $\ell$ are
(normalized) diagonal holomorphic. First, we will restate the definition given
in the introduction. Given $a,b\in\mathbb{N}^{{\mathsf{g}}},$ we will write
$a\leq b$ to mean $a_{i}\leq b_{i}$ for $i\in\\{1,\dots,{\mathsf{g}}\\}.$

######  Definition 2.13.

A non-empty set $F\subseteq\mathbb{N}^{\mathsf{g}}$ is co-invariant if
$\\{a\in\mathbb{N}^{\mathsf{g}}:a\leq b\\}\subseteq F$ for each $b\in F.$ ∎

Thus, if $F$ is co-invariant, then $F$ contains $(0,\dots,0)=\mathbf{0}.$

######  Definition 2.14.

A pair $(k,\ell)$ of (normalized) diagonal holomorphic kernels is a complete
Carathéodory pair if, for any positive integer $J$, any finite co-invariant
set of indices $\varnothing\neq F\subseteq\mathbb{N}^{\mathsf{g}}$ and any
collection of matrix coefficients $\\{c_{a}:a\in F\\}\subseteq\mathbb{M}_{J}$,
the positivity of the block upper-triangular matrix $C$ indexed by $F\times F$
with block $J\times J$ entries

$C_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise,}\end{cases}$ (2.4)

is equivalent to the existence of a collection
$\\{c_{a}:a\in\mathbb{N}^{\mathsf{g}}\setminus F\\}\subseteq\mathbb{M}_{J}$
such that the function

$\Phi(z)=\sum_{a\in\mathbb{N}^{\mathsf{g}}}z^{a}\otimes c_{a}^{*}$ (2.5)

is a contractive multiplier from $\mathcal{H}_{k}\otimes\mathbb{C}^{J}$ to
$\mathcal{H}_{\ell}\otimes\mathbb{C}^{J}.$ In the special case $k=\ell,$ the
kernel $k$ is known as a complete Carathéodory kernel. ∎

Just as in the classical Carathéodory problem above, it is possible to
reformulate Definition 2.14 so that it does not involve function theory.

######  Proposition 2.15.

A pair of diagonal holomorphic kernels $(k,\ell)$ is a complete Carathéodory
pair if and only if, if for each $J\in\mathbb{N},$ each finite co-invariant
$\varnothing\neq F\subseteq\mathbb{N}^{{\mathsf{g}}}$ and each collection
$\\{c_{a}:a\in F\\}\subseteq\mathbb{M}_{J}$ such that the block upper-
triangular matrix $C$ indexed by $F\times F$ with block $J\times J$ entries as
in (2.4) is a contraction, there exists a collection
$\\{c_{a}:a\in\mathbb{N}^{{\mathsf{g}}}\setminus F\\}$ such that the
(infinite) block matrix $\mathscr{C}$ indexed by
$\mathbb{N}^{{\mathsf{g}}}\times\mathbb{N}^{{\mathsf{g}}}$ with
$\mathscr{C}_{a,b}$ entries

$\mathscr{C}_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise,}\end{cases}$ (2.6)

is a contraction.

######  Remark 2.16.

Assume $k,\ell$ are normalized. By considering $F=\\{\mathbf{0}\\}$ and
choosing $\\{c_{\mathbf{0}}=1\\}$ we obtain the $1\times 1$ matrix
$\Big{(}c_{\mathbf{0}}\sqrt{\frac{k_{\mathbf{0}}}{\ell_{\mathbf{0}}}}\Big{)}=1,$
which is a contraction. Hence, assuming that $(k,\ell)$ is a CC pair, there
exists $c_{a}$ for $a>0$ such that the matrix $\mathscr{C}$ in (2.6) is a
contraction. In particular, its diagonal entries satisfy
$1\geq\Big{|}c_{\mathbf{0}}\sqrt{\frac{k_{a}}{\ell_{a}}}\Big{|}=\sqrt{\frac{k_{a}}{\ell_{a}}}$
and thus $k_{a}\leq\ell_{a},$ an inequality that follows from Theorem 1.9. ∎

The remainder of this subsection will be devoted to a proof sketch of
Proposition 2.15. We begin with a function-theoretic interpretation of the
contractivity of (2.4). Let

$e_{a}(z)=\sqrt{k_{a}}z^{a},\hskip 5.69046ptf_{a}=\sqrt{\ell_{a}}z^{a},\hskip
5.69046pta\in\mathbb{N}^{{\mathsf{g}}},$

so that $\\{e_{a}\\}$ and $\\{f_{a}\\}$ form orthonormal bases for
$\mathcal{H}_{k}$ and $\mathcal{H}_{\ell}$, respectively. Fix $J\in\mathbb{N}$
and a co-invariant $F\subseteq\mathbb{N}^{{\mathsf{g}}}$ and let
$\mathcal{H}_{k,F}$ and $\mathcal{H}_{\ell,F}$ denote the subspaces of
$\mathcal{H}_{k}$ and $\mathcal{H}_{\ell}$ spanned by $\\{e_{a}:a\in F\\}$ and
$\\{f_{a}:a\in F\\}$, respectively. Further, assume the collection
$\\{c_{a}:a\in\mathbb{N}^{{\mathsf{g}}}\\}\subseteq\mathbb{M}_{J}$ is such
that the function

$\Phi(z)=\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}z^{a}\otimes c_{a}^{*}$

is a multiplier
$\mathcal{H}_{k}\otimes\mathbb{C}^{J}\to\mathcal{H}_{\ell}\otimes\mathbb{C}^{J}$.
A short computation reveals that

$M^{*}_{\Phi}(f_{a}\otimes h)=\sum_{u\leq
a}\sqrt{\frac{k_{u}}{\ell_{a}}}e_{u}\otimes c_{a-u}h,\hskip 5.69046pt\text{
for all }a\in\mathbb{N}^{{\mathsf{g}}}\text{ and }h\in\mathbb{C}^{J}.$ (2.7)

Thus, $M_{\Phi}^{*}$ maps $\mathcal{H}_{\ell,F}$ into $\mathcal{H}_{k,F}.$
Moreover, the matrix representation of the restriction of $M_{\Phi}^{*}$ to
$\mathcal{H}_{\ell,F}$ is given by the matrix of equation (2.4). Since
$M_{\Phi}^{*}$ is, by assumption, a contraction, so is $C.$

Next, we will show that if the
$\mathbb{N}^{{\mathsf{g}}}\times\mathbb{N}^{{\mathsf{g}}}$ matrix
$\mathscr{C}$ given by (2.6) is a contraction, then the function

$\Phi(z)=\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}z^{a}\otimes c_{a}^{*}$ (2.8)

is a contractive multiplier
$\mathcal{H}_{k}\otimes\mathbb{C}^{J}\to\mathcal{H}_{\ell}\otimes\mathbb{C}^{J}$
(the converse follows immediately from our previous argument). First, observe
that, since $\mathscr{C}$ is a contraction, the $\ell^{2}$-norm of the first
row ($a=\mathbf{0}$) is bounded. Thus,

$\sum_{b\in\mathbb{N}^{{\mathsf{g}}}}\frac{\|c_{b}\|^{2}}{\ell_{b}}<\infty.$
(2.9)

If $k,\ell$ are defined on $\Omega,$ then, by Proposition 2.8,
$\Omega\subseteq\Omega_{\ell}$, where $\Omega_{\ell}$ is defined as in (1.5).
Thus, for every $z\in\Omega$, (2.9) and the Cauchy-Schwarz inequality gives us

$\|\Phi(z)\|^{2}\leq\bigg{(}\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}\|c_{a}\|\cdot|z|^{a}\bigg{)}^{2}\leq\bigg{(}\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}\ell_{a}|z|^{2a}\bigg{)}\bigg{(}\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}\frac{\|c_{a}\|^{2}}{\ell_{a}}\bigg{)}<\infty.$

So, $\Phi$ does actually define a function on $\Omega$. Now, arguing as
earlier in this proof, we see that $\|\mathscr{C}\|\leq 1$ is equivalent to

$f_{a}\otimes h\mapsto\sum_{u\leq a}\sqrt{\frac{k_{u}}{\ell_{a}}}e_{u}\otimes
c_{a-u}h$

defining a contractive operator from $\mathcal{H}_{\ell}\otimes\mathbb{C}^{J}$
to $\mathcal{H}_{k}\otimes\mathbb{C}^{J}$. Denote that operator by $T$. It is
then not hard to verify that

$T(\ell_{w}\otimes v)=k_{w}\otimes\Phi(w)^{*}v,$

for all $w\in\Omega$ and $v\in\mathbb{C}^{J}.$ In view of Lemma 2.1,
$M^{*}_{\Phi}=T$ is a contraction and $\Phi$ is a contractive multiplier, as
desired.

##  3\. Shimorin and Strong Shimorin Certificates

In this section, we will introduce the notion of a Shimorin certificate for a
pair of abstract (not necessarily holomorphic) kernels. Our motivation behind
Definition 3.1 was to come up with the weakest set of conditions that
guarantee the CP property for a general pair $(k,\ell)$. Further, even though
having a Shimorin certificate will turn out to be weaker than having a strong
Shimorin certificate, we will see that the two properties coincide in many
interesting cases.

###  3.1. Shimorin certificates

In this subsection we define a Shimorin certificate and collect some
consequences.

######  Definition 3.1.

Assume $k,\ell$ are kernels on the non-empty set $X.$ A family of kernels
$\\{p[z]\\}_{z\in X}$ on $X$ is a Shimorin certificate for the pair $(k,\ell)$
if, for each $z\in X,$

$\ell^{z}\succeq p[z]\ell\hskip 8.5359pt\text{ and }\hskip
8.5359ptk^{z}\preceq p[z]k.$ (3.1)

######  Remark 3.2.

A strong Shimorin certificate is a CP kernel $s$ on $X$, while a Shimorin
certificate is a family of kernels $\\{p[z]\\}$ all defined on $X$ and indexed
by $X$.

######  Example 3.3.

Given an arbitrary kernel $k$, we can always find a second kernel $\ell$ with
the property that $(k,\ell)$ has a Shimorin certificate. Indeed, choose $\ell$
to be a kernel on $X$ with $\ell(z,w)=0$ whenever $z\neq w.$ Now, for any
$t\in X,$ set

$p[t](z,w)=\begin{cases}0\hskip 17.07182pt\text{ if }z=t\text{ or }w=t,\\\
1,\hskip 18.49411pt\text{ otherwise}.\end{cases}$

Since $\ell^{t}=p[t]\ell$ for all $t$ and

$p[t](z,w)k(z,w)-k^{t}(z,w)=p[t](z,w)\frac{k(z,t)\,k(t,w)}{k(t,t)}\succeq 0,$

$\\{p[t]\\}_{t\in X}$ is a Shimorin certificate for $(k,\ell)$.

It is now easy to construct an example of a pair $(k,\ell)$ with a Shimorin
certificate, but without a strong Shimorin certificate. Let $X=\\{1,2,3\\}$
and let

$k=\begin{pmatrix}1&1&0\\\ 1&2&1\\\ 0&1&2\end{pmatrix},$

so that $k(a,b)$ is the $(a,b)$ entry of this matrix. Note $k$ is positive
definite. If $B$ and $s$ are positive kernels and $k=(1-B)s,$ then either
$B(1,3)=1$ or $s(1,3)=0.$ If $s$ is a complete Pick kernel, then, by Lemma
2.5, either $s(2,3)=0$ or $s(1,2)=0.$ Either way one obtains a contradiction,
since $k(1,2)\neq 0\neq k(2,3).$ If $B(1,3)=1,$ then as $B$ is positive and
$B(1,1),B(3,3)\leq 1$ (since $k(1,1),k(3,3),s(1,1),s(3,3)\geq 0$), we obtain
$B(1,1)=1$ and so $k(1,1)=0,$ a contradiction. Thus, choosing an $\ell$ such
that $(k,\ell)$ has a Shimorin certificate, which is possible from the
discussion above, gives the desired example.

######  Proposition 3.4.

Suppose $k,\ell,p$ are kernels over the nonempty set $X.$ If
$\ell^{z_{0}}\succeq p\ell$ and $k^{z_{0}}\preceq pk$ for some $z_{0}\in X$
and $p(x,x)<1$ for all $x\in X,$ then

$s(w,v)=\frac{k(w,z_{0})k(z_{0},v)}{k(z_{0},z_{0})}\frac{1}{1-p(w,v)}$

is a strong Shimorin certificate for $(k,\ell).$

###### Proof.

By positivity, $|p(w,v)|<1$ for all $w,v\in X$ and it is evident from the
discussion at the outset of subsection 2.2 that $s$ is a complete Pick kernel.
From $k^{z_{0}}\preceq pk,$ there exists a kernel $C$ on $X$ such that

$k(w,v)-\frac{k(w,z_{0})k(z_{0},v)}{k(z_{0},z_{0})}+C(w,v)=p(w,v)k(w,v),$

and therefore,

$k(w,v)=\frac{k(w,z_{0})k(z_{0},v)}{k(z_{0},z_{0})}\frac{\Big{(}1-\frac{k(z_{0},z_{0})}{k(w,z_{0})k(z_{0},v)}C(w,v)\Big{)}}{1-p(w,v)},$

for any $w,v\in X$. Similarly, there exists a kernel $D$ such that, for all
$w,v\in X,$

$\ell(w,v)=\frac{\frac{\ell(w,z_{0})\ell(z_{0},v)}{\ell(z_{0},z_{0})}+D(w,v)}{1-p(w,v)}.$

Defining $s$ as in the statement of the proposition and setting

$\begin{split}h(w,v)&=\frac{k(z_{0},z_{0})}{k(w,z_{0})k(z_{0},v)}C(w,v)\\\
g(w,v)&=\frac{k(z_{0},z_{0})}{k(w,z_{0})k(z_{0},v)}\bigg{(}\frac{\ell(w,z_{0})\ell(z_{0},v)}{\ell(z_{0},z_{0})}+D(w,v)\bigg{)},\end{split}$

we see that $s$ is a strong Shimorin certificate for $(k,\ell).$ ∎

We close this subsection by recording some basic restrictions that are imposed
on a pair of a kernels by the existence of a Shimorin certificate.

