# Structured Kalman Filter for Time Scale Generation in Atomic Clock Ensembles

Yuyue Yan  Member, IEEE  Takahiro Kawaguchi  Member, IEEE  Yuichiro Yano
Yuko Hanado  and Takayuki Ishizaki  Member, IEEE This work is supported by the
Ministry of Internal Affairs and Communications (MIC) under its ”Research and
Development for Expansion of Radio Resources (JPJ000254)” program. Yuyue Yan
and Takayuki Ishizaki are with the Department of Systems and Control
Engineering, Tokyo Institute of Technology, Meguro, Tokyo 152-8552 Japan
(e-mail<EMAIL_ADDRESS>ishizaki@sc.e.titech.ac.jp). Takahiro
Kawaguchi is with the Division of Electronics and Informatics, Gunma
University, Kiryu, Gunma 371-8510 Japan. Yuichiro Yano and Yuko Hanado are
with the National Institute of Information and Communications Technology,
Koganei, Tokyo 184-0015 Japan.

###### Abstract

In this article, we present a structured Kalman filter associated with the
transformation matrix for observable Kalman canonical decomposition from
conventional Kalman filter (CKF) in order to generate a more accurate time
scale. The conventional Kalman filter is a special case of the proposed
structured Kalman filter which yields the same predicted unobservable or
observable states when some conditions are satisfied. We consider an
optimization problem respective to the transformation matrix where the
objective function is associated with not only the expected value of
prediction error but also its variance. We reveal that such an objective
function is a convex function and show some conditions under which CKF is
nothing but the optimal algorithm if ideal computation is possible without
computation error. A numerical example is presented to show the robustness of
the proposed method in terms of the initial error covariance.

Atomic clocks, state-space model, prediction, confidence interval, Kalman
filter, time scale.

An atomic clock ensemble is a collection of highly accurate atomic clocks that
work together to achieve a precise and stable timekeeping system. Atomic
clocks are devices that measure time based on the vibrations of atoms with
constant resonance frequencies. Using the measurements from several atomic
clocks within an ensemble, the national metrology institutes (NMIs) all over
the world can reduce these variations and create a more reliable and robust
timekeeping system [1, 2, 3]. The advancements in atomic clock technology and
the development of accurate time scales based on clock ensembles offer
numerous benefits and applications that are crucial for the future smart
society, such as the satellite navigation [4], financial networks with high-
frequency trading and time-sensitive transactions [5], telecommunications [6].

The variations in tick rates of atomic clocks are referred to as time
deviations from the ideal clock behavior, which can be modeled as stochastic
processes. It has been revealed experimentally that the time deviations of the
atomic clocks can be characterized as random noises in a stochastic linear
differential equation [7]. Based on this, in the task of time generation, how
to properly deal with the prediction problem for time deviations is the main
issue to guarantee excellent performance of the generated time [1]. In order
to improve the accuracy of atomic time and thus to benefit modern technology
and fundamental physics, researchers in the time and frequency community have
attempted to use Kalman filters for predicting those noises [8, 1, 9, 10, 11].

The Kalman filter [12, 13, 14] is an optimal prediction algorithm minimizing
an objective function associated solely with the mean-squared prediction error
for detectable systems since in such a case the covariances of prediction
errors are supposed to be bounded and converging to a steady-state value [15,
16]. However, for an undetectable system, the covariance is growing
unboundedly and thus both the expected value and covariance should be taken
into account evaluating the performance of prediction, which implies that the
conventional Kalman filter (CKF) may not be optimal. In fact, as shown in
[17], the system model of the atomic clock ensembles is undetectable due to
its intrinsic property that the phase difference is only measurable with fine
resolution. Even though some works have been done in terms of solving
numerical instability [9, 18]111Note that the method used in [18] is the same
as that presented by Greenhall in [9]. due to such undetectability, it is
worth asking whether there is space to improve the performance of the time
scale from the conventional Kalman filter or not.

In this paper, different from the existing methods, we propose a structured
Kalman filter associated with the transformation matrix for observable Kalman
canonical decomposition of the atomic clock model. The conditions where the
proposed structured Kalman filter is reduced to the CKF in the unobservable or
observable state space are presented. In such a case, the proposed method is
understood as an alternative method of CKF algorithms avoiding numerical
instability. By defining an objective function with respect to the
transformation matrix according to the expected value and variance of _atomic
time_ (i.e., prediction error in ensemble time deviation), an optimization
problem is formulated to construct the optimal transformation matrix. We
reveal the fact that such an objective function is convex, and show some
conditions under which the CKF is nothing but the optimal algorithm minimizing
not only the square of expected value but also the variance of atomic time if
ideal computation is possible without computation error.

Notation We write $\mathbb{R}$ for the set of real numbers, $\mathbb{R}_{+}$
for the set of positive real numbers, $\mathbb{R}^{n}$ for the set of _n_
$\times 1$ real column vectors, and $\mathbb{R}$n×m for the set of _n_ $\times
m$ real matrices. Moreover, $\otimes$ denotes the Kronecker product,
$(\cdot)^{\sf T}$ denotes transpose, $(\cdot)^{\dagger}$ denotes the Moore-
Penrose inverse, and $\mathsf{diag}(\cdot)$ denotes a diagonal matrix.
Finally, $\mathbb{E}[x]$ denotes the expected value of a random variable $x$,
whereas $\mathds{1}_{n}$ and $I_{n}$ denote the ones (column) vector and the
identity matrix of dimension $n$, respectively.

##  1 Preliminaries and Problem

Consider the $n$th order linear stochastic discrete-time system for an
$m$-clock homogeneous ensemble given by

$\Sigma:\left\\{\;\begin{aligned}
\boldsymbol{x}[k+1]&=\boldsymbol{F}\boldsymbol{x}[k]+\boldsymbol{v}[k]\\\
\boldsymbol{y}[k]&=\boldsymbol{H}\boldsymbol{x}[k]+\boldsymbol{w}[k]\\\
\boldsymbol{z}[k]&=\boldsymbol{D}\boldsymbol{x}[k]\end{aligned}\right.$ (1)

where $\boldsymbol{x}[k]:=(\boldsymbol{x}_{1}^{\sf
T}[k],\ldots,\boldsymbol{x}_{n}^{\sf T}[k])^{\sf T}\in\mathbb{R}^{nm}$,
$\boldsymbol{y}[k]\in\mathbb{R}^{m-1}$, are the state and measurement,
respectively, with
$\boldsymbol{x}_{i}[k]:=(x^{1}_{i}[k],\ldots,x^{m}_{i}[k])^{\sf
T}\in\mathbb{R}^{m}$; $\boldsymbol{F}:=A\otimes I_{m}\in\mathbb{R}^{nm\times
nm}$, $\boldsymbol{H}:=C\otimes\overline{V}\in\mathbb{R}^{(m-1)\times nm}$ are
the system state transition matrix and observation matrix defined as

$\displaystyle A:=$ $\displaystyle
A_{\tau}:=\left[\;\begin{matrix}1&\tau&\frac{\tau^{2}}{2}&\cdots&\tfrac{\tau^{n-1}}{(n-1)!}\\\
0&1&\tau&\cdots&\frac{\tau^{n-2}}{(n-2)!}\\\ \vdots&&\ddots&\ddots&\vdots\\\
\vdots&&&1&\tau\\\ 0&0&\cdots&\cdots&1\end{matrix}\>\right]$ (2)
$\displaystyle C:=$
$\displaystyle\left[\;\begin{matrix}1&0&\cdots&0\end{matrix}\>\right]\in\mathbb{R}^{1\times
n}$ (3) $\displaystyle\overline{V}:=$
$\displaystyle\left[\;\begin{matrix}I_{m-1}&-\mathds{1}_{m-1}\end{matrix}\>\right]\in\mathbb{R}^{(m-1)\times
m}$ (4)

for the sampling interval $\tau$; $\boldsymbol{v}[k]\in\mathbb{R}^{nm}$,
$\boldsymbol{w}[k]\in\mathbb{R}^{m-1}$ are the system noise and observation
noise, and they are white Gaussian with the covariances [1] being

$W:=\int_{0}^{\tau}A_{t}\mathsf{diag}(q_{1}^{2},\ldots,q_{n}^{2})A_{t}^{\sf
T}dt\otimes I_{m},\quad R:=r^{2}I_{m-1}$

respectively. In such a homogeneous ensemble, each atomic clock works as an
independent oscillator with the number of waves counted as the clock reading
of the clocks, where there exists a slight difference (referred to as the time
deviation) between the actual clock reading and the ideal clock reading.

Note that in the system model (1), the vector $\boldsymbol{x}_{1}[k]$
represents the time deviations of the clocks [19] and hence the scalar
$\boldsymbol{z}[k]\in\mathbb{R}$ denotes the ensemble time deviation defined
as a weighted value from the individual time deviations with

$\displaystyle\boldsymbol{D}:=\tfrac{1}{m}C\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}})$ (5)

where the weights are considered to be all equal. Furthermore, $Q$ is the
covariance of the individual system noise where $A_{t}$ in the integrant is
understood as $A_{\tau}$ with $\tau$ replaced by $t$. In terms of the physical
meanings, the measurement $\boldsymbol{y}[k]$ is the difference of the time
deviations (or, equivalently, the difference of the clock reading) measured
independently between the clocks where clock $m$ plays the role as a reference
clock (see the detailed model in [1, 19]).

Even though the detectability condition is not satisfied in the system model
[17], CKF algorithms are still used in many practice scenarios of time scale
generation of clock ensembles [9, 1, 10]. The CKF algorithm is

$\displaystyle\boldsymbol{L}_{k}=-\boldsymbol{F}\boldsymbol{P}_{k}{{\boldsymbol{H}}^{\sf
T}\bigr{(}\boldsymbol{H}\boldsymbol{P}_{k}{\boldsymbol{H}}^{\sf
T}+R\bigr{)}^{-1}}$ (6)
$\displaystyle\boldsymbol{P}_{k+1}=(\boldsymbol{F}+\boldsymbol{L}_{k}{\boldsymbol{H}})\boldsymbol{P}_{k}\boldsymbol{F}^{\sf
T}+W$ (7)
$\displaystyle\hat{\boldsymbol{x}}[k+1]=\boldsymbol{F}\hat{\boldsymbol{x}}[k]-\boldsymbol{L}_{k}(\boldsymbol{y}[k]-\boldsymbol{H}\hat{\boldsymbol{x}}[k])$
(8)

for $k\geq 0$ with the initial $\boldsymbol{P}_{0}=pI_{nm}$ for some constant
$p\in\mathbb{R}_{+}$, where $\boldsymbol{L}_{k}$ and $\boldsymbol{P}_{k}$ are
the Kalman (observer) gain and error covariance. Together with the linear-
quadratic regulator, the Kalman filter solves the linear–quadratic–Gaussian
control problem minimizing the mean squared error
$\mathbb{E}\big{[}\|\boldsymbol{x}[k]-\hat{\boldsymbol{x}}[k]\|^{2}\big{]}$.
Using the Kalman filter, the ensemble time deviation ${\boldsymbol{z}}[k]$ is
predicted as $\hat{\boldsymbol{z}}[k]:=\boldsymbol{D}\hat{\boldsymbol{x}}[k]$
and hence generated clock reading of the $m$-clock ensemble is given as

$\displaystyle\hat{h}_{0}[k]=\tfrac{1}{m}\sum\nolimits_{i=1}^{m}h^{j}[k]-\hat{\boldsymbol{z}}[k]\in\mathbb{R}$
(9)

where $h^{j}[k]$ is the actual clock reading of clock $j$ and hence the ideal
clock reading can be written as $h_{0}[k]=\hat{h}_{0}[k]\in\mathbb{R}$ with
$\hat{\boldsymbol{z}}[k]$ replaced by ${\boldsymbol{z}}[k]$. The accuracy of
the generated clock reading is evaluated by _atomic time_ (which is referred
to as _Temps Atomique_ (TA) in Europe [1]) as

${\rm
TA}[k]:=h_{0}[k]-\hat{h}_{0}[k]=\boldsymbol{z}[k]-\hat{\boldsymbol{z}}[k]=\boldsymbol{D}(\boldsymbol{x}[k]-\hat{\boldsymbol{x}}[k])$
(10)

which is depending on how we yield the predicted state
$\hat{\boldsymbol{x}}[k]$.

_Problem_ : Consider an $m$-clock ensemble to generate an accurate time scale
for a given initial guess $\hat{\boldsymbol{x}}[0]$. A fundamental question is
whether there is a space to further improve the performance of the time scale
from the CKF algorithms, i.e., further decrease the atomic time ${\rm TA}[k]$
as much as possible.