######  Lemma 3.5.

Suppose $k,\ell,p$ are kernels on $X$ and $v,w,z\in X$ are distinct. If
$\ell^{z}\succeq p\ell,$ then

  1. (i)

$p(w,w)\leq 1$ for all $w\in X$ and thus, by positivity, $|p(w,v)|\leq 1$ for
all $w,v\in X;$

  2. (ii)

if $p(w,w)=1,$ then $\ell(z,w)=0;$

  3. (iii)

if $p(w,w)=1,$ then either $\ell(w,v)=0$ or $p(v,v)=1=p(w,v).$

If $k^{z}\preceq pk,$ then

  1. (iv)

if $k(z,w)=0,$ then $p(w,w)=1;$

  2. (v)

if $k(z,w)=0,$ then either $k(w,v)=0$ or $p(v,v)=1=p(w,v).$

If $\ell^{z}\succeq p\ell$ and $k^{z}\preceq pk$ and if $k(z,w)=0$, then

  1. (vi)

$\ell(z,w)=0.$

If $(k,\ell)$ has a Shimorin certificate and $k(z,w)=0,$ then

  1. (vii)

at least one of the following holds:

    1. (a)

$\ell(z,v)=\ell(w,v)=0$;

    2. (b)

either $k(z,v)=0$ or $k(w,v)=0$.

###### Proof.

By definition, $\ell^{z}\succeq p\ell$ means

$X\times
X\ni(x,y)\mapsto\ell(x,y)-\frac{\ell(x,z)\ell(z,y)}{\ell(z,z)}-p(x,y)\ell(x,y)$

is a kernel (PsD). In particular,
$\ell(w,w)(1-p(w,w))\geq\frac{|\ell(z,w)|^{2}}{\ell(z,z)}\geq 0.$ Thus,
$p(w,w)\leq 1.$ Moreover, if $p(w,w)=1,$ then necessarily, $\ell(z,w)=0.$
Assuming $p(w,w)=1,$ from what has already been proved $\ell(z,w)=0$ and thus
the positivity condition $\ell^{z}\succeq p\ell$ gives

$\begin{bmatrix}\ell^{z}(w,w)&\ell^{z}(w,v)\\\
\ell^{z}(v,w)&\ell^{z}(v,v)\end{bmatrix}=\begin{bmatrix}\ell(w,w)&\ell(w,v)\\\
\ell(v,w)&\ell^{z}(v,v)\end{bmatrix}\succeq\begin{bmatrix}\ell(w,w)&p(w,v)\ell(w,v)\\\
p(v,w)\ell(v,w)&p(v,v)\ell(v,v)\end{bmatrix}.$

Hence

$\begin{bmatrix}0&\big{(}-p(w,v)+1\big{)}\ell(w,v)\\\
\big{(}-p(v,w)+1\big{)}\ell(v,w)&-p(v,v)\ell(v,v)+\ell^{z}(v,v)\end{bmatrix}\succeq
0.$

Thus, either $\ell(w,v)=0$ or $p(w,v)=1.$ In the later case, by positivity,
$p(v,v)=1,$ concluding the proof of the first three items of the lemma. Items
(iv) and (v) are proved in an entirely analogous manner.

Assume $\ell^{z}\succeq p\ell$ and $k^{z}\preceq pk.$ If $k(z,w)=0,$ then, by
item (iv), $p(w,w)=1$ and thus, by item (ii), $\ell(z,w)=0,$ proving item
(vi).

Now suppose $(k,\ell)$ has a Shimorin certificate. In particular, there exist
kernels $p[z]$ and $p[w]$ such that $\ell^{z}\succeq p[z]\,\ell$ and
$k^{z}\preceq p[z]\,k$ as well as $\ell^{w}\succeq p[w]\,\ell$ and
$k^{w}\preceq p[w]\,k.$ If $k(z,v)\neq 0\neq k(w,v),$ then, by item (v),
$p[z](v,v)=p[w](v,v)=1.$ Thus, by item (ii) (twice), $\ell(z,v)=0=\ell(w,v)$
and the proof of item (vii), and thus the lemma, is complete. ∎

###  3.2. Shimorin versus strong Shimorin certificates

In this section we consider two sufficient conditions under which the
existence of a Shimorin certificate implies the existence of a strong Shimorin
certificate. The first is a general condition that applies in a number of
cases of interest. The second includes the assumption that $k$ and $\ell$ are
holomorphic kernels on a domain $\Omega\subseteq\mathbb{C}^{\mathsf{g}}.$

######  Proposition 3.6.

Let $k,\ell$ be kernels on $X.$

  1. (i)

If $(k,\ell)$ has a strong Shimorin certificate, then it also has a Shimorin
certificate. The converse does not always hold.

  2. (ii)

If $\ell$ is non-vanishing, then $(k,\ell)$ has a Shimorin certificate if and
only if $(k,\ell)$ has a strong Shimorin certificate if and only if there
exists a kernel $p$ on $X$ and a point $z_{0}\in X$ with

$\ell^{z_{0}}\succeq p\ell\hskip 11.38092pt\text{ and }\hskip
11.38092ptk^{z_{0}}\preceq pk.$ (3.2)

Before proving Proposition 3.6 we record the following lemma.

######  Lemma 3.7.

If $k,\ell$ are kernels over the set $X$ and $s$ is a strong Shimorin
certificate for $(k,\ell),$ then, with respect to the unique partition $X=\cup
X_{i}$ for $s$ from Lemma 2.5,

  1. (i)

$k(z,w)=\ell(z,w)=s(z,w)=0$ whenever $z\in X_{i}$ and $w\in X_{j}$ with $i\neq
j;$

  2. (ii)

for each $i,$ the function $s|_{X_{i}\times X_{i}}$ is non-vanishing;

  3. (iii)

for each $i,$ either $k|_{X_{i}\times X_{i}}$ is non-vanishing or identically
$0.$

######  Remark 3.8.

Under our usual assumption that $k(x,x)>0$ for all $x,$ the conclusion of item
(iii) is, for each $i,$ the function $k|_{X_{i}\times X_{i}}$ does not vanish.

###### Proof.

Since $s$ is a strong Shimorin certificate for $(k,\ell),$ there exist kernels
$h$ and $g$ on $X$ such that

$k=(1-h)s,\hskip 19.91684pt\ell=gs.$

By Lemma 2.5, there exists a unique decomposition $X=\cup_{i\in I}X_{i}$ such
that $s_{i}:=s|_{X_{i}\times X_{i}}$ never vanishes and $s|_{X_{a}\times
X_{b}}\equiv 0$ whenever $a\neq b,$ which also yields $k|_{X_{a}\times
X_{b}}\equiv 0$ and $\ell|_{X_{a}\times X_{b}}\equiv 0$. Since $s_{i}$ never
vanishes, Lemma 2.3 tells us that $1-h$ is either non-vanishing on
$X_{i}\times X_{i}$ or identically zero. As $k=(1-h)s,$ we conclude that $k$
splits into a collection of kernels $k_{i}=k|_{X_{i}\times X_{i}}$ that are
non-vanishing. ∎

###### Proof of Proposition 3.6.

Example 3.3 shows that the converse of item (i) need not hold. To prove the
remainder of item (i), suppose $(k,\ell)$ has a strong Shimorin certificate
$s.$ Thus, there exist kernels $h$ and $g$ on $X$ such that

$k=(1-h)s,\hskip 19.91684pt\ell=gs.$

By Lemma 2.5, there exists a decomposition $X=\cup_{i\in I}X_{i}$ such that
$s_{i}:=s|_{X_{i}\times X_{i}}$ never vanishes, $k(z,w)=\ell(z,w)=s(z,w)=0$
for $i\neq j$ and $z\in X_{i}$ and $w\in X_{j},$ and, for each $i,$ either
$k|_{X_{i}\times X_{i}}$ is non-vanishing or identically $0.$ Now, set
$\ell_{i}=\ell|_{X_{i}\times X_{i}}$ and $g_{i}=g|_{X_{i}\times X_{i}}$ and
observe that, for any $i\in I$ and $z\in X_{i}$, we have

$\ell^{z}_{i}\succeq\frac{s_{i}^{z}}{s_{i}}\ell_{i}\hskip 11.38092pt\text{ and
}\hskip 11.38092ptk^{z}_{i}\preceq\frac{s_{i}^{z}}{s_{i}}k_{i}.$ (3.3)

Indeed, the first inequality is equivalent to

$\frac{g_{i}(w,z){g_{i}(z,v)}}{g_{i}(z,z)}\preceq g_{i}(w,v),\hskip
8.5359ptw,v\in X_{i},$

which holds because $g$ is positive. The second inequality in (3.3) is trivial
if $1-h\equiv 0.$ If $1-h$ is non-vanishing on $X_{i}\times X_{i},$ then,
setting $t_{i}=(1-h|_{X_{i}\times X_{i}})^{-1}=s_{i}/k_{i},$ the second
inequality becomes

$\frac{t_{i}^{z}}{t_{i}}\succeq 0,$

which holds for all $z\in X_{i}$ because $t$ is a CP kernel. We now define a
Shimorin certificate for $(k,\ell)$. Given $z\in X_{i}$, we set

$p[z](w,v)=\begin{cases}\cfrac{s_{i}^{z}(w,v)}{s_{i}(w,v)},\hskip
10.81218pt\text{ if }w,v\in X_{i},\\\ 1,\hskip 41.25641pt\text{ if }w,v\in
X_{j}\text{ with }j\neq i,\\\ 0,\hskip 42.10995pt\text{
otherwise.}\end{cases}$

Combining (3.3) with the observation that $k^{z}|_{X_{i}\times
X_{i}}=k|_{X_{i}\times X_{i}}$ and $\ell^{z}|_{X_{i}\times
X_{i}}=\ell|_{X_{i}\times X_{i}}$ whenever $z\in X_{j}$ with $i\neq j$, we
conclude that $\\{p[z]\\}_{z\in X}$ is a Shimorin certificate for $(k,\ell)$
and the proof of item (i) is complete.

Since we have already shown that having a strong Shimorin certificate implies
the existence of a Shimorin certificate (which, in turn, clearly implies
(3.2)), all that remains is to show that, under the assumption that $\ell$
never vanishes, (3.2) leads to a strong Shimorin certificate, a conclusion
that follows immediately from Proposition 3.4 after noting that $\ell$ non-
vanishing implies $p(x,x)<1$, for all $x$, by Lemma 3.5 (ii). ∎

The notions of Shimorin and strong Shimorin certificate also coincide for
pairs of holomorphic kernels over domains $\Omega\subseteq\mathbb{C}^{d}$,
even if $\ell$ has zeros. To prove this claim, we need a few preliminary
lemmas.

Assume $(k,\ell)$ is a pair of kernels on $X$ with a Shimorin certificate
$\\{p[z]\\}_{z\in X}$. For $w\in X$, let

$X^{w}_{0}:=\\{z\in X:p[w](z,z)<1\\},\hskip 11.38092ptX^{w}_{1}:=\\{z\in
X:p[w](z,z)=1\\}.$ (3.4)

######  Lemma 3.9.

For every $w\in X,$ the sets $X^{w}_{0},X^{w}_{1}$ partition $X.$ Further,
$w\in X^{w}_{0},$ so $X^{w}_{0}$ is always non-empty.

###### Proof.

The proof is straightforward and is omitted. ∎

######  Remark 3.10.

As Remark 3.3 shows, it could happen that $X^{w}_{0}=\\{w\\}$ for every $w$.
On the other hand, it could also happen that $X^{w}_{0}=X$ for every $w$ (this
coincides with the existence of a strong Shimorin certificate $s$ that does
not vanish - see Proposition 3.4).

######  Lemma 3.11.

Given $w\in X,$ if $z\in X^{w}_{0}$ and $v\in X^{w}_{1}$, then $\ell(z,v)=0.$

###### Proof.

By definition, $p[w](v,v)=1.$ Further, since $p[w](z,z)<1,$ we must have
$|p[w](z,v)|<1$. Lemma 3.5 item (iii) then implies $\ell(z,v)=0$. ∎

The following lemma is well-known. We include a proof for the reader’s
convenience.