##  2 Proposed Structured Kalman Filter

This section introduces a structured Kalman filter motivated by observable
Kalman canonical decomposition for the atomic clock model. Specifically,
defining a transformation matrix $\varGamma\in\mathbb{R}^{n\times n(m-1)}$, we
consider

$\boldsymbol{x}=\underbrace{\left[\;\begin{matrix}I_{n}\otimes\overline{V}^{\dagger}&I_{n}\otimes\mathds{1}_{m}\end{matrix}\>\right]\left[\;\begin{matrix}I_{n(m-1)}&0\\\
\varGamma&I_{n}\end{matrix}\>\right]}_{\mathbb{T}}\underbrace{\left[\;\begin{matrix}\boldsymbol{\xi}_{\rm
o}\\\ \boldsymbol{\xi}_{\rm\bar{o}}\end{matrix}\>\right]}_{\boldsymbol{\xi}}$
(11)

to make decomposition of the system $\Sigma$ into the observable state
$\boldsymbol{\xi}_{\rm o}$ and unobservable state
$\boldsymbol{\xi}_{\rm\bar{o}}$ for constructing a Kalman filter and a
predictor, respectively. With such a structure of $\mathbb{T}$, the basis
selection of observable states is limited without $\varGamma$ since all
possible basis selections can be represented by introducing $\varGamma$. Our
main idea of including the matrix $\varGamma$ in the linear transformation
(11) is to keep the variability of the basis selections so that one can
discuss an optimization problem introduced later for improving the performance
of the generated atomic time ${\rm TA}[k]$. Then, we have

$\tilde{\Sigma}:\left\\{\;\begin{aligned}
\boldsymbol{\xi}[k+1]&=\left[\;\begin{matrix}\boldsymbol{F}_{\rm oo}&0\\\
\boldsymbol{F}_{\rm\bar{o}o}&\boldsymbol{F}_{\rm\bar{o}\bar{o}}\end{matrix}\>\right]\boldsymbol{\xi}[k]+\mathbb{T}^{-1}\boldsymbol{v}[k]\\\
\boldsymbol{y}[k]&=\bigl{[}\boldsymbol{H}_{\rm o}\ \ \
0\bigl{]}\boldsymbol{\xi}[k]+\boldsymbol{w}[k]\\\
\boldsymbol{z}[k]&=\boldsymbol{D}\mathbb{T}\boldsymbol{\xi}[k]\end{aligned}\right.$
(12)

where $\boldsymbol{H}_{\rm o}:=C\otimes I_{m-1}$,

$\left\\{\;\begin{aligned} \boldsymbol{F}_{\rm oo}&:=A\otimes I_{m-1},\\\
\boldsymbol{F}_{\rm\bar{o}\bar{o}}&:=A,\\\
\boldsymbol{F}_{\rm\bar{o}o}&:=-\varGamma(A\otimes
I_{m-1})+A\varGamma,\end{aligned}\right.$ (13)

and the covariance of the state noise affecting the observable state
$\boldsymbol{\xi}_{\rm o}$ is given by $W_{\rm
o}:=(I_{n}\otimes\overline{V})W(I_{n}\otimes\overline{V})^{\sf
T}=Q\otimes\overline{V}\overline{V}^{\sf T}.$

Now we introduce our structured Kalman filter as follows.

###  2.0.1 Initialization

A guess of the initial state $\hat{\boldsymbol{x}}[0]$ is transformed into
$\hat{\boldsymbol{\xi}}[0]=T^{-1}\hat{\boldsymbol{x}}[0]$.

###  2.0.2 Update for observable states

Construct

$\displaystyle\hat{\boldsymbol{L}}_{k}=-\boldsymbol{F}_{\rm
oo}\hat{\boldsymbol{P}}_{k}{\boldsymbol{H}_{\rm o}^{\sf
T}\bigr{(}\boldsymbol{H}_{\rm o}\hat{\boldsymbol{P}}_{k}\boldsymbol{H}_{\rm
o}^{\sf T}+R\bigr{)}^{-1}}$ (14)
$\displaystyle\hat{\boldsymbol{P}}_{k+1}=(\boldsymbol{F}_{\rm
oo}+\hat{\boldsymbol{L}}_{k}\boldsymbol{H}_{\rm
o})\hat{\boldsymbol{P}}_{k}\boldsymbol{F}_{\rm oo}^{\sf T}+W_{\rm o}$ (15)
$\displaystyle\hat{\boldsymbol{\xi}}_{\rm o}[k+1]=\boldsymbol{F}_{\rm
oo}\hat{\boldsymbol{\xi}}_{\rm
o}[k]-\hat{\boldsymbol{L}}_{k}(\boldsymbol{y}[k]-\boldsymbol{H}_{\rm
o}\hat{\boldsymbol{\xi}}_{\rm o}[k])$ (16)

for $k\geq 0$ with some initial condition $\hat{\boldsymbol{P}}_{0}$, where
$\hat{\boldsymbol{L}}_{k}$ and $\hat{\boldsymbol{P}}_{k}$ are the Kalman gain
and error covariance for observable state.

###  2.0.3 Prediction for unobservable states

Construct

$\displaystyle\hat{\boldsymbol{\xi}}_{\rm\bar{o}}[k+1]=\boldsymbol{F}_{\rm\bar{o}o}\hat{\boldsymbol{\xi}}_{\rm
o}[k]+\boldsymbol{F}_{\rm\bar{o}\bar{o}}\hat{\boldsymbol{\xi}}_{\rm\bar{o}}[k].$
(17)

###  2.0.4 Update in original coordinate

Update the predicted state according to the linear transformation (11), i.e.,

$\hat{\boldsymbol{x}}[k]=\mathbb{T}\hat{\boldsymbol{\xi}}[k].$ (18)

##  3 Main Results

Before we present the main results for the proposed structured Kalman filter,
we show a lemma for CKF algorithms.

###### Lemma 1

If the initial error covariance $\boldsymbol{P}_{0}$ satisfies
$\boldsymbol{P}_{0}(I_{n}\otimes\overline{V})^{\dagger}=(I_{n}\otimes\overline{V})^{\dagger}\check{P}_{0}$
for some $\check{P}_{0}$, then the error covariance of CKF satisfies
$\boldsymbol{P}_{k}(I_{n}\otimes\overline{V})^{\dagger}=(I_{n}\otimes\overline{V})^{\dagger}\check{\boldsymbol{P}}_{k}$
with

$\displaystyle\check{\boldsymbol{P}}_{k+1}=$
$\displaystyle(\boldsymbol{F}_{\rm
oo}+\check{\boldsymbol{G}}_{k}\boldsymbol{H}_{\rm
o})\check{\boldsymbol{P}}_{k}\boldsymbol{F}_{\rm oo}^{\sf T}+Q\otimes I_{m-1}$
(19) $\displaystyle\check{\boldsymbol{G}}_{k}=$
$\displaystyle-\boldsymbol{F}_{\rm
oo}\check{\boldsymbol{P}}_{k}\boldsymbol{H}_{\rm o}^{\sf
T}\bigr{(}\boldsymbol{H}_{\rm o}\check{\boldsymbol{P}}_{k}\boldsymbol{H}_{\rm
o}^{\sf T}+r(\overline{V}\overline{V}^{\sf T})^{-1}\bigr{)}^{-1}$ (20)

for $\check{\boldsymbol{P}}_{0}=pI_{nm-n}$ and any $k=0,1,2,\ldots$.

######  Proof 3.1.

The proof is shown in [20].

######  Theorem 3.2.

Consider an $m$-clock ensemble to generate an accurate time scale for a given
initial guess $\hat{\boldsymbol{x}}[0]$. If $\varGamma=0$ (resp.,
$\hat{\boldsymbol{P}}_{0}=(I_{n}\otimes\overline{V})\boldsymbol{P}_{0}(I_{n}\otimes\overline{V})^{{\sf
T}}$), then the structured Kalman filter (14)–(18) ideally yield the same
predicted unobservable state $\hat{\boldsymbol{\xi}}_{\rm\bar{o}}$ (resp.,
observable state $\hat{\boldsymbol{\xi}}_{\rm o}$) as CKF. In particular, if
$\varGamma=0$ and
$\hat{\boldsymbol{P}}_{0}=(I_{n}\otimes\overline{V})\boldsymbol{P}_{0}(I_{n}\otimes\overline{V})^{{\sf
T}}$, then the structured Kalman filter ideally reduces the same as CKF
algorithms yielding the same predicted state $\hat{\boldsymbol{x}}[k]$ for all
$k\in\\{0,1,2,\ldots\\}$.

######  Proof 3.3.

First, we prove equivalency of the unobservable states
$\hat{\boldsymbol{\xi}}_{\rm\bar{o}}[k]$ in the two methods. Note from
$\mathds{1}_{m}^{\sf T}\overline{V}^{\dagger}=0$ and Lemma 1 that

$\displaystyle 0=$ $\displaystyle(A\\!\otimes\\!\mathds{1}_{m}^{{\sf
T}})(I_{n}\\!\otimes\\!\overline{V})^{\dagger}\check{\boldsymbol{P}}_{k}\boldsymbol{H}_{\rm
o}(I_{n}\\!\otimes\\!\overline{V}\overline{V}^{\sf
T}\\!)\bigr{(}\boldsymbol{H}\boldsymbol{P}_{k}{\boldsymbol{H}}^{\sf
T}\\!\\!+\\!R\bigr{)}^{-1}$ $\displaystyle=$
$\displaystyle(A\\!\otimes\\!\mathds{1}_{m}^{{\sf
T}})\boldsymbol{P}_{k}(I_{n}\\!\otimes\\!\overline{V})^{\dagger}\boldsymbol{H}_{\rm
o}(I_{n}\\!\otimes\\!\overline{V}\overline{V}^{\sf
T}\\!)\bigr{(}\boldsymbol{H}\boldsymbol{P}_{k}{\boldsymbol{H}}^{\sf
T}\\!\\!+\\!R\bigr{)}^{-1}$ $\displaystyle=$
$\displaystyle(I_{n}\otimes\mathds{1}_{m}^{{\sf
T}})\boldsymbol{F}\boldsymbol{P}_{k}{\boldsymbol{H}}^{\sf
T}\bigr{(}\boldsymbol{H}\boldsymbol{P}_{k}{\boldsymbol{H}}^{\sf
T}+R\bigr{)}^{-1}$ $\displaystyle=$
$\displaystyle-(I_{n}\otimes\mathds{1}_{m}^{{\sf T}})\boldsymbol{L}_{k},\quad
k=0,1,2,\ldots$ (21)

Then, since the prediction error
$\boldsymbol{\epsilon}:=\boldsymbol{x}-\hat{\boldsymbol{x}}$ of CKF is

$\boldsymbol{\epsilon}[k+1]=(\boldsymbol{F}+\boldsymbol{L}_{k}\boldsymbol{H})\boldsymbol{\epsilon}[k]+\boldsymbol{L}_{k}\boldsymbol{w}[k]+\boldsymbol{v}[k]$

it follows that the prediction error
$\boldsymbol{\epsilon}_{\rm\bar{o}}:={\boldsymbol{\xi}}_{\rm\bar{o}}-\hat{\boldsymbol{\xi}}_{\rm\bar{o}}=\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}})\boldsymbol{\epsilon}$ of unobservable state in CKF is given by

$\displaystyle\boldsymbol{\epsilon}_{\rm\bar{o}}[k+1]=$ $\displaystyle
A\boldsymbol{\epsilon}_{\rm\bar{o}}[k]+\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\boldsymbol{v}[k]$ $\displaystyle+(I_{n}\otimes\mathds{1}_{m}^{{\sf
T}})\boldsymbol{L}_{k}(\boldsymbol{H}\boldsymbol{\epsilon}[k]+\boldsymbol{w}[k])$
$\displaystyle=$ $\displaystyle
A\boldsymbol{\epsilon}_{\rm\bar{o}}[k]+\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\boldsymbol{v}[k].$ (22)

For the structured Kalman filter, the predicted state $\hat{\boldsymbol{x}}$
follows

$\displaystyle\hat{\boldsymbol{x}}[k+1]=$
$\displaystyle\mathbb{T}\left(\left[\;\begin{matrix}\boldsymbol{F}_{\rm
oo}+\hat{\boldsymbol{L}}_{k}\boldsymbol{H}_{\rm o}\\!\\!&0\\\
\boldsymbol{F}_{\rm\bar{o}o}\\!\\!&\boldsymbol{F}_{\rm\bar{o}\bar{o}}\end{matrix}\>\right]\right)\mathbb{T}^{-1}\hat{\boldsymbol{x}}[k]-\boldsymbol{G}_{k}\boldsymbol{y}[k]$
$\displaystyle=$
$\displaystyle\Big{(}\boldsymbol{F}+\mathbb{T}\left[\;\begin{matrix}\hat{\boldsymbol{L}}_{k}\boldsymbol{H}_{\rm
o}&0\\\
0&0\end{matrix}\>\right]\mathbb{T}^{-1}\Big{)}\hat{\boldsymbol{x}}[k]-\boldsymbol{G}_{k}\boldsymbol{y}[k]$
$\displaystyle=$
$\displaystyle(\boldsymbol{F}+\boldsymbol{G}_{k}\boldsymbol{H})\hat{\boldsymbol{x}}[k]\\!-\\!\boldsymbol{G}_{k}(\boldsymbol{H}{\boldsymbol{x}}[k]+\boldsymbol{w}[k])$

where the matrix $\boldsymbol{G}_{k}$ is defined as

$\boldsymbol{G}_{k}:=\mathbb{T}\left[\;\begin{matrix}\hat{\boldsymbol{L}}_{k}\\\
0\end{matrix}\>\right]=\Big{[}I_{n}\otimes\overline{V}^{\dagger}+(I_{n}\otimes\mathds{1}_{m})\varGamma\Big{]}\hat{\boldsymbol{L}}_{k}.$

Then it follows from the prediction error given by

$\displaystyle{\boldsymbol{\epsilon}}[k+1]=$
$\displaystyle(\boldsymbol{F}+\boldsymbol{G}_{k}\boldsymbol{H}){\boldsymbol{\epsilon}}[k]+\boldsymbol{G}_{k}\boldsymbol{w}[k]+\boldsymbol{v}[k]$
(23)

that the prediction error $\boldsymbol{\epsilon}_{\rm\bar{o}}$ of unobservable
state in structured Kalman filter is given by

$\displaystyle\boldsymbol{\epsilon}_{\rm\bar{o}}[k+1]=$ $\displaystyle
A\boldsymbol{\epsilon}_{\rm\bar{o}}[k]+\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\boldsymbol{v}[k]$
$\displaystyle+m\varGamma\hat{\boldsymbol{L}}_{k}(\boldsymbol{H}\boldsymbol{\epsilon}[k]+\boldsymbol{w}[k])$
(24)

which reduces the same as (22) for CKF when $\varGamma=0$.

Next we prove equivalency of observable states $\hat{\boldsymbol{\xi}}_{\rm
o}[k]$ in the two methods. Note that error covariance of the observable state
${\boldsymbol{\xi}}_{\rm o}[k]$ of CKF is

$\displaystyle\boldsymbol{P}_{\rm o}[k+1]=$
$\displaystyle(I_{n}\otimes\overline{V})\boldsymbol{P}_{k+1}(I_{n}\otimes\overline{V})^{\sf
T}=\check{\boldsymbol{P}}_{k+1}(I_{n}\\!\otimes\\!\overline{V}\overline{V}^{\sf
T})$ $\displaystyle=$ $\displaystyle(\boldsymbol{F}_{\rm
oo}+\check{\boldsymbol{G}}_{k}\boldsymbol{H}_{\rm
o})\check{\boldsymbol{P}}_{k}\boldsymbol{F}_{\rm oo}^{\sf
T}(I_{n}\otimes\overline{V}\overline{V}^{\sf T})+W_{\rm o}$ $\displaystyle=$
$\displaystyle(\boldsymbol{F}_{\rm
oo}+\check{\boldsymbol{G}}_{k}\boldsymbol{H}_{\rm
o})\check{\boldsymbol{P}}_{k}(I_{n}\otimes\overline{V}\overline{V}^{\sf
T})\boldsymbol{F}_{\rm oo}^{\sf T}+W_{\rm o}$ $\displaystyle=$
$\displaystyle(\boldsymbol{F}_{\rm
oo}+\check{\boldsymbol{G}}_{k}\boldsymbol{H}_{\rm o}){\boldsymbol{P}}_{\rm
o}[k]\boldsymbol{F}_{\rm oo}^{\sf T}+W_{\rm o}$ (25)
$\displaystyle\check{\boldsymbol{G}}_{k}=$ $\displaystyle-\boldsymbol{F}_{\rm
oo}\boldsymbol{P}_{\rm o}[k]\boldsymbol{H}_{\rm o}^{\sf
T}(\overline{V}\overline{V}^{\sf T})^{-1}$
$\displaystyle\cdot\bigr{(}\boldsymbol{H}_{\rm o}\boldsymbol{P}_{\rm
o}[k]\boldsymbol{H}_{\rm o}^{\sf T}(\overline{V}\overline{V}^{\sf
T})^{-1}+r(\overline{V}\overline{V}^{\sf T})^{-1}\bigr{)}^{-1}$
$\displaystyle=$ $\displaystyle-\boldsymbol{F}_{\rm oo}\boldsymbol{P}_{\rm
o}[k]\boldsymbol{H}_{\rm o}^{\sf T}\bigr{(}\boldsymbol{H}_{\rm
o}\boldsymbol{P}_{\rm o}[k]\boldsymbol{H}_{\rm o}^{\sf
T}\\!+\\!R\bigr{)}^{-1}$ (26)

which indicate along with condition
$\hat{\boldsymbol{P}}_{0}=(I_{n}\otimes\overline{V})\boldsymbol{P}_{0}(I_{n}\otimes\overline{V})^{{\sf
T}}$ that $\boldsymbol{P}_{\rm o}[k+1]=\hat{\boldsymbol{P}}_{k+1}$ for the
error covariance $\hat{\boldsymbol{P}}_{k+1}$ defined in (15) and
$\check{\boldsymbol{G}}_{k}=\hat{\boldsymbol{L}}_{k}$ for the Kalam gain
$\hat{\boldsymbol{K}}_{k}$ defined in (14). Thus, the prediction error
$\boldsymbol{\epsilon}_{\rm o}:={\boldsymbol{\xi}}_{\rm
o}-\hat{\boldsymbol{\xi}}_{\rm
o}=\bigl{(}I_{n}\otimes\overline{V})\boldsymbol{\epsilon}$ of observable state
under CKF is the same as structured Kalman filter, which completes the proof.

######  Remark 3.4.

It follows from the definition of atomic time that ${\rm
TA}[k]=\frac{1}{m}C\boldsymbol{\epsilon}_{\rm\bar{o}}[k]$. Thus, Theorem 3.2
implies that ${\rm TA}[k]$ under CKF and structured Kalman filter are the same
when the condition $\varGamma=0$ is satisfied, and they are ideally
independent from the measurement noises
$\boldsymbol{w}[0],\ldots,\boldsymbol{w}[k]$ (see the expression of
$\boldsymbol{\epsilon}_{\rm\bar{o}}$ in (22)). In addition, it is important to
note that both of error covariance (25) along with (26) and error covariance
(15) along with (14) are converging to the same steady state. Therefore, even
for the case of
$\hat{\boldsymbol{P}}_{0}\not=(I_{n}\otimes\overline{V})\boldsymbol{P}_{0}(I_{n}\otimes\overline{V})^{{\sf
T}}$, the updated observable states $\hat{\boldsymbol{\xi}}_{\rm o}$ of those
two methods are eventually the same for $\varGamma$ if there is no computation
error. In other words, the essential difference between the CKF and our
structured Kalman filter algorithms comes from the third term of the the
unobservable prediction error $\boldsymbol{\epsilon}_{\rm\bar{o}}$ in (24).

######  Remark 3.5.

The CKF algorithms may result in numerical instability problems in practical
implementation for the undetectable system of the atomic clock ensemble. This
is because the entries of the error covariance matrix and the computational
errors of the CKF algorithms grow unboundedly [9, 18]. However, Theorem 3.2
indicates that the proposed structured Kalman filter can be used as an
alternative method of CKF algorithms avoiding numerical instability.

Recalling from (13) and (17) that the proposed method is associated with the
transformation matrix $\varGamma$, which affects the predictions of
unobservable states within the transformed system and hence influences the
performance of the atomic time. Therefore, although the proposed structured
Kalman filter with $\varGamma=0$ is an alternative to the CKF algorithms, it
may be possible to further improve the time scale by addressing an
optimization problem with respect to the transformation matrix. In such a
case, the conditions for determining the optimal $\varGamma$ are derived based
on the prediction error and the error covariance resulting from the
application of Kalman filter to the transformed system.

######  Theorem 3.6.

Consider an $m$-clock ensemble with the structured Kalman filter algorithm
(14)–(18) and the measurement $\\{\boldsymbol{y}[k]:k=0,1,\ldots,T\\}$. It
follows that the cost function

$J(\varGamma):=\sum\nolimits_{k=0}^{T}\Big{(}\delta_{1}(\mathbb{E}[{\rm
TA}[k]])^{2}+\delta_{2}\mathbb{V}[{{\rm TA}}[k]]\Big{)}$ (27)

is a convex function with respect to $\varGamma$ for any
$\delta_{1},\delta_{2}\geq 0$, $\delta_{1}+\delta_{2}\not=0$, where
$\mathbb{V}[{{\rm TA}}[k]]$ represents the variance of atomic time ${\rm
TA}[k]$ for a given $\varGamma$. Furthermore, if the initial predicted state
$\boldsymbol{\epsilon}[0]\sim\mathcal{N}(\mu_{0},Q_{0})$ satisfies
$Q_{0}=\hat{Q}_{0}\otimes pI_{m}$ and
$\mu_{0}=\hat{\mu}_{0}\otimes\mathds{1}_{m}$ for some positive definite matrix
$\hat{Q}_{0}=\hat{Q}_{0}^{{\sf T}}\in\mathbb{R}^{n\times n}$, some $p\geq 0$,
and some vector $\hat{\mu}_{0}\in\mathbb{R}^{n}$, then $\varGamma=0$ minimizes
the cost function (27) in structured Kalman filter.

######  Proof 3.7.

First of all, let $\lambda_{0}^{k},\ldots,\lambda_{k}^{k}$ be defined as
$\lambda_{k}^{k}=I$ and
$\lambda_{i}^{k}:=\prod\nolimits_{j=k}^{i+1}(\boldsymbol{F}+\boldsymbol{G}_{j}\boldsymbol{H})=\prod\nolimits_{j=k}^{i+1}[\boldsymbol{F}+(\boldsymbol{G}_{j}^{1}+\boldsymbol{G}_{j}^{2})\boldsymbol{H}]$
for $i<k$, where
$\boldsymbol{G}_{j}^{1}:=(I_{n}\otimes\overline{V}^{\dagger})\hat{\boldsymbol{L}}_{j}$
and
$\boldsymbol{G}_{j}^{2}:=(I_{n}\otimes\mathds{1}_{m})\varGamma\hat{\boldsymbol{L}}_{j}$.
Using $\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\boldsymbol{G}_{k}^{1}=0$,
$\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\boldsymbol{G}_{k}^{2}=\varGamma\hat{\boldsymbol{L}}_{k}$,
$\boldsymbol{H}\lambda_{i}^{k}=\boldsymbol{H}\prod\nolimits_{j=k}^{i+1}(\boldsymbol{F}+\boldsymbol{G}_{j}^{1}\boldsymbol{H})$,
$i<k$, it is obtained from mathematical induction [20] that

$\displaystyle\boldsymbol{D}\lambda_{i}^{k}=\tfrac{C}{m}\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\lambda_{i}^{k}=\tfrac{C}{m}\bigl{[}A\bigl{(}I_{n}\\!\otimes\\!\mathds{1}_{m}^{{\sf
T}}\bigr{)}+m\varGamma\hat{\boldsymbol{K}}_{k}\boldsymbol{H}\bigr{]}\lambda_{i}^{k-1}$
$\displaystyle=C\sum\nolimits_{\delta=0}^{k-i-1}\Big{(}A^{\delta}\varGamma\hat{\boldsymbol{K}}_{k-\delta}\underbrace{\boldsymbol{H}\prod\nolimits_{j=k-\delta-1}^{i+1}(\boldsymbol{F}+\boldsymbol{G}_{j}^{1}\boldsymbol{H})}_{\boldsymbol{\Phi}^{i}_{k-\delta}}\Big{)}$
$\displaystyle\quad+\tfrac{C}{m}A^{k-i}\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)},\quad i<k,$ (28)
$\displaystyle\boldsymbol{D}\lambda_{i}^{k}\boldsymbol{G}_{i}=C\sum\nolimits_{\delta=0}^{k-i-1}\\!\\!A^{\delta}\varGamma\hat{\boldsymbol{K}}_{k-\delta}\boldsymbol{\Phi}^{i}_{k-\delta}\boldsymbol{G}_{i}^{1}$
$\displaystyle\quad\quad\quad\quad\
+\tfrac{C}{m}A^{k-i}\bigl{(}I_{n}\otimes\mathds{1}_{m}^{{\sf
T}}\bigr{)}\varGamma\hat{\boldsymbol{K}}_{i},\quad i<k.$ (29)

which are linear to $\varGamma$.

Note from (23) that the expected value and variance of the prediction error
are $\mathbb{E}[\boldsymbol{\epsilon}[k]]=\lambda_{0}^{k}\mu_{0}$,
$\mathbb{V}[{\boldsymbol{\epsilon}}[k]]=\lambda_{0}^{k}Q_{0}(\lambda_{0}^{k})^{{\sf
T}}+\sum\nolimits_{i=1}^{k}\lambda_{i}^{k}(\boldsymbol{G}_{i}R\boldsymbol{G}_{i}^{{\sf
T}}+W)(\lambda_{i}^{k})^{{\sf T}}$, respectively. Thus, (28), (29) imply that
the expected value $\mathbb{E}[{\rm
TA}[k]]=\boldsymbol{D}\lambda_{0}^{k}\mu_{0}$ is linear to $\varGamma$,
whereas the variance $\mathbb{V}[{{\rm
TA}}[k]]=\boldsymbol{D}\mathbb{V}[{{\boldsymbol{\epsilon}}}[k]]\boldsymbol{D}^{{\sf
T}}=\boldsymbol{D}\lambda_{0}^{k}Q_{0}(\lambda_{0}^{k})^{{\sf
T}}\boldsymbol{D}^{{\sf
T}}+\sum\nolimits_{i=1}^{k}\boldsymbol{D}\lambda_{i}^{k}(\boldsymbol{G}_{i}R\boldsymbol{G}_{i}^{{\sf
T}}+W)(\boldsymbol{D}\lambda_{i}^{k})^{{\sf T}}$ is quadratic to $\varGamma$.
Now, since the variance $\mathbb{V}[{{\rm TA}}[k]]$ is nonnegative, it follows
that the function $\mathbb{V}[{{\rm TA}}[k]]$ is convex to $\varGamma$ and
hence the cost function $J(\varGamma)$ is a convex quadratic function in
$\varGamma$ for any $\delta_{1},\delta_{2}\geq 0$,
$\delta_{1}+\delta_{2}\not=0$. Then, due to the convexity and
differentiability of the cost function (27) with
$\varGamma\in\mathbb{R}^{n\times m(n-1)}$ in unbounded space, the matrix
$\varGamma=0$ is optimal if and only if there is no linear term in neither
$\mathbb{E}[{\rm TA}[k]]$ nor $\mathbb{V}[{{\rm TA}}[k]]$ for $\varGamma$.
Hence, the result is immediate since it follows from (28), (29) that possible
linear terms in $\mathbb{E}[{\rm TA}[k]]$ and $\mathbb{V}[{{\rm TA}}[k]]$
given by

$\displaystyle\sum\nolimits_{\delta=0}^{k-1}CA^{\delta}\varGamma\hat{\boldsymbol{K}}_{k-\delta}\underbrace{\boldsymbol{\Phi}^{0}_{k-\delta}\mu_{0}}_{=0}$
$\displaystyle\tfrac{1}{m}\sum\nolimits_{\delta=0}^{k-1}CA^{\delta}\varGamma\hat{\boldsymbol{K}}_{k-\delta}\underbrace{\boldsymbol{\Phi}^{0}_{k-\delta}Q_{0}\bigl{(}I_{n}\otimes\mathds{1}_{m}\bigr{)}}_{=0}{(CA^{k})}^{{\sf
T}}$

are all zero when $Q_{0}=\hat{Q}_{0}\otimes pI_{m}$ and
$\mu_{0}=\hat{\mu}_{0}\otimes\mathds{1}_{m}$ because

$\boldsymbol{\Phi}^{i}_{k-\delta}=\sum\nolimits_{j=1}^{2^{k-\delta-i-1}}\\!\\!\boldsymbol{\phi}_{j}\boldsymbol{H}\boldsymbol{F}^{\lambda_{j}}=\sum\nolimits_{j=1}^{2^{k-\delta-i-1}}\\!\\!\boldsymbol{\phi}_{j}(CA^{\lambda_{j}}\otimes\overline{V})$

for some matrices $\boldsymbol{\phi}_{j}$ and integer
$\lambda_{j}\in\\{0,1,\ldots,k-\delta-i-1\\}$. The proof is complete.