######  Lemma 3.12.

Let $\Omega$ denote a domain in $\mathbb{C}^{\mathsf{g}}.$ If
$f_{1},\dots,f_{n}:\Omega\to\mathbb{C}$ are non-zero and holomorphic, then
$\Omega\neq\cup Z(f_{j}),$ where $Z(f_{j})$ denotes the zero set of $f_{j}$
(in $\Omega$).

###### Proof.

Since the $f_{j}$’s are non-zero and holomorphic, the sets
$U_{j}=\Omega\setminus Z(f_{j})$ are open and dense in $\Omega.$ Hence, so is
$\cap_{j=1}^{n}U_{j}.$ In particular, $\cap_{j=1}^{n}U_{j}\neq\varnothing.$
Equivalently, $\cup_{j=1}^{n}Z(f_{j})\neq\Omega.$ ∎

The following is our main result in this subsection.

######  Proposition 3.13.

Assuming $(k,\ell)$ are non-zero holomorphic kernels on the (connected) domain
$\Omega\subseteq\mathbb{C}^{{\mathsf{g}}},$ the pair $(k,\ell)$ has a Shimorin
certificate if and only if it has a strong Shimorin certificate, in which case
$k$ is non-vanishing.

###### Proof.

We first show the existence of a Shimorin certificate implies $k$ is non-
vanishing. Assuming $k$ has a Shimorin certificate and arguing by
contradiction, suppose $z,w\in\Omega$ and $k(z,w)=0$ and consider the sets

$\begin{split}\Omega_{1}&=\\{v\in\Omega:k_{z}(v)=k(v,z)=0\\},\\\
\Omega_{2}&=\\{v\in\Omega:k_{w}(v)=k(v,w)=0\\},\\\
\Omega_{3}&=\\{v\in\Omega:\ell_{z}(v)=\ell(v,z)=0\\}.\end{split}$

Since the kernels $k,\ell$ are holomorphic in their first argument, the sets
$\Omega_{i}$ are the zero sets for (non-zero) holomorphic functions on
$\Omega$ and, by Lemma 3.5 (vii) it follows that $\cup_{i}\Omega_{i}=\Omega,$
contradicting the conclusion of Lemma 3.12.

Since the existence of a strong Shimorin certificate implies the existence of
a Shimorin certificate (Proposition 3.6 (i)), what remains to be proved is
that the existence of a Shimorin certificate implies that of a strong Shimorin
certificate. In view of Proposition 3.4, it suffices to show there is a point
$z\in X$ and a (PsD) kernel $p$ on $X$ such that $\ell^{z}\succeq p\ell$ and
$k^{z}\preceq pk$ and with $p(x,x)<1$ for all $x\in X.$ To this end, fix
$w\in\Omega$ and consider the decomposition
$\Omega=\Omega^{w}_{0}\cup\Omega^{w}_{1}$ as in (3.4). Arguing by
contradiction, suppose $\Omega^{w}_{1}$ is non-empty and choose
$v\in\Omega^{w}_{1}.$ Lemma 3.11 then tells us that $\Omega^{w}_{0}$ is
contained in the zero set of $\ell_{v},$ while $\Omega^{w}_{1}$ is contained
in the zero set of $\ell_{w},$ both of which are non-zero holomorphic
functions $\ell_{v},\ell_{w}:\Omega\to\mathbb{C}.$ Since the union of these
two sets is equal to all of $\Omega,$ Lemma 3.12 gives a contradiction. Thus,
$\Omega^{w}_{1}$ is empty and $\Omega=\Omega^{w}_{0}$ and consequently,
$p[w](z,z)<1$ for all $z\in\Omega.$ Hence, by Proposition 3.4, $(k,\ell)$ has
a strong Shimorin certificate. ∎

##  4\. Sufficient Conditions for the CP property

This section contains a new proof of Theorem 1.3, one based upon a general
version of the Leech factorization Theorem. It also contains a generalization
of Theorem 1.3, Theorem 4.5, proved using a one-step extension argument that
replaces the strong Shimorin certificate assumption of Theorem 1.3 with the
weaker assumption of the existence of a Shimorin certificate.

###  4.1. Theorem 1.3

This subsection begins with our Leech-based proof of Theorem 1.3 and concludes
with a corollary that generalizes a result of [5].

###### Proof of Theorem 1.3.

Let $s$ denote a strong certificate for $(k,\ell).$ Thus, $s$ is a CP kernel
and there exist kernels $h$ and $g$ on $X$ such that $k=(1-h)s$ and $\ell=sg$.
From Lemma 3.7, there exists a decomposition $X=\cup_{i\in I}X_{i}$ such that
$k(z,w)=s(z,w)=\ell(z,w)=0$ whenever $z\in X_{i}$ and $w\in X_{j}$ with $i\neq
j.$ Further, $s|_{X_{i}\times X_{i}}$ is non-vanishing and $k|_{X_{i}\times
X_{i}}$ is non-vanishing. Since $(k,\ell)$ has the CP property if and only if
every restriction $(k|_{X_{i}\times X_{i}},\ell|_{X_{i}\times X_{i}})$ does,
we assume, without loss of generality, that both $k$ and $s$ are non-vanishing
on $X.$ In particular, $|h(z,z)|<1$ for $z\in X.$

There exist Hilbert spaces $\mathcal{L}_{1},\mathcal{L}_{2}$ and functions
$\Psi:X\to\mathcal{B}(\mathcal{L}_{1},\mathbb{C}),$ and
$G:X\to\mathcal{B}(\mathcal{L}_{2},\mathbb{C})$ such that
$h(z,w)=\Psi(z)\Psi(w)^{*}$ and $g(z,w)=G(z)G(w)^{*}$. Since
$(1-\Psi\Psi^{*})s=k\succeq 0$, it follows that
$\Psi\in\operatorname{Mult}(\mathcal{H}_{s}\otimes\mathcal{L}_{1},\mathcal{H}_{s})$
is contractive. Further, $\|\Psi(z)^{*}\|^{2}=|h(z,z)|<1.$

Now, assume $z_{1},\dots,z_{n}\in X$ and $W_{1},\dots,W_{n}$ are $N\times N$
matrices that satisfy

$\big{[}\ell(z_{i},z_{j})I_{N\times
N}-k(z_{i},z_{j})W_{i}W_{j}^{*}\big{]}_{i,j=1}^{n}\succeq 0.$ (4.1)

Rewriting (4.1) as

$g(z_{i},z_{j})s(z_{i},z_{j})I_{N\times
N}+h(z_{i},z_{j})s(z_{i},z_{j})W_{i}W_{j}^{*}-s(z_{i},z_{j})W_{i}W_{j}^{*}\succeq
0,$

gives

$\big{(}H(z_{i})H(z_{j})^{*}-W_{i}W_{j}^{*}\big{)}s(z_{i},z_{j})\succeq 0,$

where

$H(z_{i})=\begin{bmatrix}I_{N\times N}\otimes
G(z_{i})&W_{i}\otimes\Psi(z_{i})\end{bmatrix}\in\mathcal{B}\big{(}(\mathbb{C}^{N}\otimes\mathcal{L}_{2})\oplus(\mathbb{C}^{N}\otimes\mathcal{L}_{1}),\mathbb{C}^{N}\big{)},$

for all $i.$ Applying the generalized version of Leech’s Theorem ([2, Theorem
8.57]) produces a contractive multiplier

$\Phi=\begin{bmatrix}\Phi_{1}\\\
\Phi_{2}\end{bmatrix}\in\operatorname{Mult}\big{(}\mathcal{H}_{s}\otimes\mathbb{C}^{N},\mathcal{H}_{s}\otimes\big{(}(\mathbb{C}^{N}\otimes\mathcal{L}_{2})\oplus(\mathbb{C}^{N}\otimes\mathcal{L}_{1})\big{)}\big{)}$

such that $W_{i}=H(z_{i})\Phi(z_{i}),$ for all $i.$ This last equality can be
rewritten as

$\begin{split}W_{i}&=\big{(}I_{N\times N}\otimes
G(z_{i})\big{)}\Phi_{1}(z_{i})+\big{(}W_{i}\otimes\Psi(z_{i})\big{)}\Phi_{2}(z_{i})\\\
&=\big{(}I_{N\times N}\otimes
G(z_{i})\big{)}\Phi_{1}(z_{i})+W_{i}\big{(}I_{N\times
N}\otimes\Psi(z_{i})\big{)}\Phi_{2}(z_{i}),\end{split}$ (4.2)

for all $i.$

Since $\Phi$ is a contractive multiplier, $\|\Phi_{2}(z)\|\leq 1$ for all $z.$
Since also $\|\Psi(z)^{*}\|<1,$ it follows that, for each $z,$ the $N\times N$
matrix $(I_{N\times N}\otimes\Psi(z))\Phi_{2}(z)$ is a strict contraction and
therefore $I_{N\times N}-(I_{N\times N}\otimes\Psi(z))\Phi_{2}(z)$ is
pointwise invertible. Define $R:X\to\mathbb{M}_{N}$ by

$R(z)=\big{(}I_{N\times N}\otimes G(z)\big{)}\Phi_{1}(z)\big{(}I_{N\times
N}-\big{(}I_{N\times N}\otimes\Psi(z)\big{)}\Phi_{2}(z)\big{)}^{-1}.$

Solving equation 4.2 for $W_{i}$ gives

$W_{i}=R(z_{i})$

for all $i.$ Extending $H$ to all of $X$ point-wise by setting

$H(z)=\begin{bmatrix}I_{N\times N}\otimes
G(z)&R(z)\otimes\Psi(z)\end{bmatrix},$

it follows from the definition of $R$ that $R(z)=H(z)\Phi(z),$ for all $z\in
X.$ Hence

$\begin{split}\ell(z,w)I_{N\times
N}-k(z,w)R(z)R(w)^{*}&=\big{(}H(z)H(w)^{*}-R(z)R(w)^{*}\big{)}s(z,w)\\\
&=H(z)\big{(}(I-\Phi(z)\Phi(w)^{*})s(z,w)\big{)}H(w)^{*}\succeq 0,\end{split}$

since $(I-\Phi(z)\Phi(w)^{*})s(z,w)\succeq 0$. Thus,
$R\in\operatorname{Mult}(\mathcal{H}_{k}\otimes\mathbb{C}^{N},\mathcal{H}_{\ell}\otimes\mathbb{C}^{N})$
is an interpolating contractive multiplier and our proof is complete. ∎

As a consequence of the above proof, we obtain the following parametrization
of multipliers between spaces with a strong Shimorin certificate, which
generalizes [5, Proposition 4.10].

######  Corollary 4.1.

Let $k,s,\ell,\Psi,G$ be as in the previous proof. A function
$R:X\to\mathbb{M}_{N}$ is a contractive multiplier from
$\mathcal{H}_{k}\otimes\mathbb{C}^{N}$ to
$\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}$ if and only if there exists a
contractive multiplier

$\Phi=\begin{bmatrix}\Phi_{1}\\\
\Phi_{2}\end{bmatrix}\in\operatorname{Mult}\big{(}\mathcal{H}_{s}\otimes\mathbb{C}^{N},\mathcal{H}_{s}\otimes\big{(}(\mathbb{C}^{N}\otimes\mathcal{L}_{2})\oplus(\mathbb{C}^{N}\otimes\mathcal{L}_{1})\big{)}\big{)}$

satisfying

$R(z)=\big{(}I_{N\times N}\otimes G(z)\big{)}\Phi_{1}(z)\big{(}I_{N\times
N}-\big{(}I_{N\times N}\otimes\Psi(z)\big{)}\Phi_{2}(z)\big{)}^{-1},$

for all $z\in X.$

######  Remark 4.2.

A standard modification of the above argument can be used to generalize [2,
Theorem 8.57] to pairs of kernels possessing a strong Shimorin certificate.
Such a generalization has already been obtained by Shimorin as a corollary of
his commutant lifting theorem; see [41, Corollary 2.3].

######  Remark 4.3.

We will shortly prove Theorem 4.5 (sufficiency for Shimorin certificates),
which (in view of Proposition 3.6) contains Theorem 1.3 as a special case.
However, we still chose to include the Leech factorization proof of Theorem
1.3 because it is of independent interest and also yields Corollary 4.1.

###  4.2. Shimorin Certificates and the CP property for pairs

Next, we will briefly investigate how the CP property for $(k,\ell)$ places
restrictions on the linear independence of kernel functions before turning to
generalizing Theorem 1.3 to pairs $(k,\ell)$ that have a Shimorin certificate.

In the case of a single CP kernel $k$ on a set $X,$ it is known (see [2, Lemma
7.5]) that if $k_{z},k_{w}$ are linearly independent for every choice of
distinct points $z,w\in X$ then the same must hold for any finite collection
$\\{k_{z_{1}},\dots,k_{z_{n}}\\},$ where $z_{1},\dots,z_{n}\in X$ are $n$
distinct points. Moving to the two-kernel setting, Remark 3.3 tells us that,
even if, for each $n$ and distinct point $z_{1},\dots,z_{n}\in X$ the
collection $\\{\ell_{z_{1}},\dots,\ell_{z_{n}}\\}$ is linearly independent, we
cannot draw any conclusions concerning $k_{z_{1}},\dots,k_{z_{n}}$ without
further knowledge of $(k,\ell)$. But what if $z_{1},\dots,z_{n}\in X$ are
distinct, but $\\{\ell_{z_{1}},\dots,\ell_{z_{n}}\\}$ is linearly dependent?

######  Lemma 4.4.

Assume $(k,\ell)$ is a CP pair on $X$ and $n\geq 2.$ Further, assume
$z_{1},\dots,z_{n}$ are distinct points in $X,$ the vectors
$\ell_{z_{1}},\dots,\ell_{z_{n-1}}$ are linearly independent and there exist
scalars $c_{1},\dots c_{n-1}\in\mathbb{C}$ such that

$\ell_{z_{n}}=\sum_{i=1}^{n-1}c_{i}\ell_{z_{i}}.$ (4.3)

If $1\leq j\leq n-1$ and $c_{j}\neq 0$, then $k_{z_{n}},k_{z_{j}}$ are
linearly dependent.

###### Proof.

Assume that there exist scalars $c_{1},\dots c_{n-1}$, not all of which can be
zero, such that (4.3) holds. Given $e_{1},\dots,e_{n-1}\in\mathbb{C},$ define

$R:\underset{1\leq i\leq n-1}{\text{span}}\\{\ell_{z_{i}}\\}\to\underset{1\leq
i\leq n-1}{\text{span}}\\{k_{z_{i}}\\}$

by $R\ell_{z_{i}}=e_{i}k_{z_{i}}$ (where we suppress the dependence of $R$ on
$e$). By the independence of $\ell_{z_{1}},\dots,\ell_{z_{n-1}},$ there exists
$\delta>0$ such that $R$ is a contraction whenever $|e_{i}|<\delta$ for all
$i.$ Thus, for any such choice of $e_{1},\dots,e_{n-1},$ the CP property of
$(k,\ell)$ implies the existence of $a\in\mathbb{C}$ such that the extension

$R_{a}:\underset{1\leq i\leq
n}{\text{span}}\\{\ell_{z_{i}}\\}\to\underset{1\leq i\leq
n}{\text{span}}\\{k_{z_{i}}\\}$

of $R$ determined by $R_{a}\ell_{z_{n}}=ak_{z_{n}}$ is a contraction. Thus,
(4.3) gives us

$\sum_{i=1}^{n-1}e_{i}c_{i}k_{z_{i}}=R\bigg{(}\sum_{i=1}^{n-1}c_{i}\ell_{z_{i}}\bigg{)}=R\big{(}\ell_{z_{n}}\big{)}\in\text{span}\\{k_{z_{n}}\\},$
(4.4)

for any $e_{1},\dots,e_{n-1}$ with modulus less than $\delta.$ Now, assume
$c_{j}\neq 0$ and choose $e_{j}\neq 0$ sufficiently small with $e_{i}=0$
whenever $i\neq j.$ The equality (4.4) implies
$k_{z_{j}}\in\text{span}\\{k_{z_{n}}\\},$ as desired. ∎

We will conclude this section by showing that the existence of a Shimorin
certificate is sufficient to guarantee the CP property. The proof employs the
standard one-point-extension argument in combination with Parrott’s Lemma,
which were also the main ingredients in Shimorin’s proof of Theorem 1.3.

######  Theorem 4.5.

If $(k,\ell)$ is a pair of kernels that possesses a Shimorin certificate, then
$(k,\ell)$ has the CP property.

###### Proof.

We will first prove the theorem under the assumption that for each $n$ and
distinct points $z_{1},\dots,z_{n},$ the kernel functions
$\ell_{z_{1}},\dots,\ell_{z_{n}},$ are linearly independent.

Assume $z_{1},\dots,z_{n}$ are given distinct points in $X$ and
$W_{1},\dots,W_{n}$ are $N\times N$ matrices that satisfy

$\big{[}\ell(z_{i},z_{j})I_{N\times
N}-k(z_{i},z_{j})W_{i}W_{j}^{*}\big{]}_{i,j=1}^{n}\succeq 0.$ (4.5)

We will first show that, for any $z_{n+1}\in
X\setminus\\{z_{1},\dots,z_{n}\\},$ there exists a matrix $W_{n+1}$ so that

$\big{[}\ell(z_{i},z_{j})I_{N\times
N}-k(z_{i},z_{j})W_{i}W_{j}^{*}\big{]}_{i,j=1}^{n+1}\succeq 0.$ (4.6)

Let $\\{u^{\alpha}\\}_{\alpha=1}^{N}$ be a basis for $\mathbb{C}^{N}$ and set

$\mathcal{M}^{k}_{n}=\text{span}\\{k_{z_{i}}\otimes u^{\alpha}:1\leq i\leq
n,1\leq\alpha\leq N\\},$
$\mathcal{M}^{\ell}_{n}=\text{span}\\{\ell_{z_{i}}\otimes u^{\alpha}:1\leq
i\leq n,1\leq\alpha\leq N\\}.$

Define $R:\mathcal{M}_{n}^{\ell}\to\mathcal{M}_{n}^{k}$ by

$R:\ell_{z_{i}}\otimes u^{\alpha}\mapsto k_{z_{i}}\otimes W^{*}_{i}u^{\alpha}$
(4.7)

and extend linearly. The operator $R$ is a contraction if and only if for
every choice of scalars $\\{a^{\alpha}_{i}\\}$ we have

$\big{\langle}\big{(}I-R^{*}R\big{)}\sum_{j,\beta}a^{\beta}_{j}\ell_{z_{j}}\otimes
u^{\beta},\sum_{i,\alpha}a^{\alpha}_{i}\ell_{z_{i}}\otimes
u^{\alpha}\big{\rangle}$
$=\sum_{i,j,\alpha,\beta}a^{\beta}_{j}\overline{a}^{\alpha}_{i}\Big{(}\ell(z_{i},z_{j})\langle
u^{\beta},u^{\alpha}\rangle-k(z_{i},z_{j})\big{\langle}W_{i}W^{*}_{j}u^{\beta},u^{\alpha}\big{\rangle}\Big{)}\geq
0,$

which is equivalent to (4.5). Thus, $\|R\|\leq 1$.

Now, by the independence assumption on $\ell_{z_{1}},\dots,\ell_{z_{n+1}},$
for each choice of $W=W_{n+1}$ there is an extension $R_{W}$ of $R$ on
$\mathcal{M}_{n+1}^{\ell}$ uniquely determined by

$R_{W}:\ell_{z_{n+1}}\otimes u^{\alpha}\mapsto k_{z_{n+1}}\otimes
W^{*}u^{\alpha}.$ (4.8)

When does there exist $W$ so that $\|R_{W}\|\leq 1$ (which is equivalent to
(4.6))? The answer is given by the following lemma, the proof of which rests
on an application of Parrott’s Lemma to an appropriate matrix decomposition of
$R_{W}.$ The single-kernel version of this argument is well-known (see e.g.
[38], [1], [2] and the unpublished work of Agler), and there are essentially
no new wrinkles in this generalized two-kernel version. Thus, we relegate the
details to Appendix A.

######  Lemma 4.6.

In the above setting, there exists $W$ so that $\|R_{W}\|\leq 1$ if and only
if

$\big{[}\ell^{z_{n+1}}(z_{i},z_{j})I_{N\times
N}-k^{z_{n+1}}(z_{i},z_{j})W_{i}W_{j}^{*}\big{]}_{i,j=1}^{n}\succeq 0.$

Now, let $J$ denote the matrix all of whose entries are $1$ with respect to
$\\{u^{\alpha}\\}$. By assumption, there exists a Shimorin certificate
$\\{p[x]\\}_{x\in X}$ for $(k,\ell).$ Thus,

$\begin{split}&\ell^{z_{n+1}}(z_{i},z_{j})I_{N\times
N}-k^{z_{n+1}}(z_{i},z_{j})W_{i}W_{j}^{*}\\\ =&I_{N\times
N}\cdot\big{[}\ell^{z_{n+1}}(z_{i},z_{j})\otimes
J\big{]}-W_{i}W_{j}^{*}\cdot\big{[}k^{z_{n+1}}(z_{i},z_{j})\otimes J\big{]}\\\
\succeq&I_{N\times
N}\cdot\big{[}p[z_{n+1}](z_{i},z_{j})\ell(z_{i},z_{j})\otimes
J\big{]}-W_{i}W_{j}^{*}\cdot\big{[}p[z_{n+1}](z_{i},z_{j})k(z_{i},z_{j})\otimes
J\big{]}\\\ =&\big{[}\ell(z_{i},z_{j})I_{N\times
N}-k(z_{i},z_{j})W_{i}W_{j}^{*}\big{]}\cdot\big{[}p[z_{n+1}](z_{i},z_{j})\otimes
J\big{]},\end{split}$

which is positive, being the Schur product of (4.5) with a positive matrix.
Thus, Lemma 4.6 tells us that, whenever $z_{1},\dots,z_{n}\in X$ and
$W_{1},\dots,W_{n}$ are $N\times N$ matrices that satisfy (4.5) and $z_{n+1}$
is any distinct point in $X,$ there exists a matrix $W_{n+1}$ so that (4.6)
holds. In other words, we have shown that one can always extend a multiplier
defined on a finite subset of $X$ to any other point without increasing the
norm. A transfinite induction (as in [38, Section 3]) leads to a contractive
multiplier satisfying the initial conditions. We omit the details.

Finally, we prove the general case, where it is possible to have linearly
dependent kernel functions. As before, assume $z_{1},\dots,z_{n}$ are given
distinct points in $X$ and $W_{1},\dots,W_{n}$ are $N\times N$ matrices that
satisfy (4.5). Without loss of generality we assume that
$\\{\ell_{z_{1}},\dots,\ell_{z_{m}}\\}$ is a maximal linearly independent
subset of $\\{\ell_{z_{1}},\dots,\ell_{z_{n}}\\}$. By Zorn’s lemma, there
exists a maximal subset $Y\subseteq X$ with the properties that no finite
subcollection of $\\{\ell_{v}:v\in Y\\}$ is linearly dependent and
$\\{z_{1},\dots,z_{m}\\}\subseteq Y.$ Let $\widetilde{k}=k|_{Y\times Y}$ and
$\widetilde{\ell}=\ell|_{Y\times Y}$. Since $(k,\ell)$ has the CP property,
the same must be true for $(\widetilde{k},\widetilde{\ell})$. Note that (4.5)
continues to hold if we restrict to the points $\\{z_{1},\dots,z_{m}\\}$.
Thus, in view of what has already been proved, there exists
$\widetilde{\Phi}:Y\to\mathbb{M}_{N}$ such that $\widetilde{\Phi}$ is a
contractive multiplier
$\mathcal{H}_{\widetilde{k}}\otimes\mathbb{C}^{N}\to\mathcal{H}_{\widetilde{\ell}}\otimes\mathbb{C}^{N}$
that satisfies $\widetilde{\Phi}(z_{i})=W_{i}$ for $1\leq i\leq m.$

Since the span of $\\{\ell_{x}:x\in X\\}$ is dense in $\mathcal{H}_{\ell}$, by
construction, the span of $\\{\ell_{y}:y\in Y\\}$ is dense in
$\mathcal{H}_{\ell}$ and thus $\widetilde{\Phi}$ induces a contractive map
$M^{*}:\mathcal{H}_{\ell}\otimes\mathbb{C}^{N}\to\mathcal{H}_{k}\otimes\mathbb{C}^{N}$
determined by $M^{*}\ell_{y}\otimes u=k_{y}\otimes\widetilde{\Phi}(y)^{*}u$
for $y\in Y$ and $u\in\mathbb{C}^{N}.$ Now, we argue that the function
$\widetilde{\Phi}:Y\to\mathbb{M}_{N}$ has a unique extension to a function
$\Phi:X\to\mathbb{M}_{N}$ such that $M^{*}=M_{\Phi}^{*}.$ The function $\Phi$
so extended satisfies $\Phi(z_{j})=W_{j}$ for all $j$ and thus solves the
original interpolation problem. Indeed, given $x\in X\setminus Y,$ we know (by
definition of $Y$) that there exist unique points $v_{1},\dots,v_{p}\in Y$ and
unique non-zero scalars $c_{1},\dots,c_{p}$ such that

$\ell_{x}=\sum_{i=1}^{p}c_{i}\ell_{v_{i}}.$

In view of Lemma 4.4, there exist, for each $1\leq i\leq p,$ (non-zero)
scalars $d_{1},\dots,d_{p}$ such that $k_{v_{i}}=d_{i}k_{x}.$ Thus,

$M^{*}\ell_{x}\otimes u=M^{*}\bigg{(}\sum_{i=1}^{p}c_{i}\ell_{v_{i}}\otimes
u\bigg{)}=\sum_{i=1}^{p}c_{i}k_{v_{i}}\otimes\widetilde{\Phi}(v_{i})^{*}u=k_{x}\otimes\bigg{[}\sum_{i=1}^{p}c_{i}d_{i}\widetilde{\Phi}(v_{i})^{*}u\bigg{]}.$