######  Remark 3.8.

Theorem 3.6 along with Theorem 3.2 indicates that when the conditions
$Q_{0}=\hat{Q}_{0}\otimes pI_{m}$ and
$\mu_{0}=\hat{\mu}_{0}\otimes\mathds{1}_{m}$ are satisfied, there is no more
space to further narrow the confidence interval of atomic time from the CKF
algorithms. In practice, if the homogeneous atomic clocks utilized in the
ensemble are located at the same laboratory, then the conditions for $Q_{0}$
and $\mu_{0}$ are usually realistic. This is because a reference signal such
as UTC$[k]$ (Coordinated Universal Time) is usually received as an external
source at the initial time $k=0$ for determining the initial guess
$\hat{\boldsymbol{x}}[0]$, where UTC$[k]$ can be regarded as the approximated
ideal time with a tiny time deviation. In such a case, the initial predicted
error of the time deviation of the clocks can be regarded as the same with the
variances being 0. However, if the clocks are located in different
laboratories with independent receivers for UTC data, then the conditions may
not hold since the received UTC$[0]$ may be diverse to the laboratories due to
the synchronous requested timing in practice.

##  4 Numerical Simulations

This section provides an application example to demonstrate our results and
compares the performance of the atomic time among the proposed method and the
CKF algorithms with and without the existing covariance correction method by
Greenhall [9, 18]. In particular, we consider a third-order atomic clock
ensemble with $m=5$ clocks. The coefficients of the model are set to
$q_{1}^{2}=2.9394e-10$, $q_{2}^{2}=1.1785e-16$, and $q_{3}^{2}=4.5574e-35$.
The sampling interval is set to $\tau=1$. The variances of measurement noises
are set to $r_{i}^{2}=1e-12$, $i=1,2,3,4,5$. In the following statements, we
verify the results of Theorems 3.2 and 3.6, and give a discussion for the
robustness of the proposed structured Kalman filter concerning the initial
prediction error covariance.

###  4.0.1 Case 1

Letting the initial state and the initial guess be set to
$\boldsymbol{x}[0]=\hat{\boldsymbol{x}}[0]=1e-28\mathds{1}_{15}$, and letting
$\boldsymbol{P}_{0}=0.1I$, the confidence interval of the atomic time ${\rm
TA}[k]$ is shown in Fig. 1 to compare the proposed structured Kalman filter
and the CKF algorithm with and without Greenhall’s correction. It follows from
Theorem 3.2 that the proposed method with
$\hat{\boldsymbol{P}}_{0}=0.1(I_{n}\otimes\overline{V}\overline{V}^{{\sf T}})$
and $\varGamma=0$ generates the same performance as the CKF algorithms if
there is no computation error. which can be verified by the fact shown in Fig.
1(a) that the black line representing the boundaries of the confidence
interval of CKF algorithms in theory222The ideal confidence interval of CKF is
generated by ${\rm TA}[k]=\frac{1}{m}C\boldsymbol{\epsilon}_{\rm\bar{o}}$
using (22). The difference of CKF between real implementation (green) and
ideal case (black) comes from the computation errors in numerical calculations
which are diverging unboundedly along with the time. coincides with the one of
the proposed method with $\varGamma=0$. It can be further seen from the figure
that the proposed method can narrow the confidence interval from the CKF
algorithms with Greenhall’s correction [9, 18], meaning that numerical
stability is improved. Furthermore, the optimal $\varGamma$ is found as 0 in
the optimization problem (27), which verifies Theorem 3.6.

###  4.0.2 Case 2

Let the initial state $\boldsymbol{x}[0]$ be set to a deterministic value
around ${x}_{i}^{j}[0]\in(-6e-8,6e-8)$, whereas the initial predicted value is
set to $\hat{\boldsymbol{x}}[0]=1e-28\mathds{1}_{15}$. In this case, the
expected value $\mu_{0}$ of initial prediction error does not satisfy the
sufficient conditions in Theorem 3.6. The similar fact that the proposed
method solves the numerical instability of CKF algorithms and the equivalence
between the proposed method with
$\hat{\boldsymbol{P}}_{0}=(I_{n}\otimes\overline{V})\boldsymbol{P}_{0}(I_{n}\otimes\overline{V})^{{\sf
T}}$ and the CKF algorithms can also be observed from the 98$\%$-confidence
interval of ${\rm TA}[k]$ shown in Fig. 1(b) with $\boldsymbol{P}_{0}=1e-2I$.
Those observations also verify Theorem 3.2. In terms of the optimal
transformation matrix, we use $\delta_{1}=1.0$, $\delta_{2}=5.4117$,
$\hat{\boldsymbol{P}}_{0}=1e-4I$, and $T=1000$ to build the optimization
problem (27) and find the optimal transformation matrix $\varGamma$, which is
found as a nonzero matrix. It can be seen from the yellow region which
represents the 98$\%$-confidence interval of ${\rm TA}[k]$ under the optimal
$\varGamma$ in Fig. 1(b) is better maintained around zero than the case with
$\varGamma=0$.

Figure 1: 98$\%$-confidence interval of the _atomic time_ ${\rm TA}[k]$ under
different algorithms. The black lines in (a) and (b) represent the ones under
the CKF algorithms in ideal case. The smooth line in the middle of the
confidence interval represents the expected value of ${\rm TA}[k]$ in 10
stochastic paths. Both of (a) and (b) verify Theorem 3.2, whereas (a) further
verifies Theorem 3.6. The atomic time can be better maintained around 0 under
the proposed optimal method than the CKF algorithms without numerical
instability issue.

###  4.0.3 Discussion

The precise optimal solution of the optimization problem (27) depends on the
initial error covariance $\hat{\boldsymbol{P}}_{0}$ and the time $T$. However,
ideally, for a given initial guess $\hat{\boldsymbol{x}}[0]$, the performance
of both the CKF and our structured Kalman filter with a fixed $\varGamma$ does
not significantly depend on the initial error covariance
$\hat{\boldsymbol{P}}_{0}$ when $T$ is large enough. This is because the
atomic time ${\rm
TA}[k+1]=\frac{1}{m}C\boldsymbol{\epsilon}_{\rm\bar{o}}[k+1]$ depends on the
prediction error $\boldsymbol{\epsilon}_{\rm o}[k]$ and hence depends on the
error covariance $\hat{\boldsymbol{P}}_{k-1}$ of the observable state. Here,
even though $\hat{\boldsymbol{P}}_{k}$ is ideally converging to the same
steady-state value for any initial $\hat{\boldsymbol{P}}_{0}$, the value of
$\hat{\boldsymbol{P}}_{k}$ may be diverging in the CKF algorithms for
practical implementation due to the numerical instability and hence the atomic
time ${\rm TA}[k]$ may be significantly different from each other for
different initial $\hat{\boldsymbol{P}}_{0}$ in the application. In fact, this
phenomenon does at appear in our proposed method. Hence, the proposed optimal
structured Kalman filter should show better robustness in terms of
$\hat{\boldsymbol{P}}_{0}$ than the CKF algorithms. To verify the robustness
of the proposed method concerning the initial error covariance
$\hat{\boldsymbol{P}}_{0}$, we use
$\hat{\boldsymbol{P}}_{0}=0.01I,0.02I,0.04I$ in real implementation in the
simulation. It can be seen from the trajectories of the atomic time ${\rm
TA}[k]$ shown in Fig. 2 that CKF algorithm with Greenhall’s covariance
correction method [9, 18] under $\boldsymbol{P}_{0}=0.01I,0.02I,0.04I$ shows
worse robustness in Fig. 2 since its generated atomic time ${\rm TA}[k]$
(solid lines in blue, green, and red) significantly depend on the initial
$\boldsymbol{P}_{0}$. This fact can be also observed from the aspect of
frequency stability of the predicted ensemble time deviations using the
overlapping Allan deviations shown in Fig. 3. It can be seen from the figure
that the frequency stability of CKF with Greenhall’s correction (orange lines)
changes a lot when ${\boldsymbol{P}}_{0}=0.01I,0.02I,0.04I$ but it remains the
same under our proposed structured Kalman filter (blue lines).

The impact of the value of $\delta_{2}$ in the atomic time under the optimal
proposed method with $\delta_{1}=1$ is illustrated in Fig. 1(c). It can be
seen from the figure that if $\delta_{2}$ is too small (e.g., $\delta_{2}=0$),
the mean of the atomic time is very close to zero while the variance is large.
Slightly increasing $\delta_{2}$ can narrow the confidence interval without a
big change in the mean value (see the case with $\delta_{2}=5.4117$). However,
if the $\delta_{2}$ is large enough, then increasing $\delta_{2}$ can make the
mean of the atomic time worse (see the blue solid line representing the mean
of the atomic time with $\delta_{2}=1000$).

Figure 2: The _atomic time_ ${\rm TA}[k]$ under the proposed structured Kalman
filter and CKF with Greenhall’s correction correction method by Greenhall [9,
18]. The black line represents the theoretical expression of CKF. Figure 3:
Overlapping Allan deviations of the predicted ensemble time deviation under
the optimal PKF and CKF with Greenhall’s covariance correction method [9, 18]
(rhombus, square, and circle markers: $\boldsymbol{P}_{0}=0.01I,0,02I,0,04I$).
The proposed optimal structured Kalman filter show better robustness in terms
of $\hat{\boldsymbol{P}}_{0}$ than Greenhall’s CKF algorithms.

##  5 Conclusion

In this paper, we addressed a prediction problem of atomic clock ensembles for
generating an accurate time scale and proposed a structured Kalman filter
associated with the transformation matrix for observable Kalman canonical
decomposition from CKF. We presented the conditions where the proposed
structured Kalman filter is reduced to the same as the CKF in the unobservable
or observable state space. Due to undetectability of the clock model, we
considered an objective function associated with not only the expected value
of atomic time (prediction error in ensemble time deviation) but also its
variance to find out the optimal transformation matrix in observable Kalman
canonical decomposition. We revealed that such an objective function is a
convex function and showed some conditions under which CKF is nothing but the
optimal algorithm minimizing the objective function. A numerical example was
presented in the paper to show the robustness of the proposed optimal
structured Kalman filter in terms of the initial error covariance. It was
found that the proposed method may further narrow the confidence interval from
CKF as long as the transformation matrix can be properly picked.