Hence, setting
$\Phi^{*}(x)=\sum_{i=1}^{p}c_{i}d_{i}\widetilde{\Phi}(v_{i})^{*},$ we have
$M^{*}=M_{\Phi}^{*}$ and this condition uniquely determines $\Phi.$ By
construction, $\Phi(z_{j})=W_{j}$ for $1\leq j\leq m.$ While not as
transparent, it is also true that $\Phi(z_{j})=W_{j}$ for $j>m.$ Indeed,
applying the argument above to $j>m=p$ and
$\ell_{z_{j}}=\sum_{i}c_{i}\ell_{z_{i}}$ yields

$k_{z_{j}}\otimes\Phi^{*}(z_{j})u=M_{\Phi}^{*}\big{(}\ell_{z_{j}}\otimes
u\big{)}=k_{z_{j}}\otimes\sum c_{i}d_{i}W_{i}^{*}u.$

On the other hand, the matrices $W_{j}$ for $j>m$ are determined by the
$W_{j}$ for $j\leq m.$ To verify this claim, let $\mathcal{E}$ denote the span
of $\\{\ell_{j}:1\leq j\leq n\\}$ and $\mathcal{F}$ the span of
$\\{k_{j}:1\leq j\leq n\\}.$ The inequality of equation 4.5 is equivalent to
the statement that the mapping
$T^{*}:\mathcal{E}\otimes\mathbb{C}^{N}\to\mathcal{F}\otimes\mathbb{C}^{N}$
determined by $T^{*}\big{(}\ell_{z_{j}}\otimes u\big{)}=k_{z_{j}}\otimes
W_{j}^{*}u$ is well-defined and a contraction. In particular, arguing as above
with the same notations,

$k_{z_{j}}\otimes W_{j}^{*}u=T^{*}\big{(}\ell_{z_{j}}\otimes
u\big{)}=\sum_{i=1}^{m}c_{i}k_{z_{i}}\otimes W_{i}^{*}u=k_{z_{j}}\otimes\sum
c_{i}d_{i}W_{i}^{*}u.$

Thus $W_{j}^{*}=\sum_{i=1}^{m}c_{i}d_{i}W_{i}^{*}=\Phi(z_{j})^{*}.$ ∎

##  5\. The CC Property for Diagonal Holomorphic Pairs

Throughout this section, we work exclusively with diagonal holomorphic kernels
$k,\ell.$ Unless otherwise noted, they are assumed normalized.

###  5.1. One-step extensions

Proposition 5.2 below, whose proof depends upon the following lemma,
interprets Proposition 2.15 in terms of one-step extensions.

Recall, for $a=(a_{1},\dots,a_{\mathsf{g}})\in\mathbb{N}^{\mathsf{g}},$ the
length of $a$ is $|a|=\sum|a_{j}|.$

######  Lemma 5.1.

Assume $\varnothing\neq F\subseteq\mathbb{N}^{{\mathsf{g}}}$ is finite and co-
invariant and $d\in\mathbb{N}^{{\mathsf{g}}}$ satisfies

  1. (i)

$d\notin F;$

  2. (ii)

if $a\in\mathbb{N}^{{\mathsf{g}}}$ and $|a|<|d|,$ then $a\in F.$

Setting $F^{+}:=(F\cup\\{d\\})\setminus\\{\mathbf{0}\\},$ the set
$F^{+}\cup\\{\mathbf{0}\\}$ is co-invariant and if $a,b\in F^{+}$ and $a\leq
b,$ then $b-a\in F.$

###### Proof.

If $a,b\in F$, we get the desired conclusion because $F$ is co-invariant. If
$b=d,$ then, since $|a|\geq 1,$ we must have $|b-a|<|b|=|d|,$ hence $b-a\in F$
in view of our hypothesis for $d.$ Finally, if $a=d,$ then we cannot have
$b\in F$ (since $d\leq b$ would imply that $d\in F$, by co-invariance), and so
we must have $b-a=d-d=\textbf{0}.$

Now, to check co-invariance, assume $b\in F^{+}$ and let
$a\in\mathbb{N}^{{\mathsf{g}}}$ satisfy $a\leq b$. If $b\in F,$ then we obtain
$a\in F$ since $F$ is co-invariant. Thus, either $a\in F^{+}$ or
$a=\\{\mathbf{0}\\}$. Further, if $b=d,$ then either $a=d$ (hence $a\in
F^{+}$) or $|a|<|d|,$ which implies $a\in F^{+}\cup\\{\mathbf{0}\\}$ in view
of our hypothesis for $d.$ ∎

######  Proposition 5.2.

A pair $(k,\ell)$ of diagonal holomorphic kernels has the complete
Carathéodory property if and only if for each $J\in\mathbb{N},$ each finite
co-invariant $F\subseteq\mathbb{N}^{{\mathsf{g}}}$ and each collection
$\\{c_{a}:a\in F\\}\subseteq\mathbb{M}_{J}$ such that the block (upper-
triangular) matrix $C$ indexed by $F\times F$ with block $J\times J$ entries,

$C_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise,}\end{cases}$ (5.1)

is a contraction and for every $d\in\mathbb{N}^{{\mathsf{g}}}$ such that

  1. (i)

$d\notin F;$ and

  2. (ii)

if $a\in\mathbb{N}^{{\mathsf{g}}}$ and $|a|<|d|,$ then $a\in F,$

the matrix $C^{+}$ indexed by $F^{+}\times F^{+},$ where
$F^{+}=(F\cup\\{d\\})\setminus\\{\mathbf{0}\\}$, and given by

$C^{+}_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise,}\end{cases}$ (5.2)

is also a contraction.

######  Remark 5.3.

The condition of, universally, passing from a contraction $C$ to a contraction
$C^{+}$ is the one-step extension property.

######  Remark 5.4.

Observe Lemma 5.1 is implicitly used in defining $C^{+}.$

######  Remark 5.5.

The existence of a $d$ satisfying the conditions of items (i) and (ii) of
Proposition 5.2 is not in doubt so long as $F\neq\mathbb{N}^{\mathsf{g}}.$
Indeed, let $m=\min\\{|f|:f\notin F\\}$ and choose $d$ such that $d\notin F$
and $|d|=m.$ In particular, by Lemma 5.1, the set $G=F\cup\\{d\\}$ is co-
invariant.

###### Proof of Proposition 5.2.

Suppose $(k,\ell)$ has the CC property and fix a co-invariant $F$ and
$d\in\mathbb{N}^{{\mathsf{g}}}\setminus F$ is such that $F\cup\\{d\\}$ is also
co-invariant. Let $F^{+}=(F\cup\\{d\\})\setminus\\{\mathbf{0}\\}$ and assume
$\\{c_{a}:a\in F\\}$ is such that the corresponding matrix $C$ in equation
(5.1) is a contraction. By assumption, there exists $c_{d}$ such that the
matrix indexed by $(F\cup\\{d\\})\times(F\cup\\{d\\})$

$\widetilde{C}_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise.}\end{cases}$

is also a contraction. By considering the submatrix indexed by $F^{+}\times
F^{+},$ we find that the matrix $C^{+}$ (indexed by $F^{+}\times F^{+}$),

$C^{+}_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a>0,\\\ \mathbf{0},\hskip
48.36958pt\text{otherwise,}\end{cases}$ (5.3)

is a contraction.

It remains to prove, if $(k,\ell)$ satisfies the one-step extension property
as in the statement of the proposition, then $(k,\ell)$ has the CC property.
Accordingly, suppose $F$ is co-invariant, $d$ satisfying the conditions of
items (i) and (ii) of the proposition and $C$ as in equation (5.1) is $C$ is a
contraction. Thus, by the one-step-extension assumption, $C^{+}$ is also a
contraction. With $c_{d}$ to be determined, consider the enlarged
$(F\cup\\{d\\})\times(F\cup\\{d\\})$ matrix

$\widetilde{C}_{a,b}=\begin{cases}c_{b-a}\sqrt{\cfrac{k_{a}}{\ell_{b}}},\hskip
14.22636ptb\geq a,\\\ \mathbf{0},\hskip
29.87547pt\text{otherwise.}\end{cases}$

Partitioning

$\widetilde{C}=\begin{pmatrix}A&X\\\ B&D\end{pmatrix},$

where the scalar entry $X$ is to be determined and

$\begin{pmatrix}A\\\
B\end{pmatrix}=\begin{pmatrix}\widetilde{C}_{a,b}\end{pmatrix}_{b\neq
d}=\begin{pmatrix}C\\\ \mathbf{0}\end{pmatrix},$

(where the partitionings on the left and right are not the same) and

$\begin{pmatrix}B&D\end{pmatrix}=\begin{pmatrix}\widetilde{C}_{a,b}\end{pmatrix}_{a\neq
d}=\begin{pmatrix}\mathbf{0}&C^{+}\end{pmatrix},$

an application of Parrott’s Lemma produces an $X$ such that $\widetilde{C}$ is
a contraction. Setting $c_{d}=\sqrt{\ell_{d}}X$ shows there is a choice of
$c_{d}$ for which $\widetilde{C}$ is a contraction.

Since, by Lemma 5.1, $F\cup\\{d\\}$ is also co-invariant, we may now proceed
by induction; first, we extend $F$ finitely many (possibly zero) times so that
it contains every $a\in\mathbb{N}^{{\mathsf{g}}}$ with $|a|\leq 1,$ then we
perform another finite number of extensions to include every
$a\in\mathbb{N}^{{\mathsf{g}}}$ with $|a|\leq 2,$ and so on. Indeed, choosing
the new point $d\in\mathbb{N}^{{\mathsf{g}}}$ so that it satisfies items (i)
and (ii) at each step guarantees that the extended index set $F\cup\\{d\\}$
will always be co-invariant. By induction we obtain
$\\{c_{a}\in\mathbb{M}_{J}:a\notin F\\}$ so that the infinite matrix
$\mathscr{C}$ given by (2.6) is a contraction as well, in the sense that each
finite submatrix is a contraction, which is what the CC property requires. ∎

###  5.2. Sufficient conditions for the CC property

For the purposes of this section, we will temporarily recast Definition 1.2 in
terms of formal power series. Domain consideration issues will not trouble us
until Section 6.

Given diagonal holomorphic kernels $(k,\ell)$, a formal Shimorin certificate
for $(k,\ell)$ is a formal power series

$t(x)=\sum_{|a|>0}t_{a}x^{a},$

where $t_{a}\geq 0$ for all $a$, for which there exist

$g(x)=\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}g_{a}x^{a}\hskip 5.69046pt\text{ and
}\hskip 5.69046pth(x)=\sum_{a\in\mathbb{N}^{{\mathsf{g}}}}h_{a}x^{a}$

such that $g_{a},h_{a}\ \geq 0$ for all $a$ and

$\ell(1-t)=g,\ \ \ 1-k(1-t)=h,$

in the sense of formal power series. These conditions are equivalent to
$g_{\mathbf{0}}=\ell_{\mathbf{0}}$ and $h_{\mathbf{0}}=1-k_{\mathbf{0}}$ and,
for $|a|>0,$

$\ell_{a}=g_{a}+\sum_{0<u\leq a}t_{u}\ell_{a-u}$ (5.4)

and

$k_{a}+h_{a}=\sum_{0<u\leq a}t_{u}k_{a-u}.$ (5.5)

We will now show that existence of a formal Shimorin certificate guarantees
the CC property. Even though this result is subsumed by Theorem 6.4 (to be
proved in the next section), we have chosen to include the proof as it
requires only a short detour and might be of independent interest.

######  Theorem 5.6.

Suppose the kernels $k(z,w)=1+\sum_{|a|\geq 1}k_{a}z^{a}\overline{w}^{a}$ and
$\ell(z,w)=1+\sum_{|a|\geq 1}\ell_{a}z^{a}\overline{w}^{a}$ are diagonal and
holomorphic. If there is a formal Shimorin certificate for $(k,\ell),$ then
$(k,\ell)$ has the CC property.

Before turning to the proof of Theorem 5.6, we record the following lemma.

######  Lemma 5.7.

Assume $F\subseteq\mathbb{N}^{{\mathsf{g}}}$ is co-invariant. If the block
matrix $C=(C_{a,b})_{a,b\in F}$ with entries from $\mathbb{M}_{J}$ is
positive, then, for each $v\in\mathbb{N}^{{\mathsf{g}}}$, the matrix
$\widetilde{C}=(\widetilde{C}_{a,b})_{a,b\in F}$ defined by

$\widetilde{C}_{a,b}=\begin{cases}C_{a-v,b-v},\hskip 8.5359pta,b\geq v\\\
\mathbf{0},\hskip 31.2982pt\text{otherwise},\end{cases}$

is positive.