## References

  * [1] L. Galleani and P. Tavella, “Time and the Kalman filter,” _IEEE Control Syst. Mag._ , vol. 30, no. 2, pp. 44–65, 2010.
  * [2] Y. C. Chan, W. A. Johnson, S. K. Karuza, A. M. Young, and J. C. Camparo, “Self-monitoring and self-assessing atomic clocks,” _IEEE Trans. Instrum. Meas._ , vol. 59, no. 2, pp. 330–334, 2009.
  * [3] Y. Liu, B. Xu, D. Shen, M. Ouyang, and X. Zhu, “Improving the ensemble time scale by eliminating frequency drifts of hydrogen masers,” _IEEE Trans. Instrum. Meas._ , vol. 71, pp. 1–8, 2021.
  * [4] Y. Wu, X. Zhu, Y. Huang, G. Sun, and G. Ou, “Uncertainty derivation and performance analyses of clock prediction based on mathematical model method,” _IEEE Trans. Instrum. Meas._ , vol. 64, no. 10, pp. 2792–2801, 2015\.
  * [5] D. Mulvin, “The media of high-resolution time: Temporal frequencies as infrastructural resources,” _The Information Society_ , vol. 33, no. 5, pp. 282–290, 2017.
  * [6] P. K. Seidelmann and J. H. Seago, “Time scales, their users, and leap seconds,” _Metrologia_ , vol. 48, no. 4, p. S186, 2011.
  * [7] C. Zucca and P. Tavella, “The clock model and its relationship with the allan and related variances,” _IEEE Trans. Ultrason. Ferroelectr. Freq. Control_ , vol. 52, no. 2, pp. 289–296, 2005.
  * [8] L. Galleani and P. Tavella, “On the use of the Kalman filter in timescales,” _Metrologia_ , vol. 40, no. 3, p. S326, 2003.
  * [9] C. A. Greenhall, “A review of reduced Kalman filters for clock ensembles,” _IEEE Trans. Ultrason. Ferroelectr. Freq. Control_ , vol. 59, no. 3, pp. 491–496, 2012.
  * [10] C. Trainotti, G. Giorgi, and C. Günther, “Detection and identification of faults in clock ensembles with the generalized likelihood ratio test,” _Metrologia_ , 2022.
  * [11] A. I. Mostafa, G. G. Hamza, and A. Zekry, “Enhancing the frequency stability of national time scale using emd,” _IEEE IEEE Instrum. Meas. Mag._ , vol. 23, no. 2, pp. 53–60, 2020.
  * [12] W. Ma, J. Qiu, J. Liang, and B. Chen, “Linear Kalman filtering algorithm with noisy control input variable,” _IEEE Trans. Circuits Syst. II: Express Br._ , vol. 66, no. 7, pp. 1282–1286, 2019.
  * [13] D.-J. Xin and L.-F. Shi, “Kalman filter for linear systems with unknown structural parameters,” _IEEE Trans. Circuits Syst. II: Express Br._ , vol. 69, no. 3, pp. 1852–1856, 2022.
  * [14] X. Zhang, K. Fan, W. Ma, J. Duan, J. Liang, and R. Ji, “A novel fractional Kalman filter algorithm with noisy input,” _IEEE Trans. Circuits Syst. II: Express Br._ , vol. 70, no. 3, pp. 1239–1243, 2023.
  * [15] C. De Souza, M. Gevers, and G. Goodwin, “Riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices,” _IEEE Trans. Autom. Control._ , vol. 31, no. 9, pp. 831–838, 1986\.
  * [16] Y. S. Shmaliy, S. Zhao, and C. K. Ahn, “Unbiased finite impluse response filtering: An iterative alternative to Kalman filtering ignoring noise and initial conditions,” _IEEE Control Systems Magazine_ , vol. 37, no. 5, pp. 70–89, 2017.
  * [17] Y. Yan, N. J. Jensen, T. Kawaguchi, Y. Yano, Y. Hanado, and T. Ishizaki, “Possibility of prediction improvements for atomic clock ensembles: Basis selection in undetectable systems,” in _Proc. IFAC World Congress 2023_ , to appear.
  * [18] M. Gödel, T. D. Schmidt, and J. Furthner, “Kalman filter approaches for a mixed clock ensemble,” in _Proc. EFTF Conf. & IEEE Intl. Freq. Cont. Symp._ IEEE, 2017, pp. 666–672.
  * [19] Y. Yan, T. Kawaguchi, Y. Yano, Y. Hanado, and T. Ishizaki, “Relations between generalized JST algorithm and Kalman filtering algorithm for time scale generation,” _arXiv preprint arXiv:2308.12548_ , 2023.
  * [20] ——, “Structured Kalman filter for time scale generation in atomic clock ensembles,” _arXiv preprint arXiv:2305.05894_ , 2023.


#  A class of weighted composition operators whose Range and Null spaces are
complemented

Anurag Kumar Patel Anurag Kumar Patel
Department of Mathematics
Banaras Hindu University
Varanasi 221005, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that the null space of a weighted composition operator
on $\ell_{p}~{}(1\leq p<\infty)$ is a complemented subspace. We also give a
necessary and sufficient condition for a weighted composition operator on
$\ell_{p}$ whose range space is of finite co-dimension. Thereafter, we
characterize a class of weighted composition operators whose range space is a
complemented subspace. Lastly, we characterize weighted composition operators
on $\ell_{p}$ which are Fredholm operators.

###### Key words and phrases:

Banach space, Quotient spaces, Composition operators, Fredholm operators

###### 1991 Mathematics Subject Classification:

46B25, 54B15, 47B33, 47A53

##  1\. Introduction

Composition operators appeared implicitly in the B. O. Koopman’s formulations
of classical mechanics in 1931[8]. J. V. Ryff[22] initially examined these
operators in the realm of Banach spaces of analytic functions. However, the
term “Composition Operator” was coined by Nordgren in 1968 [15]. The study of
composition operators has been extensively pursued in at least two distinct
contexts: (i) composition operators defined on the space of holomorphic
functions and (ii) composition operators defined on $L^{p}(\mu)$ associated
with a $\sigma$\- finite measure space. Composition operators represent
diverse class of operators which illuminate the whole operator theory by
providing nice examples and counter examples. Further, several important
problems in operator theory such as the invariant subspace problem can be
simplified in terms of composition operators [2]. Thus the study of
composition operators contributes significantly to our understanding of
various issues in operator theory. Consequently, the study of composition
operators is an active area of research. For more details in above matter we
suggest following references [3, 4, 6, 7, 10, 11, 16, 23].

Weighted composition operators are meaningful generalization of composition
operators with many interesting properties. For example, all the isometries of
$L_{p}(\mu)$ spaces are weighted composition operators [24]. Further, weighted
composition operators were among the prominent examples of Hypercyclic
operators [19].

Notation and Terminology : Let $\mathbb{N}$ denote the set of all positive
integers. For $1\leq p<\infty$, $\ell_{p}$ denotes the Banach space of all
$p$-summable real or complex sequences and $\ell_{\infty}$ denotes the Banach
space of all bounded sequences of complex numbers. Let support of a function
$u:X\to\mathbb{C}$ be denoted as $S(u)=\\{x\in X:u(x)\neq 0\\}.$ For Banach
spaces $X$ and $Y$, we denote by $\mathcal{B}(X,Y)$ the set of all bounded
linear operators from $X$ in to $Y$. For $T\in\mathcal{B}(X,Y),$ we denote the
range and null space by $\mathcal{R}(T)$ and $N(T),$ respectively. For given a
subset $A$ of a non-empty set $X$, the characteristic function of $A$ is
denoted by $\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{A}$ and is defined as

$\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{A}(x)=\begin{cases}{1},&x\in A\\\
0,&\text{otherwise.}\end{cases}$

Also $|A|$ denotes the cardinality of the set A and $\text{dim}(V)$ denotes
the dimension of any vector space $V$.

Weighted composition operator: Let $X$ be a non-empty set and $V(X)$ be a
linear space of complex valued functions on $X$ under point wise addition and
scalar multiplication. If $u:X\to\mathbb{C}$ and $\phi$ is a self-map on $X$
into itself such that composition $u\cdot f\circ\phi$ belongs to $V(X$) for
each $f\in V(X)$, then $u$ and $\phi$ induces a linear transformation
$uC_{\phi}$ on $V(X),$ defined as

$uC_{\phi}(f)=u\cdot f\circ\phi\quad\forall~{}f\in V(X).$

The transformation $uC_{\phi}$ is known as weighted composition
transformation. If $V(X)$ is a Banach space or Hilbert space and $uC_{\phi}$
is a bounded linear operator on $V(X),$ then $uC_{\phi}$ is called weighted
composition operator. In particular, if $u$ becomes the identity map 1, the
corresponding operator $uC_{\phi}$ on $V(X)$ becomes $C_{\phi}$ and is said to
be a weighted composition operator.

######  Definition 1.1.

Let $u:X\to\mathbb{C}$ and $\phi$ be a self-map on the set of natural numbers
$\mathbb{N}$. Then $u$ and $\phi$ induces a linear transformation $uC_{\phi}$
on $\ell_{p}$ , defined by

$uC_{\phi}\left(\sum_{n=1}^{\infty}x_{n}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\right)=\sum_{n=1}^{\infty}u(n)x_{n}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n)}$

where $\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}$ stands for the
characteristic function of the set $\\{n\\}$. Further,

$(uC_{\phi})^{n}f=u\cdot(u\circ\phi)\cdots(u\circ\phi_{n-1})\cdot(f\circ\phi_{n})$

where $\phi_{n}$ denotes $n^{th}$ iterate of $\phi.$

Let $(X,\mathfrak{B},\mu)$ be a $\sigma$-finite measure space,
$L^{p}(X,\mathfrak{B},\mu)$ be the Banach space of all p-summable complex-
valued measurable functions on X and $L^{\infty}(X,\mathfrak{B},\mu)$ denote
the Banach space of all essentially bounded complex-valued measurable
functions on X. Let $u\in L^{\infty}(\mu)$ and $\phi:X\to X$ be a measurable
transformation. Now we state a necessary and sufficient conditions on $u$ and
$\phi$ so that $uC_{\phi}$ become a weighted composition operator on
$L^{p}(X,\mathfrak{B},\mu)$.

The following results can be found in [25, 16].

######  Theorem 1.2.

Necessary and sufficient condition for a complex valued measurable function
$u$ and a measurable transformation $\phi$ to induce a bounded operator on
$L^{p}(\mu)(1\leq p<\infty)$ defined by $uC_{\phi}(f)=u\cdot f\circ\phi$ are
$\mu\phi^{-1}<<\mu$ and $\frac{d\mu_{u^{p}},\phi}{d\mu}$ is essentially
bounded, where

$\mu_{u^{p},\phi}(E)=\int_{\phi^{-1}(E)}|u|^{p}d\mu$

and $\frac{d\mu_{u^{p}},\phi}{d\mu}$ represents the Radon-Nikodym derivative.

In case when $X=\mathbb{N}$ and $\mu$ is the counting measure on X, then
$L^{p}(\mathbb{N},\mu)=\ell^{p}.$ In this setting theorem 1.2 gets the
following form.

######  Theorem 1.3.

[23] Let $u:\mathbb{N}\to\mathbb{C}$ and $\phi:\mathbb{N}\to\mathbb{N}.$ Then
the weighted composition operator $uC_{\phi}$ on $\ell_{p}$ is bounded if and
only if there exist a positive constant $K>0$ such that

$\sup_{n\geq 1}\left(\sum_{m\in\phi^{-1}(n)}|u(m)|^{p}\right)\leq K.$

In this case

$\|uC_{\phi}\|=\sup_{n\geq 1}\left(\sum_{m\in\phi^{-1}(n)}|u(m)|^{p}\right).$

The following remark follows directly from the above theorem.

######  Remark 1.4.

Let $u:\mathbb{N}\to\mathbb{C}$ and $\phi:\mathbb{N}\to\mathbb{N}$ be such
that $uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then $u\in\ell_{\infty}.$

In this paper, we study complemented subspace problem for null space and range
space of a weighted composition operator on $\ell_{p}.$ To that end, we give a
brief description of the complemented subspace problem as follows.

######  Definition 1.5.

[14] Let $X$ be a Banach space. Then a closed subspace $E$ of $X$ is called
complemented if there is a closed subspace $F\subseteq X$(called a
complementary subspace of $E$) such that $E\oplus F=X.$

Complemented subspace problem. The problem concerning complemented subspaces
lie at the core of Banach space theory, representing a foundational aspect
with a history spanning over fifty years. These concepts are pivotal in
shaping and advancing the overall theory of Banach spaces. In 1937, Murray
[14] proved, for the first time, that $\ell_{p},~{}p\neq 2,~{}p>1$ has non-
complemented subspace. This significant fact has been generalized by many
mathematicians cf. [18, 20]. In 1960, Pelczynski [17] showed that complemented
subspaces of $\ell_{1}$ are isomorphic to $\ell_{1}$. Kothe [9] generalized
this result to the non-separable case. In 1967, Lindenstrauss [12] proved that
every infinite dimensional complemented subspace of $\ell_{\infty}$ is
isomorphic to $\ell_{\infty}$. This also holds if $\ell_{\infty}$ is replaced
by $\ell_{p},~{}1\leq p<\infty,~{}c_{0}$ or $c.$ In 1971, Lindenstrauss and
Tzafriri [13] proved that every infinite dimensional Banach space which is not
isomorphic to a Hilbert space contains a closed non-complemented subspace. In
1993, Gowers and Maurey [5] showed that there exists a Banach space $X$
without non-trivial complemented subspaces.

We now state following well known theorems, which are used in our subsequent
results.

######  Theorem 1.6.

[21] Let $X$ be a Banach space. Then every finite dimensional subspace $E$ of
$X$ is complemented.

######  Theorem 1.7.

[21] Let $X$ be a Banach space and $E$ be its closed subspace. If

$\text{codim}(E)=\text{dim}(X\textfractionsolidus E)<\infty,$

then $E$ is complemented.

The paper is arranged in the following manner.
In the second section, we find the dimension of null space of a weighted
composition operator $uC_{\phi}$ on $\ell_{p}.$ We also show that the null
space of $uC_{\phi}$ on $\ell_{p}$ is always complemented. Further, we also
give a necessary and sufficient condition for the null space of a weighted
composition operator on $\ell_{p}$ to be of infinite co-dimension.

In the third section, we discuss the above aspects for the range space of
weighted composition operators. In the last section we characterize weighted
composition operators on $\ell_{p}$ which are Fredholm operators.

Main results

##  2\. Weighted composition operators on $\ell_{p}~{}(1\leq p<\infty)$ spaces
with finite dimensional null space

######  Theorem 2.1.

Let $u:\mathbb{N}\to\mathbb{C}$ and $\phi:\mathbb{N}\to\mathbb{N}$ be such
that $uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Suppose $S(u)$ is invariant under
$\phi.$ Then,

$N((uC_{\phi})^{n})=\\{f\in\ell_{p}:f|_{\phi_{n}(S(u))}=0\\}.$

###### Proof.

Suppose that $f\in N((uC_{\phi})^{n}).$ Then $(uC_{\phi})^{n}(f)=0.$
Therefore,

$u\cdot(u\circ\phi)\cdots(u\circ\phi_{n-1})\cdot(f\circ\phi_{n})=0.$

Let $g=u\cdot(u\circ\phi)\cdots(u\circ\phi_{n-1})\cdot(f\circ\phi_{n}).$ Now,
since $S(u)$ is invariant under $\phi,$ then for each $m\in S(u)$

$g(m)=0\text{ implies that }f(\phi_{n}(m))=0.$

Therefore

$f|_{\phi_{n}(S(u))}=0.$

Conversely, suppose $f|_{\phi_{n}(S(u))}=0.$ Then, $f(\phi_{n}(m))=0$ for each
$m\in S(u).$
Hence $g(m)=0$ for all $m\in\mathbb{N}.$ This implies that
$(uC_{\phi})^{n}(f)=0.$ Therefore

$f\in N((uC_{\phi})^{n}).$

∎

The following remark is a particular case of the above Theorem

######  Remark 2.2.

If $u:\mathbb{N}\to\mathbb{C}$ is bounded away from zero and
$uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then

$N((uC_{\phi})^{n})=\\{f\in\ell_{p}:f|_{\phi_{n}(\mathbb{N})}=0\\}.$

######  Theorem 2.3.

Let $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}\in\mathcal{B}(\ell_{p}).$
Suppose that $S(u)$ is invariant under $\phi.$ Then, for each $n\geq 1,$

$\text{dim}\left(N((uC_{\phi})^{n})\right)=|\mathbb{N}-\phi_{n}(S(u))|.$

###### Proof.

Let $n\geq 1$ and $A=\mathbb{N}-\phi_{n}(S(u)).$ If $A$ is empty. Then,
$\phi_{n}(S(u))=\mathbb{N}$ and hence, by Theorem 2.1,
$N((uC_{\phi})^{n})=\\{0\\}.$ Therefore,

$\text{dim}(N((uC_{\phi})^{n}=0.$

Now, suppose $A$ is a non-empty set. Since for each $n\in A,$

$\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}|_{\phi_{n}(S(u))}=0.$

Hence by Theorem 2.1, $\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\in
N((uC_{\phi})^{n})\text{ for each }n\in A.$

We claim that $\mathfrak{B}=\\{\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}:n\in
A\\}$ is a basis for $N((uC_{\phi})^{n}).$ We first prove that $\mathfrak{B}$
is linearly independent. Let $\\{n_{1},n_{2},...n_{k}\\}\subset A$ and
$\lambda_{1},\lambda_{2},...,\lambda_{k}\in\mathbb{F}$ be such that
$\sum_{i=1}^{k}\lambda_{i}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n_{i}}=0.$
Let
$g=\sum_{i=1}^{k}\lambda_{i}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n_{i}}.$
Then,

$g(n_{j})=\lambda_{j}=0\text{ for each }1\leq j\leq k.$

Hence, $\mathfrak{B}$ is linearly independent.

Now, we prove that $\mathfrak{B}$ is a spanning set for $N((uC_{\phi})^{n}).$
Let $f=\sum_{n=1}^{\infty}f(n)\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\in
N((uC_{\phi})^{n}).$ Then $f=\sum\limits_{n\in
A}f(n)\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}$ since
$f|_{\phi_{n}(S(u))}=0.$ Therefore $\mathfrak{B}$ is a spanning set for
$N((uC_{\phi})^{n}).$ ∎

######  Corollary 2.4.

Let $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}\in\mathcal{B}(\ell_{p}).$
Suppose that $S(u)$ is invariant under $\phi.$ Then dimension of
$N((uC_{\phi})^{n})$ is finite if and only if $\mathbb{N}-\phi_{n}(S(u))$ is a
finite set. In this case,

$\text{dim}(N((uC_{\phi})^{n}))=|\mathbb{N}-\phi_{n}(S(u))|.$

###### Proof.

Let $A=\mathbb{N}-\phi_{n}(S(u)).$ Then by the Theorem 2.3,
$\mathfrak{B}=\\{\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}:n\in A\\}$ is a
basis for $N((uC_{\phi})^{n})$ . Hence $\text{dim}(N((uC_{\phi})^{n}))$ is
finite if and only if $A$ is finite. This completes the proof. ∎

We now give two examples, one in which the dimension of the null space of
$uC_{\phi}$ is finite and one in which the dimension of the null space of
$uC_{\phi}$ is infinite.

######  Example 2.5.

Let $u:\mathbb{N}\to\mathbb{C}$ be defined as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&n=1,2\\\
\frac{1}{n}&\mbox{if}&n>2\end{array}\right.$

and $\phi:\mathbb{N}\to\mathbb{N}$ be defined as
$\phi(n)=n~{}\forall~{}n\in\mathbb{N}.$ Clearly $S(u)=\\{n:n\geq 3\\}.$ Hence
$\phi(S(u))=\\{n:n\geq 4\\}$ and $\mathbb{N}-\phi(S(u))=\\{1,2,3\\}.$
Therefore by Corollary 2.4, $\text{dim}(N(uC_{\phi}))=3.$

######  Example 2.6.

Let $p_{n}$ denote the $n^{th}$ prime. Define $u:\mathbb{N}\to\mathbb{C}$ as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&1\leq n\leq 10\\\
\frac{1}{n}&\mbox{if}&n>10\end{array}\right.$

and define $\phi:\mathbb{N}\to\mathbb{N}$ as $\phi(n)=p_{n}.$ Then
$S(u)=\\{n:n>10\\}$ and $\phi(S(u))\subseteq S(u).$ Hence by Corollary 2.4,
$\text{dim}(N(uC_{\phi}))=|\mathbb{N}-\phi(S(u))|=\infty.$

The following lemma gives a necessary and sufficient condition on $\phi$ for
null space of $C_{\phi}$ to be an invariant subspace of $C_{\phi}.$

######  Lemma 2.7.

Let $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}$ is a bounded linear operator
on $\ell_{p}.$ Suppose $S(u)$ is invariant under $\phi.$ Then
$N((uC_{\phi})^{2})=N(uC_{\phi})$ if and only if $\phi_{2}(S(u))=\phi(S(u))$.

###### Proof.

Suppose $N((uC_{\phi})^{2})=N(uC_{\phi}).$ Then it follows from 2.1 that

$\phi_{2}(S(u))=\phi(S(u)).$

Conversely, if $\phi_{2}(S(u))=\phi(S(u)).$ Then

$\displaystyle N((uC_{\phi})^{2})$
$\displaystyle=\\{f\in\ell_{p}:f|_{\phi_{2}(S(u))=0}\\}$
$\displaystyle=\\{f\in\ell_{p}:f|_{\phi(S(u))=0}\\}$
$\displaystyle=N(uC_{\phi}).$

∎

######  Lemma 2.8.

Suppose $u:\mathbb{N}\to\mathbb{C}$ is bounded away from zero and
$uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then $\text{dim}(N(uC_{\phi}))<\infty$ if
and only if for each $n\geq 1$

$\text{dim}(N((uC_{\phi})^{n}))<\infty.$

###### Proof.

Suppose $\text{dim}(N(uC_{\phi}))<\infty.$ Then by Corollary 2.4,
$\mathbb{N}-\phi(S(u))$ is a finite set. Hence this together with the fact
that
$\phi_{k}(\mathbb{N})-\phi_{k+1}(S(u))\subseteq\phi_{k}\left(\mathbb{N}-\phi(S(u))\right)$
implies that $\phi_{k}(\mathbb{N})-\phi_{k+1}(S(u))$ is finite set for each
$k\geq 1$. Now

$\mathbb{N}-\phi_{n}(S(u))=\bigcup_{k=1}^{n}\left(\phi_{k-1}(\mathbb{N})-\phi_{k}(S(u))\right)$

is a finite union of finite sets. Hence $\mathbb{N}-\phi_{n}(S(u))$ is a
finite set. Therefore by Remark 2.2,

$\text{dim}(N((uC_{\phi})^{n}))<\infty.$

Conversely, suppose that $\text{dim}(N((uC_{\phi})^{n})))<\infty$ for each
$n\geq 1.$ Then

$\text{dim}(N(C_{\phi}))<\infty.$

∎

The following corollary is the negation of the Lemma 2.8

######  Corollary 2.9.

Let $u:\mathbb{N}\to\mathbb{C}$ be such that
$uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then $\text{dim}(N(uC_{\phi}))=\infty$ if
and only if $\text{dim}(N((uC_{\phi})^{n})))=\infty$ for all $n\in\mathbb{N}.$

The following theorem shows that the null space of a weighted composition
operator is always a complemented subspace.

######  Theorem 2.10.

Suppose $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}\in\mathcal{B}(\ell_{p}).$
Then $N(uC_{\phi})$ is complemented.

###### Proof.

Let $uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then

$N(uC_{\phi})=\\{f\in\ell_{p}:f|_{\phi(S(u))}=0\\}.$

Let $M=\\{f\in\ell_{p}:f|_{\mathbb{N}-\phi(S(u))}=0\\}.$ Clearly, $M$ is a
closed set.
Let $f\in N(uC_{\phi})\cap M.$ Then

$f|_{\phi(S(u))}=0\text{ and }f|_{\mathbb{N}-\phi(S(u))}=0.$

This implies that $f=0.$ Hence $N(uC_{\phi})\cap M=\\{0\\}.$

Now let
$f=\sum\limits_{n\in\mathbb{N}}f_{n}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\in\ell_{p}.$
Then

$f=\sum\limits_{n\in(\mathbb{N}-\phi(S(u)))}f_{n}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}+\sum\limits_{n\in\phi(S(u))}f_{n}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}=g+h,$

where $g\in N(uC_{\phi})$ and $h\in M.$ Hence $f\in N(uC_{\phi})+M.$ Therefore
$N(uC_{\phi})$ is complemented.

∎

In the following theorem we find the codimension of $N(uC_{\phi})$ in
$\ell_{p}.$

######  Theorem 2.11.

Let $u:\mathbb{N}\to\mathbb{C}$ be such that
$uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then

$\text{dim}(\ell_{p}\textfractionsolidus
N(uC_{\phi}))=\left|\phi(S(u))\right|.$

###### Proof.

Observe that for each $n\in\phi(S(u)),$

$\phi^{-1}(n)\cap S(u)\neq\emptyset.$

Hence by Theorem 2.1

$\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\notin N(uC_{\phi}))\quad\text{ for
each }n\in\phi(S(u)).$

We claim that
$\mathfrak{B}=\\{\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}+N(uC_{\phi}):n\in\phi(S(u))\\}$
is a basis for $\ell_{p}\textfractionsolidus N(uC_{\phi}).$ Let
$\lambda_{1},\lambda_{2},...,\lambda_{k}\in\mathbb{F}$ and
$n_{1},n_{2},...n_{k}\in\phi(S(u))$ be such that

$\sum_{i=1}^{k}\lambda_{i}(\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n_{i}}+N(uC_{\phi}))=N(uC_{\phi}).$

Then
$\sum_{i=1}^{k}\lambda_{i}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n_{i}}\in
N(uC_{\phi}).$

Let
$g=\sum_{i=1}^{k}\lambda_{i}\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n_{i}}.$
Since $g\in N(uC_{\phi}),$ then $g|_{\phi(S(u))}=0.$ Therefore for each $1\leq
j\leq k,$

$g(n_{j})=\lambda_{j}=0.$

This implies that $\mathfrak{B}$ is linearly independent.

Now let
$f+N(uC_{\phi})=\sum\limits_{n\in\mathbb{N}}f(n)\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}+N(uC_{\phi})\in\ell_{p}\textfractionsolidus
N(uC_{\phi}).$ Then

$\displaystyle f+N(uC_{\phi})$
$\displaystyle=\sum\limits_{n\in\phi(S(u))}f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}+\sum\limits_{n\notin\phi(S(u))}f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}+N(uC_{\phi})$
$\displaystyle=\sum\limits_{n\in\phi(S(u))}f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}+N(uC_{\phi}).$

Hence $\mathfrak{B}$ is a spanning set for $\ell_{p}\textfractionsolidus
N(uC_{\phi}).$ Therefore

$\text{dim}(\ell_{p}\textfractionsolidus
N(uC_{\phi}))=\left|\phi(S(u))\right|.$

∎

The following lemma is the direct consequence of the above Theorem 2.11. In
this lemma, we show that codimension of $N(uC_{\phi})$ in $\ell_{p}$ is
infinite if and only if $\phi(S(u))$ is an infinite set. Hence the converse of
the Theorem 1.7 is not true.

######  Lemma 2.12.

Suppose $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}\in\mathcal{B}(\ell_{p}).$
Then $\ell_{p}\textfractionsolidus N(uC_{\phi})$ is infinite dimensional if
and only if $\phi(S(u))$ is an infinite set.

Now we give below two examples- one in which co-dimension of $N(uC_{\phi})$ is
infinite and another- in which $N(uC_{\phi})$ has finite co-dimension.