######  Remark 5.8.

The statement of Lemma 5.7 has used the assumption $F$ is coinvariant to
guarantee that $a,b\geq v$ implies $a-v,b-v\in F.$

###### Proof.

Since $C$ is positive, there exists $M\geq 1$ and $G:F\to\mathbb{M}_{J\times
M}$ such that $C_{a,b}=G(a)G(b)^{*}$ for all $a,b\in F.$ Define
$\widetilde{G}:F\to\mathbb{M}_{J\times M}$ by setting
$\widetilde{G}(a)=G(a-v)$ if $a\geq v$ and $G(a)=\mathbf{0}$ otherwise. Then,
$\widetilde{C}_{a,b}=\widetilde{G}(a)\widetilde{G}(b)^{*},$ for all $a,b\in
F$, which implies that $\widetilde{C}$ is positive. ∎

###### Proof of Theorem 5.6.

By assumption, there exist power series $t,g,h$ satisfying the conditions of
equations (5.5) and (5.4). We will prove that $(k,\ell)$ has the CC property
as per Proposition 5.2. Suppose $J\in\mathbb{N},$ the set
$F\subseteq\mathbb{N}^{{\mathsf{g}}}$ is co-invariant, the collection of
coefficients $\\{c_{a}:a\in F\\}\subseteq\mathbb{M}_{J}$ is such that the
block matrix $C=(c_{a,b})_{a,b\in F}$ defined as in (1.8) is a contraction and
$d\in\mathbb{N}^{{\mathsf{g}}}$ satisfies

  1. (i)

$d\notin F;$

  2. (ii)

if $a\in\mathbb{N}^{{\mathsf{g}}}$ and $|a|<|d|,$ then $a\in F.$

It suffices to show that the matrix $C^{+}$ indexed by $F^{+}\times F^{+}$
(where $F^{+}=(F\cup\\{d\\})\setminus\\{\mathbf{0}\\}$) and defined as in
(5.2) is a contraction.

Now, $C$ being a contraction implies that $X:=I-C^{*}C$ is positive. Letting
$I_{J}$ denote the $J\times J$ identity matrix, we obtain, for every $a,b\in
F,$

$X_{a,b}=\delta_{a,b}I_{J}-(C^{*}C)_{a,b}$ $=\delta_{a,b}I_{J}-\sum_{u\in
F}(C^{*})_{a,u}c_{u,b}$ $=\delta_{a,b}I_{J}-\sum_{u\in F}(c_{u,a})^{*}c_{u,b}$
$=\delta_{a,b}I_{J}-\sum_{u\in
F}c^{*}_{a-u}c_{b-u}\frac{k_{u}}{\sqrt{\ell_{a}\ell_{b}}}$
$=\delta_{a,b}I_{J}-\sum_{u\leq
a,b}c^{*}_{a-u}c_{b-u}\frac{k_{u}}{\sqrt{\ell_{a}\ell_{b}}},$

where the last equality holds because $F$ is co-invariant. Taking, for
convenience, the Schur product of $X$ with the dyad
$(\sqrt{\ell_{a}}\sqrt{\ell_{b}})_{a,b\in F},$ we obtain that the matrix with
entries

$\sqrt{\ell_{a}\ell_{b}}X_{a,b}=\ell_{a}\delta_{a,b}I_{J}-\sum_{u\leq
a,b}k_{u}c^{*}_{a-u}c_{b-u},$ (5.6)

$a,b\in F,$ is positive. Our goal now is to show that
$X^{+}:=I-(C^{+})^{*}C^{+}$ is positive. Arguing as before, $X^{+}$ is
positive if and only if the matrix with entries

$\sqrt{\ell_{a}\ell_{b}}X^{+}_{a,b}=\ell_{a}\delta_{a,b}I_{J}-\sum_{u\in
F^{+}}k_{u}c^{*}_{a-u}c_{b-u}$ $=\ell_{a}\delta_{a,b}I_{J}-\sum_{0<u\leq
a,b}k_{u}c^{*}_{a-u}c_{b-u},$ (5.7)

$a,b\in F^{+}$, is positive. In view of (5.5), (5.4),

$\begin{split}&\sqrt{\ell_{a}\ell_{b}}X^{+}_{a,b}\\\
&=g_{a}\delta_{a,b}I_{J}+\sum_{0<v\leq
a}t_{v}\ell_{a-v}\delta_{a,b}I_{J}-\sum_{0<u\leq a,b}\sum_{0<v\leq
u}t_{v}k_{u-v}c^{*}_{a-u}c_{b-u}+\sum_{0<u\leq
a,b}h_{u}c^{*}_{a-u}c_{b-u}\end{split}$ $=g_{a}\delta_{a,b}I_{J}+\sum_{0<v\leq
a}t_{v}\ell_{a-v}\delta_{a,b}I_{J}-\sum_{0<v\leq a,b}\sum_{v\leq u\leq
a,b}t_{v}k_{u-v}c^{*}_{a-u}c_{b-u}+\sum_{0<u\leq a,b}h_{u}c^{*}_{a-u}c_{b-u}$
$=g_{a}\delta_{a,b}I_{J}+\sum_{0<u\leq
a,b}h_{u}c^{*}_{a-u}c_{b-u}+\sum_{0<v\leq
a}t_{v}\Big{(}\ell_{a-v}\delta_{a,b}I_{J}-\sum_{v\leq u\leq
a,b}k_{u-v}c^{*}_{a-u}c_{b-u}\Big{)}$ $=g_{a}\delta_{a,b}I_{J}+\sum_{0<u\leq
a,b}h_{u}c^{*}_{a-u}c_{b-u}+\sum_{0<v\leq
a}t_{v}\Big{(}\ell_{a-v}\delta_{a,b}I_{J}-\sum_{w\leq
a-v,b-v}k_{w}c^{*}_{a-v-w}c_{b-v-w}\Big{)}$
$=g_{a}\delta_{a,b}I_{J}+\sum_{0<u\leq
a,b}h_{u}c^{*}_{a-u}c_{b-u}+\sum_{0<v\leq
a}t_{v}\sqrt{\ell_{a-v}\ell_{b-v}}X_{a-v,b-v},$

for all $a,b\in F^{+},$ where for the last equality we have used (5.6) (note
that, since $v>0,$ both $a-v,b-v\in F$ assuming they are defined). Thus, we
obtain

$X^{+}=G+H+\sum_{v\in F^{+}}t_{v}X_{v},$ (5.8)

where the $\mathbb{M}_{J}$-block matrices $G=(G_{a,b}),H=(H_{a,b}),$ and
$X_{v}=\big{(}(X_{v})_{a,b}\big{)}$ are indexed by $F^{+}\times F^{+}$ and
defined as follows:

  1. (i)

$G_{a,b}=\frac{g_{a}}{\ell_{a}}\delta_{a,b}I_{J}$;

  2. (ii)

$H_{a,b}=\sum_{0<u\leq
a,b}\frac{h_{u}}{\sqrt{\ell_{a}\ell_{b}}}c^{*}_{a-u}c_{b-u}$;

  3. (iii)

$(X_{v})_{a,b}=X_{a-v,b-v}$, for any $v\in F^{+},$ where
$X_{a-v,b-v}=\mathbf{0}$ if either $v\nleq a$ or $v\nleq b$,

for all $a,b\in F^{+}.$ Clearly, $G$ is positive, while a short computation
reveals that $H=(C_{h})^{*}C_{h},$ where

$(C_{h})_{a,b}=\frac{\sqrt{h_{a}}}{\sqrt{\ell_{b}}}c_{b-a},$

for all $a,b\in F^{+}.$ Thus, $H$ must be positive as well. Finally, every
$X_{v}$ is positive because of Lemma 5.7 and the fact that $X$ was positive to
begin with. Hence, (5.8) allows us to conclude that $X^{+}$ is positive, as
desired. ∎

###  5.3. Existence of a master certificate and a necessary condition for the
CP property

Let $k$ be a normalized diagonal holomorphic kernel with coefficients
$\\{k_{a}\\}_{a\in\mathbb{N}^{{\mathsf{g}}}}.$ We recall Definition 1.5 from
the introduction: the master certificate associated with $k$ is the formal
power series in ${\mathsf{g}}$ complex variables defined as

$\vartheta(x)=\sum_{b\in\mathbb{N}^{\mathsf{g}}}\vartheta_{b}x^{b},$

where $\vartheta_{\mathbf{0}}=0$, $\vartheta_{e_{j}}=k_{e_{j}}$ for all $j$
(recall that $e_{j}\in\mathbb{N}^{\mathsf{g}}$ denotes the element with $1$ in
the $j$-th entry and $0$ elsewhere) and

$\vartheta_{b}=\max\Bigg{\\{}0,k_{b}-\sum_{\begin{subarray}{c}w+u=b,\\\
w,u\neq\mathbf{0}\end{subarray}}\vartheta_{w}k_{u}\Bigg{\\}},$ (5.9)

for all $b\in\mathbb{N}^{{\mathsf{g}}}$ with $|b|>1.$ In this subsection, we
show that, for any diagonal holomorphic $\ell,$ if $(k,\ell)$ has the CC
property, then $\vartheta$ is a formal Shimorin certificate for $(k,\ell).$
Thus, in combination with our results from the previous subsection, we obtain
a complete characterization of those pairs of diagonal holomorphic kernels
possessing the CC property. Recall the standing assumption that $k$ and $\ell$
are normalized.

Two things are immediate from the construction of $\vartheta$. First,
$k_{b}\geq\vartheta_{b}\geq 0$ for all $b$ and second

$k_{b}-\sum_{\begin{subarray}{c}w+u=b,\\\
w\neq\mathbf{0}\end{subarray}}k_{u}\vartheta_{w}\leq 0$ (5.10)

for all $\mathbf{0}\neq b\in\mathbb{N}^{\mathsf{g}},$ so that, as a formal
power series, the coefficients of $h=1-k(1-\vartheta)$ are all non-negative.
These observations are summarized in the following proposition.

######  Theorem 5.9.

The master certificate $\theta$ associated to a normalized diagonal
holomorphic kernel $k$ satisfies $k_{b}\geq\vartheta_{b}$ for all $b$ and the
coefficients of the formal power series $h=1-k(1-\vartheta)$ are all non-
negative.

######  Theorem 5.10.

Suppose $\ell$ is a normalized diagonal holomorphic kernel. If $(k,\ell)$ has
the CC property, then the coefficients of the formal power series
$g=\ell(1-\vartheta)$ are all non-negative; that is, $h_{0}=0$ and

$g_{a}=\ell_{a}-\sum_{\begin{subarray}{c}u\leq a,\\\
u\neq\mathbf{0}\end{subarray}}\vartheta_{u}\ell_{a-u}\geq 0$

for $\mathbf{0}\neq a\in\mathbb{N}^{\mathsf{g}}.$

The proof of Theorem 5.10 is postponed in favor of the following key result.

######  Theorem 5.11.

A pair of diagonal holomorphic kernels $(k,\ell)$ has the CC property if and
only if the master certificate $\vartheta$ for $k$ is a formal Shimorin
certificate for $(k,\ell).$

###### Proof.

Suppose $(k,\ell)$ has the CP property. By Theorems 5.9 and 5.10, the formal
power series $h=1-k(1-\vartheta)$ and $g=\ell(1-\vartheta)$ have non-negative
coefficients. Hence, $\vartheta$ is a formal certificate for $(k,\ell).$

The converse is Theorem 5.6. ∎

The proof of Theorem 5.10 consumes the remainder of this subsection. It uses
the Lemmas 5.12 and 5.13 immediately below.

######  Lemma 5.12.

Suppose $(k,\ell)$ is a normalized diagonal holomorphic CC pair. Suppose
further, $\varnothing\neq F\subseteq\mathbb{N}^{{\mathsf{g}}}$ is finite,
$d\in\mathbb{N}^{{\mathsf{g}}}\setminus F$ and both $F$ and
$F^{+}=F\cup\\{d\\}$ are co-invariant. If $\\{v_{a}:a\in F\\}$ is a set of
non-negative real numbers such that

$\ell_{a}\geq\sum_{u\leq a}v_{a-u}k_{u}\geq 0,$ (5.11)

for each $a\in F,$ then for all $a\in F^{+},$

$\ell_{a}\geq\sum_{u<a}v_{u}k_{a-u}=\sum_{0<u\leq a}v_{a-u}k_{u}.$ (5.12)

###### Proof.

Let $c_{a}$ denote the $F\times F$ matrix with $(a,\mathbf{0})$ entry
$\sqrt{v_{a}}$ and also let $E_{1,1}$ denote the $F\times F$ matrix with $1$
in the $(\mathbf{0},\mathbf{0})$ entry and $0$ everywhere else. Note

$c_{a}^{*}c_{b}=\begin{cases}\mathbf{0}&\mbox{ if }a\neq b\\\
v_{a}E_{1,1}&\mbox{ if }a=b.\end{cases}$

The conditions of equation (5.11) imply the corresponding $F\times F$ block
matrix $C=C(\\{c_{a}\\})$ as in (5.1) is a contraction, since

$(C^{*}C)_{a,b}=\sum_{u\leq
a,b}c_{a-u}^{*}c_{b-u}\frac{k_{u}}{\sqrt{\ell_{a}\ell_{b}}}=\begin{cases}\Big{(}\sum_{u\leq
a}v_{a-u}\frac{k_{u}}{\ell_{a}}\Big{)}E_{1,1}&\hskip 3.41418pta=b\\\
\mathbf{0}&\mbox{ otherwise}.\end{cases}$

In view of the discussion preceding the proof of Proposition 5.2, the
$F^{+}\times F^{+}$ matrix $C^{+}$ as in (5.3) is a contraction. Moreover, for
$a,b\in F^{+}$