######  Example 2.13.

Define $u:\mathbb{N}\to\mathbb{C}$ as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&n\text{ is odd }\\\
\frac{1}{n^{3}}&\mbox{if}&n\text{ is even }\end{array}\right.$

and define $\phi:\mathbb{N}\to\mathbb{N}$ as $\phi(n)=n^{2}.$ Then, we have
$S(u)=\\{n:n\text{ is even natural number }\\}$ and $\phi(S(u))$ is an
infinite set. Hence by Lemma 2.12,

$\text{dim}(\ell_{p}\textfractionsolidus N(uC_{\phi}))=\infty.$

######  Example 2.14.

Define $u:\mathbb{N}\to\mathbb{C}$ as $u(n)=\frac{1}{n^{2}}$ and
$\phi:\mathbb{N}\to\mathbb{N}$ as $\phi(n)=1.$ Then, we have $S(u)=\mathbb{N}$
and $\phi(S(u))=\\{1\\}.$ Hence by Theorem 2.11,

$\text{dim}(\ell_{p}\textfractionsolidus N(uC_{\phi}))=1.$

##  3\.  Weighted composition operator on $\ell_{p}~{}(1\leq p<\infty)$ spaces
whose range space is of finite co-dimension

In this section, we characterize weighted composition operators on $\ell_{p}$
space whose range space has finite co-dimension. Thereafter, we use the
forgoing characterization to give a class of weighted composition operators
whose range space is a complemented subspace.
We begin with the following lemma, which shows that the range space of a
weighted composition operator $uC_{\phi}$ on $\ell_{p}$ is always infinite
dimensional.

######  Theorem 3.1.

Suppose $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}\in\mathcal{B}(\ell_{p}).$
Then dimension of $\mathcal{R}(uC_{\phi})$ is infinite if
$\\{n:\phi^{-1}(n)\cap S(u)\neq\emptyset\\}$ is an infinite subset of
$\mathbb{N}.$

###### Proof.

Let $A=\\{n:\phi^{-1}(n)\cap S(u)\neq\emptyset\\}.$ Observe that for each
$n\in A$

$u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n)}=uC_{\phi}(\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n})\in\mathcal{R}(uC_{\phi}).$

We claim that
$\mathfrak{B}=\\{u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n)}:n\in
A\\}$ is linearly independent. Let
$\lambda_{1},\lambda_{2},...,\lambda_{k}\in\mathbb{F}$ and
$n_{1},n_{2},...n_{k}\in A$ be such that

$\lambda_{1}u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n_{1})}+\lambda_{2}u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n_{2})}+...+\lambda_{k}u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n_{k})}=0.$

Let
$g=\lambda_{1}u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n_{1})}+\lambda_{2}u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n_{2})}+...+\lambda_{k}u\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{\phi^{-1}(n_{k})}.$
Since $\phi^{-1}(n_{i})\cap S(u)\neq\emptyset$ for each $1\leq i\leq k.$
Therefore for $m\in\phi^{-1}(n_{i})\cap S(u),$

$\displaystyle g(m)$
$\displaystyle=\lambda_{i}u(m)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{\phi^{-1}(n_{i})}(m)$
$\displaystyle=\lambda_{i}u(m)$

Since $g=0$ and $m\in S(u),$ therefore

$g(m)=0~{}\text{ implies }~{}\lambda_{i}=0~{}\text{ for each }1\leq i\leq k.$

Hence $\mathfrak{B}$ is a linearly independent subset of
$\mathcal{R}(uC_{\phi}).$ ∎

The following corollary is the direct consequence of the above theorem

######  Corollary 3.2.

Suppose $u:\mathbb{N}\to\mathbb{C}$ is bounded away from zero and
$uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then $\mathcal{R}(uC_{\phi})$ is infinite
dimensional.

######  Theorem 3.3.

Let $u:\mathbb{N}\to\mathbb{C}$ and $\phi:\mathbb{N}\to\mathbb{N}$ be such
that $S(u)$ is invariant under $\phi$ and both $C_{\phi}$ and $uC_{\phi}$ are
bounded linear operators on $\ell_{p}.$ Suppose $M_{k}={\phi^{-1}(k)}\bigcap
S(u)$ for each $k\in\mathbb{N}.$ Then range space $\mathcal{R}(uC_{\phi})$ of
$uC_{\phi}$ is given by

$\mathcal{R}(uC_{\phi})=\left\\{f\in\ell_{p}:\sum_{k\in
S(u)}|M_{k}|\left|\left(\frac{f}{u}\right)(k)\right|^{p}<\infty,f(k)=0\text{
whenever }k\in\mathbb{N}-S(u)\right.\\\ \left.\text{ and
}\right.\left.\left.\left(\frac{f}{u}\right)\right|_{M_{k}}=\text{ constant
for each }k\in\mathbb{N}\right\\}$

###### Proof.

Let $f\in\mathcal{R}(uC_{\phi}).$ Then there exists $g\in\ell_{p}$ such that

$uC_{\phi}(g)=f.$

This implies that

$f(k)=0\text{ whenever }k\in\mathbb{N}-S(u)\text{ and
}\left(\frac{f}{u}\right)(k)=(g\circ\phi)(k)\text{ for each }k\in S(u).$

Now, as $\\{|M_{k}|:k\in\mathbb{N}\\}$ is a bounded subset of $\mathbb{N}$ and
$g\circ\phi\in\ell_{p},$ therefore it follows that

$\sum_{k\in S(u)}|M_{k}|\left|\left(\frac{f}{u}\right)(k)\right|^{p}<\infty.$

Further, let $m_{1},m_{2}\in M_{k}.$ Then
$(g\circ\phi)(m_{1})=(g\circ\phi)(m_{2}).$ Hence

$\left(\frac{f}{u}\right)(m_{1})=\left(\frac{f}{u}\right)(m_{2}).$

This implies that

$\left(\frac{f}{u}\right)\Big{|}_{M_{k}}=\text{ constant }.$

Conversely, suppose $f\in\ell_{p},~{}\sum\limits_{k\in
S(u)}|M_{k}|\left|\left(\frac{f}{u}\right)(k)\right|^{p}<\infty,$

$f(k)=0\text{ whenever }k\in\mathbb{N}-S(u)\text{ and
}\left(\frac{f}{u}\right)\Big{|}_{M_{k}}=\text{ constant for each
}k\in\mathbb{N}.$

Define $g:\mathbb{N}\to\mathbb{C}$ as follow

$g(k)=\begin{cases}\left(\frac{f}{u}\right)(m),&\text{ for some }m\in
M_{k}\text{ if }M_{k}\text{ is non-empty }\\\ 0,&\text{ if }M_{k}\text{ is
empty. }\end{cases}$

Then $g$ is well defined as $\left(\frac{f}{u}\right)$ is constant on $M_{k}$
for each $k\in\mathbb{N}.$ From the above conditions on $f,$ it can be easily
seen that $g$ belongs to $\ell_{p}.$
Further,

$\displaystyle uC_{\phi}(g)(k)=$ $\displaystyle u(k)g(\phi(k))$
$\displaystyle=$ $\displaystyle f(k)\text{ for all }k\in\mathbb{N}.$

Thus $uC_{\phi}(g)=f.$ Hence $f\in\mathcal{R}(uC_{\phi}).$ ∎

The following theorem characterizes the weighted composition operators on
$\ell_{p}$ whose range has finite co-dimension.

######  Theorem 3.4.

Let $u:\mathbb{N}\to\mathbb{C}$ and $\phi:\mathbb{N}\to\mathbb{N}$ be such
that $C_{\phi}$ and $uC_{\phi}$ are both bounded linear operators on
$\ell_{p}.$ Then $\mathcal{R}(uC_{\phi})$ has finite co-dimension in
$\ell_{p}$ if and only if
$\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}$ is a non-empty finite set.

###### Proof.

Let $A=\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}.$ Suppose $A$ is a non-
empty finite set. Let $A=\\{n_{1},n_{2},...,n_{k}\\}$ and suppose for each
$i\in\\{1,2,...,k\\}$

$A_{i}=\phi^{-1}(n_{i})\cap S(u)=\\{m_{j}^{i}:j=1,2,...,l_{i}\\}$

Observe that
$\sum\limits_{n\notin\phi^{-1}(A)}u(n)f(n)\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\in\mathcal{R}(uC_{\phi}).$
We claim that
$\mathfrak{B}=\\{\mathcal{R}(uC_{\phi})+u(m_{j}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{j}^{i}}:i\in\\{1,2,...,k\\}~{}\text{and}~{}j=1,2,...,l_{i}-1\\}$
forms a Hamel basis for $\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi}).$
We first prove that $\mathfrak{B}$ is linearly independent.
Let $\lambda_{j}^{i}\in\mathbb{F}$ for all $1\leq i\leq k$ and $1\leq j\leq
l_{i}-1$ be such that

$\sum_{i=1}^{k}\left(\sum_{j=1}^{l_{i}-1}\lambda_{j}^{i}(\mathcal{R}(uC_{\phi})+u(m_{j}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{j}^{i}})\right)=\mathcal{R}(uC_{\phi}).$

Then

$\sum_{i=1}^{k}\sum_{j=1}^{l_{i}-1}\lambda_{j}^{i}u(m_{j}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{j}^{i}}\in\mathcal{R}(uC_{\phi}).$

Let
$g=\sum_{i=1}^{k}\sum_{j=1}^{l_{i}-1}\lambda_{j}^{i}u(m_{j}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{j}^{i}}.$
Then, for each $1\leq i\leq k\text{ and }1\leq j\leq l_{i}-1$

$\left(\frac{g}{u}\right)({m_{j}^{i}})=\lambda^{i}_{j}\text{ and
}\left(\frac{g}{u}\right)({m^{i}_{l_{i}}})=0.$

Now $g\in\mathcal{R}(uC_{\phi}),$ therefore
$\left(\frac{g}{u}\right)\Big{|}_{A_{i}}=\text{constant}$ for all $1\leq i\leq
k.$ This implies that

$\lambda^{i}_{j}=0\text{ for all }1\leq i\leq k\text{ and }1\leq j\leq
l_{i}-1.$

Now, we show $\mathfrak{B}$ generates
$\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi}).$ Let
$\mathcal{R}(uC_{\phi})+f\in\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi})\text{
where }f=(f(1),f(2),f(3),...).$ Then

$\displaystyle\mathcal{R}(uC_{\phi})+(f(1),f(2),f(3),...)$
$\displaystyle=\mathcal{R}(uC_{\phi})+\sum\limits_{n\notin\phi^{-1}(A)}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}+\sum\limits_{n\in\phi^{-1}(A)}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}$
$\displaystyle=\mathcal{R}(uC_{\phi})+\sum\limits_{n\in
A_{1}}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}+\sum\limits_{n\in
A_{2}}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}+...$
$\displaystyle\quad+\sum\limits_{n\in
A_{k}}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}.$

Note that

$\sum\limits_{n\in
A_{i}}u(n)f(m_{l_{i}}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{n}\in\mathcal{R}(uC_{\phi})\text{
for each }1\leq i\leq k.$

Therefore

$\displaystyle\mathcal{R}(uC_{\phi})+\sum\limits_{n\in\mathbb{N}}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}$
$\displaystyle=\mathcal{R}(uC_{\phi})+\sum_{i=1}^{k}\left(\sum\limits_{n\in
A_{i}}u(n)f(n)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}-\sum\limits_{n\in
A_{i}}u(n)f(m_{l_{i}}^{i})\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}\right)$
$\displaystyle=\mathcal{R}(uC_{\phi})+\sum_{i=1}^{k}\left(\sum\limits_{n\in
A_{i}\setminus\\{m_{l_{i}}^{i}\\}}u(n)\left(f(n)-f(m_{l_{i}}^{i})\right)\raisebox{0.0pt}[6.19998pt][6.19998pt]{$\chi$}_{n}\right)$

which is a linear combination of elements from $\mathfrak{B}.$
Therefore $\mathfrak{B}$ forms a Hamel basis for
$\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi})$ with dimension

$\text{dim}\left.(\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi})\right.)=\sum\limits_{n\in
A}\left(|\phi^{-1}(n_{i})\cap S(u)|-1\right).$

Conversely, suppose the set $A$ is not finite. Then there exists a sequence
$\\{n_{i}\\}_{i=1}^{\infty}$ of positive integers such that
$|\phi^{-1}(n_{i})\cap S(u)|>1$ for each $i\geq 1.$ Let

$A_{i}=\phi^{-1}(n_{i})\cap S(u)=\\{m_{j}^{i}:j=1,2,...,l_{i}\\}\text{ for
each }i\geq 1.$

We claim that
$\mathfrak{B}=\left\\{\mathcal{R}(uC_{\phi})+u(m_{1}^{1})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{1}^{1}},\mathcal{R}(uC_{\phi})+u(m_{1}^{2})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{1}^{2}},...,\mathcal{R}(uC_{\phi})+u(m_{1}^{k})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{1}^{k}},...\right\\}$
is a linearly independent set in
$\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi}).$

Let $\lambda_{1},\lambda_{2},...,\lambda_{k}\in\mathbb{F}$ be such that
$\sum_{i=1}^{k}\lambda_{i}(\mathcal{R}(uC_{\phi})+u(m_{1}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{1}^{i}})=\mathcal{R}(uC_{\phi}).$
Then

$\sum_{i=1}^{k}\lambda_{i}u(m_{1}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{1}^{i}}\in\mathcal{R}(uC_{\phi}).$

Let
$g=\sum_{i=1}^{k}\lambda_{i}u(m_{1}^{i})\raisebox{0.0pt}[6.2pt][6.2pt]{$\chi$}_{m_{1}^{i}}.$
Then, for each $1\leq i\leq k$

$\left(\frac{g}{u}\right)({m_{1}^{i}})=\lambda_{i}\text{ and
}g\left(\frac{g}{u}\right)({m^{i}_{2}})=0.$

Now $g\in\mathcal{R}(uC_{\phi}),$ therefore
$\left(\frac{g}{u}\right)\Big{|}_{A_{i}}=\text{constant}$ for each $i\geq 1.$
This implies that

$\lambda_{i}=0~{}\text{ for all }~{}1\leq i\leq k.$

This completes the proof. ∎

The following corollary is an immediate consequence of the above Theorem 3.4.