$\begin{split}((C^{+})^{*}\,C^{+})_{a,b}&=\begin{cases}\Big{(}\sum\\{c_{a-u}^{*}c_{a-u}\frac{k_{u}}{\ell_{a}}\mid
F_{N}^{+}\ni u\leq a\\}\Big{)}E_{1,1}&\mbox{ if }a=b\\\ \mathbf{0}&\mbox{
otherwise}.\end{cases}\\\
&=\begin{cases}\Big{(}\sum_{u<a}v_{u}\frac{k_{a-u}}{\ell_{a}}\Big{)}E_{1,1}&\mbox{
if }a=b\\\ \mathbf{0}&\mbox{ otherwise},\end{cases}\end{split}$

from which the inequalities of equation (5.12) follow. ∎

######  Lemma 5.13.

Given normalized diagonal holomorphic $k,\ell$ in ${\mathsf{g}}$ variables,
$N\in\mathbb{N}$ and $d\in\mathbb{N}^{{\mathsf{g}}}$ with $|d|=N+1$ and
$S\subseteq\\{a\leq d\\}$, define $v_{a}$ for $a\leq d$ recursively as
follows. Let $v_{\mathbf{0}}=0$ if $0\in S,$ and $v_{\mathbf{0}}=1$ if
$0\notin S$ and, assuming $0\leq M\leq N$ and $v_{u}$ have been defined for
$|u|\leq M$ and $u\leq d,$ let

$v_{a}=\begin{cases}0&a\in S\\\ \ell_{a}-\sum_{u<a}v_{u}k_{a-u}&a\notin
S\end{cases}$

for $|a|=M+1$ and $a\leq d.$ If $(k,\ell)$ is a CC pair, then

  1. (i)

$v_{a}\geq 0$ for all $a\leq d,$ and

  2. (ii)

$\ell_{a}-\sum_{u\leq a}v_{u}k_{a-u}\geq 0$ for all $a\leq d.$

###### Proof.

Let $d,S$ and $v_{a}$ be as above. We will proceed by induction on $|a|.$ By
definition, $v_{\mathbf{0}}\geq 0$ and
$\ell_{\mathbf{0}}-v_{\mathbf{0}}k_{\mathbf{0}}=1-v_{\mathbf{0}}\geq 0,$ so
the result holds if $|a|=0.$ Now, let $0\leq M\leq N$ and assume that the
conditions of both item (i) and item (ii) hold whenever $|a|\leq M$ and $a\leq
d.$ We will show that they continue to hold if $|a|=M+1$ and $a\leq d$.

Fix an arbitrary $a\in\mathbb{N}^{{\mathsf{g}}}$ with $|a|=M+1$ and $a\leq d.$
Set $F=\\{u\in\mathbb{N}^{{\mathsf{g}}}:|u|\leq M\text{ and }u\leq d\\}$.
Clearly, both $F$ and $F\cup\\{a\\}$ are co-invariant. Set
$F^{+}=(F\cup\\{a\\})\setminus\\{\mathbf{0}\\}$. The collection of non-
negative numbers $\\{v_{u}:u\in F\\}$ satisfies (5.11) by our inductive
hypothesis. By Lemma 5.12, it follows that

$\ell_{a}-\sum_{u<a}v_{u}k_{a-u}\geq 0.$

Hence $v_{a}\geq 0$ so that (i) holds for $a.$ Moreover, if $a\notin S,$ then
$v_{a}=\ell_{a}-\sum_{u<a}v_{u}k_{a-u}$ and

$\ell_{a}-\sum_{u\leq a}v_{u}k_{a-u}=\ell_{a}-\sum_{u<a}v_{u}k_{a-u}-v_{a}=0.$

On the other hand, if $a\in S$, then $v_{a}=0,$ in which case

$\ell_{a}-\sum_{u\leq a}v_{u}k_{a-u}=\ell_{a}-\sum_{u<a}v_{u}k_{a-u}\geq 0.$

Thus, in either case, the inequality of item (ii) holds for $a,$ completing a
proof by induction. ∎

###### Proof of Theorem 5.10.

Our goal is to show

$\ell_{d}-\sum_{\mathbf{0}<u\leq d}\vartheta_{u}\ell_{d-u}\geq 0$ (5.13)

for all non-zero $d\in\mathbb{N}^{{\mathsf{g}}}.$

Assume first that $|d|=1$. Choosing $F=\\{\mathbf{0}\\}$, $v_{\mathbf{0}}=1$
and $d=e_{j}$ in Lemma 5.12, it follows that

$\ell_{e_{j}}\geq k_{e_{j}}=\vartheta_{e_{j}}.$

Thus,

$(\ell(1-\vartheta))_{e_{j}}=\ell_{e_{j}}-\vartheta_{e_{j}}\geq 0$

for $1\leq j\leq{\mathsf{g}}.$

Now, assume $|d|\geq 2.$ Put $S=\\{a\leq d:\vartheta_{d-a}=0\\}$ and define
$v_{a}$ recursively as in Lemma 5.13, for every $a\leq d.$ We have $v_{a}\geq
0$ by construction. Set

$\alpha_{a}=\begin{cases}0,\hskip 17.07182pt\vartheta_{d-a}=0\\\ 1,\hskip
17.07182pt\vartheta_{d-a}>0.\end{cases}$

Thus, $v_{\mathbf{0}}=\alpha_{\mathbf{0}}$ and
$\alpha_{a}\vartheta_{d-a}=\vartheta_{d-a}$ as well as,

$v_{a}=\alpha_{a}\bigg{(}\ell_{a}-\sum_{u<a}v_{u}k_{a-u}\bigg{)},$

for all $|a|\geq 1$. In particular, $v_{\mathbf{0}}\theta_{d}=\theta_{d}.$
Moreover,

$\vartheta_{d-a}=\alpha_{a}\bigg{(}k_{d-a}-\sum_{\mathbf{0}<u<d-a}k_{u}\vartheta_{d-a-u}\bigg{)}$

assuming $|d-a|\geq 2$, while $\vartheta_{e_{j}}=a_{d-e_{j}}k_{e_{j}}$ for all
$j$, since $\theta_{e_{j}}=k_{e_{j}}>0$ and hence $a_{d-e_{j}}=1$ (recall also
that $\vartheta_{\mathbf{0}}=0=a_{d}\cdot 0$).

To prove (5.13), we will show, by induction, that for all $d,$

$\sum_{v<d}v_{a}k_{d-a}=\sum_{\mathbf{0}<u\leq d}\theta_{u}\ell_{d-u}.$ (5.14)

First, observe that

$\displaystyle\sum_{a<d}v_{a}k_{d-a}=$
$\displaystyle\sum_{a+e_{j}=d}\alpha_{a}\bigg{(}\ell_{a}-\sum_{b<a}v_{b}k_{a-b}\bigg{)}k_{e_{j}}+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-2\end{subarray}}v_{b}k_{d-b}$ $\displaystyle=$
$\displaystyle\sum_{a+e_{j}=d}\alpha_{a}\ell_{a}k_{e_{j}}-\sum_{a+e_{j}=d}\sum_{b<a}\alpha_{a}v_{b}k_{a-b}k_{e_{j}}+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}k_{d-b}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}k_{d-b}$ $\displaystyle=$
$\displaystyle\sum_{a+e_{j}=d}\vartheta_{e_{j}}\ell_{a}-\sum_{a+e_{j}=d}\sum_{b<a}v_{b}k_{d-b-
e_{j}}\vartheta_{e_{j}}+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}k_{d-b}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}k_{d-b}$ $\displaystyle=$
$\displaystyle\sum_{a+e_{j}=d}\vartheta_{e_{j}}\ell_{a}-\sum_{a+e_{j}=d}\sum_{b+e_{i}=a}v_{b}k_{e_{i}}\vartheta_{e_{j}}-\sum_{a+e_{j}=d}\sum_{\begin{subarray}{c}b<a,\\\
|b|\leq|a|-2=|d|-3\end{subarray}}v_{b}k_{d-b-e_{j}}\vartheta_{e_{j}}$
$\displaystyle\qquad+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}k_{d-b}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}k_{d-b}.$ (5.15)

Summing the second and fifth terms on the right hand side in (5.3) gives

$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}k_{d-b}-\sum_{a+e_{j}=d}\sum_{b+e_{i}=a}v_{b}k_{e_{i}}\vartheta_{e_{j}}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}k_{d-b}-\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}\sum_{\begin{subarray}{c}b+e_{i}+e_{j}={\mathsf{g}},\\\
1\leq i,j\leq{\mathsf{g}}\end{subarray}}k_{e_{i}}\vartheta_{e_{j}}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}e_{i}+e_{j}=d-b,\\\
1\leq i,j\leq d\end{subarray}}k_{e_{i}}\vartheta_{e_{j}}\bigg{)}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}\alpha_{b}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}e_{i}+e_{j}=d-b,\\\
1\leq i,j\leq{\mathsf{g}}\end{subarray}}k_{e_{i}}\vartheta_{e_{j}}\bigg{)}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}\vartheta_{d-b},$ (5.16)

and summing the third and fourth terms on the right hand side in (5.3) gives

$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}k_{d-b}-\sum_{a+e_{j}=d}\sum_{\begin{subarray}{c}b<a,\\\
|b|\leq|d|-3\end{subarray}}v_{b}k_{d-b-e_{j}}\vartheta_{e_{j}}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}k_{d-b}-\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}\sum_{\begin{subarray}{c}d-b-e_{j}>\mathbf{0},\\\
1\leq i\leq{\mathsf{g}}\end{subarray}}v_{b}k_{d-b-e_{j}}\vartheta_{e_{j}}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}d-b-
e_{j}>\mathbf{0},\\\ 1\leq i\leq{\mathsf{g}}\end{subarray}}k_{d-b-
e_{j}}\vartheta_{e_{j}}\bigg{)}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}d-b-w>\mathbf{0},\\\
|w|=1\end{subarray}}k_{d-b-w}\vartheta_{w}\bigg{)}.$ (5.17)

Combining (5.3)-(5.3)-(5.3), we obtain

$\sum_{a<d}v_{a}k_{d-a}$
$=\sum_{a+e_{j}=d}\vartheta_{e_{j}}\ell_{a}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-2\end{subarray}}v_{b}\vartheta_{d-b}+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-3\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}|w|=1,\\\
w<d-b\end{subarray}}k_{d-b-w}\vartheta_{w}\bigg{)}.$ (5.18)

Notice that if $|d|=2,$ then (5.18) becomes (recall that
$k_{\mathbf{0}}=\ell_{\mathbf{0}}=1$)

$\sum_{a<d}v_{a}k_{d-a}=\sum_{a+e_{j}=d}\vartheta_{e_{j}}\ell_{a}=\sum_{\mathbf{0}<u\leq
d}\vartheta_{u}\ell_{d-u}.$ (5.19)

Now, assume $|d|\geq 3.$ In order to arrive at $\eqref{interr}$, we will show
that, for every $2\leq m\leq|d|-1$,

$\displaystyle\sum_{a<d}v_{a}k_{d-a}=$
$\displaystyle\sum_{\begin{subarray}{c}u<d,\\\ 0<|u|\leq
m-1\end{subarray}}\vartheta_{u}\ell_{d-u}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}v_{b}\vartheta_{d-b}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-1-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}u<d-b,\\\
0<|u|<m\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)}.$ (5.20)

In view of (5.18), we see that (5.3) holds for $m=2.$ If $|d|=3,$ we are done.
Thus we further assume $|d|\geq 4$. We move to the inductive step.
Accordingly, suppose (5.3) holds for some fixed $2\leq m\leq|d|-2$. We will
show that (5.3) also holds for $m+1.$ First, observe that

$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}v_{b}\vartheta_{d-b}$
$\displaystyle=\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}\vartheta_{d-b}\alpha_{b}\bigg{(}\ell_{b}-\sum_{u<b}v_{u}k_{b-u}\bigg{)}$
$\displaystyle=\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}\alpha_{b}\vartheta_{d-b}\ell_{b}-\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}\sum_{u<b}\alpha_{b}\vartheta_{d-b}v_{u}k_{b-u}$
$\displaystyle=\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}\vartheta_{d-b}\ell_{b}-\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-m\end{subarray}}\sum_{u<b}\vartheta_{d-b}v_{u}k_{b-u}$
$\displaystyle=\sum_{\begin{subarray}{c}u<d,\\\
|u|=m\end{subarray}}\vartheta_{u}\ell_{d-u}-\sum_{\begin{subarray}{c}w<d,\\\
|w|=m\end{subarray}}\sum_{u<d-w}\vartheta_{w}v_{u}k_{d-w-u}$
$\displaystyle=\sum_{\begin{subarray}{c}u<d,\\\
|u|=m\end{subarray}}\vartheta_{u}\ell_{d-u}-\sum_{\begin{subarray}{c}u<d,\\\
|u|\leq|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}w<d-u,\\\
|w|=m\end{subarray}}\vartheta_{w}v_{u}k_{d-w-u}.$ (5.21)

Combining (5.3) with (5.3) (which holds for $m$ by our inductive hypothesis)
yields

$\displaystyle\sum_{a<d}v_{a}k_{d-a}=$
$\displaystyle\sum_{\begin{subarray}{c}u<d,\\\ 0<|u|\leq
m-1\end{subarray}}\vartheta_{u}\ell_{d-u}+\sum_{\begin{subarray}{c}u<d,\\\
|u|=m\end{subarray}}\vartheta_{u}\ell_{d-u}-\sum_{\begin{subarray}{c}u<d,\\\
|u|\leq|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}w<d-u,\\\
|w|=m\end{subarray}}v_{u}\vartheta_{w}k_{d-w-u}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-1-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}0<|u|<m,\\\
u<d-b\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}u<d,\\\ 0<|u|\leq
m\end{subarray}}\vartheta_{u}\ell_{d-u}-\sum_{\begin{subarray}{c}u<d,\\\
|u|\leq|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}w<d-u,\\\
|w|=m\end{subarray}}v_{u}\vartheta_{w}k_{d-w-u}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-2-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}u<d-b,\\\
0<|u|<m+1\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}u<d-b,\\\
0<|u|<m\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-2-m\end{subarray}}\sum_{\begin{subarray}{c}u<d-b,\\\
|u|=m\end{subarray}}v_{b}\vartheta_{u}k_{d-b-u}.$ (5.22)

But since $|b|=|d|-1-m$ and $u<d-b$ imply $|u|<m+1,$ we have

$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}u<d-b,\\\
0<|u|<m\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\mathbf{0}<u<d-b}\vartheta_{u}k_{d-b-u}\bigg{)}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}u<d-b,\\\
|u|=m\end{subarray}}v_{b}\vartheta_{u}k_{d-b-u}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}v_{b}\vartheta_{d-b}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}u<d-b,\\\
|u|=m\end{subarray}}v_{b}\vartheta_{u}k_{d-b-u}.$

Thus, (5.3) becomes

$\displaystyle\sum_{a<d}v_{a}k_{d-a}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}u<d,\\\ 0<|u|\leq
m\end{subarray}}\vartheta_{u}\ell_{d-u}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}v_{b}\vartheta_{d-b}-\sum_{\begin{subarray}{c}u<d,\\\
|u|\leq|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}w<d-u,\\\
|w|=m\end{subarray}}v_{u}\vartheta_{w}k_{d-w-u}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-2-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}u<d-b,\\\
0<|u|<m+1\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}\sum_{\begin{subarray}{c}u<d-b,\\\
|u|=m\end{subarray}}v_{b}\vartheta_{u}k_{d-b-u}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-2-m\end{subarray}}\sum_{\begin{subarray}{c}u<d-b,\\\
|u|=m\end{subarray}}v_{b}\vartheta_{u}k_{d-b-u}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}u<d,\\\ 0<|u|\leq
m\end{subarray}}\vartheta_{u}\ell_{d-u}+\sum_{\begin{subarray}{c}b<d,\\\
|b|=|d|-1-m\end{subarray}}v_{b}\vartheta_{d-b}$
$\displaystyle+\sum_{\begin{subarray}{c}b<d,\\\
|b|\leq|d|-2-m\end{subarray}}v_{b}\bigg{(}k_{d-b}-\sum_{\begin{subarray}{c}u<d-b,\\\
0<|u|<m+1\end{subarray}}\vartheta_{u}k_{d-b-u}\bigg{)},$

concluding the proof of the inductive step. Thus, (5.3) holds for every $2\leq
m\leq|d|-1.$ Choosing $m=|d|-1$ then yields (as the 3rd term disappears)

$\displaystyle\sum_{a<d}v_{a}k_{d-a}$
$\displaystyle=\sum_{\mathbf{0}<u<d}\vartheta_{u}\ell_{d-u}+\sum_{|b|=0}v_{b}\vartheta_{d-b}$
$\displaystyle=\sum_{\mathbf{0}<u<d}\vartheta_{u}\ell_{d-u}+v_{\mathbf{0}}\vartheta_{d}$
$\displaystyle=\sum_{\mathbf{0}<u<d}\vartheta_{u}\ell_{d-u}+\vartheta_{d}$
$\displaystyle=\sum_{\mathbf{0}<u\leq d}\vartheta_{u}\ell_{d-u},$

and $\eqref{interr}$ is proved. We may now conclude

$\ell_{d}-\sum_{\begin{subarray}{c}u\leq d,\\\
u\neq\mathbf{0}\end{subarray}}\vartheta_{u}\ell_{d-u}=\ell_{d}-\sum_{a<d}v_{a}k_{d-a}=v_{d}\geq
0,$

so (5.13) holds. ∎

###  5.4. CP implies CC

The last ingredient that will be needed for our proof of Theorem 1.6
(contained in Section 6) is a direct passage from the CP to the CC property.
Establishing such a passage will be our main objective for this subsection.
The idea is to apply the complete Pick pair assumption to tuples of points
near $\mathbf{0}$ and then take a limit - letting these points tend to
$\mathbf{0}$.

We will require the following elementary Hilbert space lemma, the proof of
which is omitted.

######  Lemma 5.14.

Let $H$ denote a Hilbert space and assume $M\subset H$ is a non-trivial
finite-dimensional subspace. Further, suppose $\gamma:[0,\delta]\to H$ is a
continuous function such that $\gamma(t)$ does not lie in $M$, for any $t.$
Then, letting $P_{t}$ denote the projection onto
$\textup{span}\big{(}M\cup\\{\gamma(t)\\}\big{)}$, we have $P_{t}\to P_{0}$ in
operator norm as $t\to 0.$

We will work with a lexicographic order on $\mathbb{N}^{{\mathsf{g}}}$. Given
$a,b\in\mathbb{N}^{{\mathsf{g}}},$ the expression $a\prec b$ means that either
$|a|<|b|$ or $|a|=|b|$ and there exists $0\leq r\leq{\mathsf{g}}-1$ such that

$a_{1}=b_{1}$ $\vdots$ $a_{r}r=b_{r}$ $a_{r+1}<b_{r+1},$

where $r=0$ simply means $a_{1}<b_{1}.$ Clearly,
$(\mathbb{N}^{{\mathsf{g}}},\prec)$ is totally ordered. Set
$\mathbb{N}^{{\mathsf{g}}}=\\{a^{0},a^{1},\dots\\}$ accordingly.

Now, let $\ell=1+\sum_{a\in\mathbb{N}^{{\mathsf{g}}},\\\
|a|>0}\ell_{a}(z\overline{w})^{a}$ be a normalized diagonal holomorphic kernel
on some domain $\Omega\subseteq\mathbb{C}^{{\mathsf{g}}}$ containing
$\mathbf{0}$. Set $p_{a^{j}}(z)=z^{a^{j}},$ for all $j$. For $m\geq 0$ and
$\Lambda\subseteq\Omega\setminus\\{\mathbf{0}\\}$, let $M_{m,\Lambda}$ and
$M_{\Lambda}$ denote the spans of $\\{p_{a^{j}}:0\leq j\leq
m\\}\cup\\{\ell_{\lambda}:\lambda\in\Lambda\\}$ and
$\\{\ell_{\lambda}:\lambda\in\Lambda\\}$, respectively, with $P_{m,\Lambda}$
and $P_{\Lambda}$ the associated projections. We will also write
$M^{0}_{m,\Lambda}$ and $M^{0}_{\Lambda}$ for the spans of $\\{p_{a^{j}}:1\leq
j\leq m+1\\}\cup\\{\ell_{\lambda}-1:\lambda\in\Lambda\\}$ and
$\\{\ell_{\lambda}-1:\lambda\in\Lambda\\}$, respectively, with
$P^{0}_{m,\Lambda}$ and $P^{0}_{\Lambda}$ denoting the associated projections.
For what is to follow, it should be kept in mind that, since all coefficients
$\ell_{a}$ are non-zero, the polynomials are contained in
$\mathcal{H}_{\ell}$. Hence, there exist no (finite) collections of linearly
dependent kernel functions, and this continues to be the case even if we have
vanishing derivatives up to a certain order.

######  Lemma 5.15.

Fix $m\geq-1$. With notation as above, for any normalized diagonal holomorphic
kernel $\ell$ and any finite
$\Lambda\subseteq\Omega\setminus\\{\mathbf{0}\\},$ there exists a continuous
map $v:(0,\delta]\to\Omega$ such that, with
$\Lambda_{t}=\Lambda\cup\\{v(t)\\}$, we have

$P_{m,\Lambda_{t}}\to P_{m+1,\Lambda}\hskip 8.5359pt\text{ and }\hskip
8.5359ptP^{0}_{m,\Lambda_{t}}\to P^{0}_{m+1,\Lambda}$

in operator norm as $t\to 0$, where $P_{-1,\Lambda_{t}}\equiv P_{\Lambda_{t}}$
and $P^{0}_{-1,\Lambda_{t}}\equiv P^{0}_{\Lambda_{t}}$.

###### Proof.

We assume $m\geq 0$ (the case $m=-1$ can be treated analogously). Set
$b=a^{m+1},c=a^{m+2}$ and choose $r_{j}=|c|^{{\mathsf{g}}-j}+N$, where
$N>\sum(b_{j}+c_{j})|c|^{{\mathsf{g}}-j}$. Put
$r=(r_{1},\dots,r_{{\mathsf{g}}}).$ Observe that, for each $\rho>m+2$ and
$|a^{\rho}|=|c|,$ the inequality $c\prec a^{\rho}$ yields

$\sum c_{j}r_{j}=\sum c_{j}|c|^{{\mathsf{g}}-j}\,+\,N\,|c|<\sum
a^{\rho}_{j}|c|^{{\mathsf{g}}-j}\,+\,N\,|c|=\sum a^{\rho}_{j}r_{j}.$

Our choice of $N$ guarantees that the same inequality continues to hold when
$|c|<|a^{\rho}|.$ One can similarly show $\langle r,b\rangle<\langle
r,c\rangle$, and thus we obtain

$\langle r,b\rangle<\langle r,c\rangle<\langle r,a^{\rho}\rangle,$ (5.23)

for all $\rho>m+2.$

Now, fix $\delta>0$ sufficiently small and, for $0<t\leq\delta,$ set
$v=v(t)=(t^{r_{1}},\dots,t^{r_{{\mathsf{g}}}})\in\Omega$. Define
$\gamma:[0,\delta]\to\mathcal{H}_{\ell}$ and
$\gamma^{0}:[0,\delta]\to\mathcal{H}_{\ell}$ as

$\gamma(t)=\begin{cases}\cfrac{\ell_{v}-\sum_{j=0}^{m}\ell_{a^{j}}p_{a^{j}}v^{a^{j}}}{\sqrt{\ell(v,v)-\sum_{j=0}^{m}\ell_{a^{j}}v^{2a^{j}}}},\hskip
17.07182pt0<t\leq\delta,\\\ \sqrt{\ell_{a^{m+1}}}p_{a^{m+1}},\hskip
65.44142ptt=0,\end{cases}$

and

$\gamma^{0}(t)=\begin{cases}\cfrac{\ell_{v}-\sum_{j=0}^{m+1}\ell_{a^{j}}p_{a^{j}}v^{a^{j}}}{\sqrt{\ell(v,v)-\sum_{j=0}^{m+1}\ell_{a^{j}}v^{2a^{j}}}},\hskip
17.07182pt0<t\leq\delta,\\\ \sqrt{\ell_{a^{m+2}}}p_{a^{m+2}},\hskip
65.44142ptt=0.\end{cases}$

Clearly, both $\gamma$ and $\gamma^{0}$ are continuous in $(0,\delta]$. We
will show that they are continuous at $0$ as well. Since the proofs are
essentially identical, we will only examine $\gamma$. For $0<t\leq\delta,$ we
have

$\displaystyle\gamma(t)=\cfrac{\sum_{j=m+1}^{\infty}\ell_{a^{j}}p_{a^{j}}v^{a^{j}}}{\sqrt{\sum_{j=m+1}^{\infty}\ell_{a^{j}}v^{2a^{j}}}}=\cfrac{\cfrac{\sum_{j=m+1}^{\infty}\ell_{a^{j}}p_{a^{j}}v^{a^{j}}}{v^{a^{m+1}}}}{\cfrac{\sqrt{\sum_{j=m+1}^{\infty}\ell_{a^{j}}v^{2a^{j}}}}{v^{a^{m+1}}}}.$
(5.24)

Now, (5.23) tells us that

$\frac{v^{a^{j}}}{v^{a^{m+1}}}=\frac{t^{\langle r,a^{j}\rangle}}{t^{\langle
r,a^{m+1}\rangle}}=t^{\langle r,a^{j}\rangle-\langle r,a^{m+1}\rangle}$

has a positive exponent, for every $j>m+1.$ Thus, by elementary power series
arguments, we have

$\cfrac{\sqrt{\sum_{j=m+1}^{\infty}\ell_{a^{j}}v^{2a^{j}}}}{v^{a^{m+1}}}\longrightarrow\sqrt{\ell_{a^{m+1}}}\hskip
8.5359pt\text{ as }t\to 0.$ (5.25)

Further, we have

$\displaystyle\Bigg{|}\Bigg{|}\frac{\sum\limits_{j\geq