######  Corollary 3.5.

Let $u:\mathbb{N}\to\mathbb{C}$ be such that
$uC_{\phi}\in\mathcal{B}(\ell_{p}).$ Then

$\text{dim}(\ell_{p}\textfractionsolidus\mathcal{R}((uC_{\phi})^{n}))<\infty$

if and only if $\\{k\in\mathbb{N}:|\phi^{-1}_{n}(k)\cap S(u)|>1\\}$ is a
finite set.

Now we give an example which satisfies Theorem 3.4.

######  Example 3.6.

Let $u:\mathbb{N}\to\mathbb{C}$ be defined as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&1\leq n\leq 5\\\
1+\frac{1}{n}&\mbox{if}&n>5\end{array}\right.$

and $\phi:\mathbb{N}\to\mathbb{N}$ be defined as

$\phi(n)=\left\\{\begin{array}[]{rcl}1&\mbox{if}&1\leq n\leq 7\\\
n+1&\mbox{if}&n>7\end{array}\right.$

Then $\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}=\\{1\\}.$ Hence by
Theorem 3.4,

$\text{dim}(\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi}))=1.$

The next theorem gives a class of weighted composition operators whose range
is a complemented subspace.

######  Theorem 3.7.

Let $u:\mathbb{N}\to\mathbb{C}$ be such that
$uC_{\phi}\in\mathcal{B}(\ell_{p})$ and $\mathcal{R}(uC_{\phi})$ is closed.
Then $\mathcal{R}(uC_{\phi})$ is complemented if
$\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}$ is a finite set.

###### Proof.

Let $A=\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}$ be a finite set. Then
by Theorem 3.4,

$\text{dim}(\ell_{p}\textfractionsolidus\mathcal{R}(uC_{\phi}))<\infty.$

Therefore by Theorem 1.7, $\mathcal{R}(uC_{\phi})$ is complemented if $A$ is a
finite set. ∎

######  Remark 3.8.

If $\\{k\in\mathbb{N}:|\phi^{-1}_{n}(k)\cap S(u)|>1\\}$ is a finite set. Then
by above Lemma 3.5,

$\text{dim}(\ell_{p}\textfractionsolidus\mathcal{R}((uC_{\phi})^{n}))<\infty.$

Consequently by Theorem 3.4, $\mathcal{R}((uC_{\phi})^{n})$ is complemented.

######  Corollary 3.9.

Suppose $u:\mathbb{N}\to\mathbb{C}$ and $uC_{\phi}\in\mathcal{B}(\ell_{p}).$
If $\\{k\in\mathbb{N}:|\phi^{-1}_{n}(k)|\cap S(u)>1\\}$ is finite, then

$\text{dim}\left(\mathcal{R}((uC_{\phi})^{n-1})\textfractionsolidus\mathcal{R}((uC_{\phi})^{n})\right)<\infty.$

###### Proof.

Let $\\{k\in\mathbb{N}:|\phi^{-1}_{n}(k)\cap S(u)|>1\\}$ is finite, then by
Lemma 3.5

$\text{dim}\left(\ell_{p}\textfractionsolidus\mathcal{R}((uC_{\phi})^{n})\right)<\infty.$

Since $\mathcal{R}((uC_{\phi})^{n-1})\subseteq\ell_{p},$ therefore
$\mathcal{R}((uC_{\phi})^{n-1})\textfractionsolidus\mathcal{R}((uC_{\phi})^{n})\subseteq\ell_{p}\textfractionsolidus\mathcal{R}((uC_{\phi})^{n})$
and hence

$\text{dim}\left(\mathcal{R}((uC_{\phi})^{n-1})\textfractionsolidus\mathcal{R}((uC_{\phi})^{n})\right)<\infty.$

∎

######  Remark 3.10.

If $\phi:\mathbb{N}\to\mathbb{N}$ and $n\in\mathbb{N}$ are such that
$\\{k\in\mathbb{N}:|\phi^{-1}_{n}(k)\cap S(u)|>1\\}$ is finite. Then from the
Theorem 1.7, $\mathcal{R}((uC_{\phi})^{n-1})$ is complemented in
$\mathcal{R}((uC_{\phi})^{n}).$

##  4\.  Weighted composition operator as Fredholm operator on $\ell_{p}$
spaces $(1\leq p<\infty)$

In this section, using the results of previous sections, we characterize
weighted composition operators which are Fredholm operators. To that end, we
first define Fredholm operator.

Fredholm operator. [1] Let $X$ and $Y$ be Banach spaces. An operator
$T\in\mathcal{B}(X,Y)$ is called a Fredholm operator if $\mathcal{R}(T)$ is
closed, $\text{dim}(N(T))$ and $\text{dim}(\mathcal{R}(T))$ are both finite.
In this case, the (Fredholm) index of $T$ is defined as

$\text{ind}(T)=\text{dim}(N(T))-\text{codim}(\mathcal{R}(T)).$

The following theorem characterizes weighted composition operators on
$\ell_{p}$ which are Fredholm operators.

######  Theorem 4.1.

Let $u:\mathbb{N}\to\mathbb{C}$ be such that
$uC_{\phi}\in\mathcal{B}(\ell_{p})$ and $\mathcal{R}(uC_{\phi})$ is closed.
Then $uC_{\phi}$ is a Fredholm operator if and only if
$|\mathbb{N}-\phi(S(u))|<\infty$ and $|A|<\infty,$ where

$A=\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}.$

In this case, Fredholm index of $uC_{\phi}$ is

$\text{ind}(uC_{\phi})=|\mathbb{N}-\phi(S(u))|-\sum\limits_{n\in
A}\left(|\phi^{-1}(n)|-1\right).$

###### Proof.

Suppose $uC_{\phi}\in\mathcal{B}(\ell_{p})$ is a Fredholm operator. Then

$\text{dim}(N(uC_{\phi}))<\infty\text{ and
}\text{codim}(\mathcal{R}(uC_{\phi}))<\infty.$

Consequently by Corollary 2.4, $|\mathbb{N}-\phi(S(u))|<\infty$ and, by
Theorem 3.4, $|A|<\infty.$

Conversely, suppose $|\mathbb{N}-\phi(S(u))|<\infty$ and $|A|<\infty.$ Then by
Corollary 2.4,

$\text{dim}(N(uC_{\phi}))<\infty$

and, by Theorem 3.4

$\text{codim}(\mathcal{R}(uC_{\phi}))<\infty.$

Therefore $uC_{\phi}$ is a Fredholm operator. In this case

$\displaystyle\text{ind}(uC_{\phi})$
$\displaystyle=\text{dim}(N(uC_{\phi}))-\text{codim}(\mathcal{R}(uC_{\phi}))$
$\displaystyle=|\mathbb{N}-\phi(S(u))|-\sum\limits_{n\in
A}\left(|\phi^{-1}(n)\cap S(u)|-1\right).$

∎

Now we construct examples of weighted composition operators on $\ell_{p}$
which are Fredholm operators with positive, zero and negative indices
respectively.

######  Example 4.2.

Let $u:\mathbb{N}\to\mathbb{C}$ be defined as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&1\leq n\leq 3\\\
1+\frac{1}{n}&\mbox{if}&n>3\end{array}\right.$

and $\phi:\mathbb{N}\to\mathbb{N}$ be defined as

$\phi(n)=\left\\{\begin{array}[]{rcl}1&\mbox{if}&1\leq n\leq 5\\\
n&\mbox{if}&n>5\end{array}\right.$

Then $S(u)=\\{n:n\geq 4\\},~{}\mathbb{N}-\phi(S(u))=\\{2,3,4,5\\}$ and

$A=\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}=\\{1\\}.$

Hence

$\displaystyle\sum\limits_{n\in A}\left(|\phi^{-1}(n)\cap S(u)|-1\right)$
$\displaystyle=|\phi^{-1}(1)\cap\\{n:n\geq 4\\}|-1$
$\displaystyle=|\\{4,5\\}|-1$ $\displaystyle=1.$

Therefore by Theorem 4.1, $uC_{\phi}$ is a Fredholm operator with
$\text{ind}(uC_{\phi})=3.$

######  Example 4.3.

Let $u:\mathbb{N}\to\mathbb{C}$ be defined as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&1\leq n\leq 2\\\
1+\frac{1}{n}&\mbox{if}&n>2\end{array}\right.$

and $\phi:\mathbb{N}\to\mathbb{N}$ be defined as

$\phi(n)=\left\\{\begin{array}[]{rcl}1&\mbox{if}&1\leq n\leq 5\\\
n-2&\mbox{if}&n>5\end{array}\right.$

Then $S(u)=\\{n:n\geq 3\\},~{}\mathbb{N}-\phi(S(u))=\\{2,3\\}$ and

$A=\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}=\\{1\\}.$

Hence

$\displaystyle\sum\limits_{n\in A}\left(|\phi^{-1}(n)\cap S(u)|-1\right)$
$\displaystyle=|\phi^{-1}(1)\cap\\{n:n\geq 3\\}|-1$
$\displaystyle=|\\{3,4,5\\}|-1$ $\displaystyle=2.$

Therefore by Theorem 4.1, $uC_{\phi}$ is a Fredholm operator with
$\text{ind}(uC_{\phi})=0.$

######  Example 4.4.

Let $u:\mathbb{N}\to\mathbb{C}$ be defined as

$u(n)=\left\\{\begin{array}[]{rcl}0&\mbox{if}&1\leq n\leq 2\\\
1+\frac{1}{n}&\mbox{if}&n>2\end{array}\right.$

and $\phi:\mathbb{N}\to\mathbb{N}$ be defined as

$\phi(n)=\left\\{\begin{array}[]{rcl}1&\mbox{if}&1\leq n\leq 5\\\
n-3&\mbox{if}&n>5\end{array}\right.$

Then $S(u)=\\{n:n\geq 3\\},~{}\mathbb{N}-\phi(S(u))=\\{2\\}$ and

$A=\\{n\in\mathbb{N}:|\phi^{-1}(n)\cap S(u)|>1\\}=\\{1\\}.$

Hence

$\displaystyle\sum\limits_{n\in A}\left(|\phi^{-1}(n)\cap S(u)|-1\right)$
$\displaystyle=|\phi^{-1}(1)\cap\\{n:n\geq 3\\}|-1$
$\displaystyle=|\\{3,4,5\\}|-1$ $\displaystyle=2.$

Therefore by Theorem 4.1, $uC_{\phi}$ is a Fredholm operator with
$\text{ind}(uC_{\phi})=-1.$

Acknowledgments

The author wishes to thank his doctoral advisor Harish Chandra of the
Department of Mathematics, Banaras Hindu University for helpful discussions
and comments.Thanks are also due to C.S.I.R. New Delhi for providing financial
assistance through NET-JRF with grant  09/013(0891)/2019-EMR-I.

## References

  * [1] T. Bühler, Dietmar A. Salamon. Functional analysis. Vol. 191, American Mathematical Soc., 2018.
  * [2] C. C. Cowen, Linear fractional composition operators on $H^{2}$, Integral equations and operator theory, 11.2 (1988), 151-160.
  * [3] C. C. Cowen, B. Maccluer, Composition operators on spaces of analytic functions, CRC Press. Boca Raton, 1995.
  * [4] E. Flytzanis, L. Kanakis, Measure preserving composition operators, Journal of functional analysis, 73.1, (1987), 113-121.
  * [5] T. W. Gowers, B. Maurey, The unconditional basic sequence problem, Journal of the American Mathematical Society 6, no. 4, (1993), 851-874.
  * [6] M.R. Jabbarzadeh, Weighted composition operators between $L^{p}$-spaces, Bull. Korean Math. Soc. 42, (2005), 369–378.
  * [7] M. R. Jabbarzadeh, E. Pourreza, A note on weighted composition operators on $L^{p}$-spaces, Bull. Iranian Math. Soc. 29 (2003), 47–54.
  * [8] B. O. Koopman, Hamiltonian system and transformations in Hilbert space, Proc. Nat. Acad. Sci., USA 17(1931), 315-318.
  * [9] G. Köthe, Hebbare lokalkonvexe Räume, Mathematische Annalen, 165(3), (1966), 181-195.
  * [10] R. Kumar, Weighted composition operators between two $L^{p}$-spaces, Mat. Vesnik 61, (2009), 111–118.
  * [11] J. Lampertt, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), 459466.
  * [12] J. Lindenstrauss, On complemented subspaces of m, Israel Journal of Mathematics 5, no. 3, (1967), 153-156.
  * [13] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. Springer-Verlag, Berlin-New York, 1977.
  * [14] F. J. Murray, On complementary manifolds and projections in spaces $L_{p}$ and $l_{p},$ Transactions of the American Mathematical Society 41, no.1(1937), 138-152.
  * [15] E. A. Nordgren, Composition operators, Canad. J. Math., 20 (1968), 442-449.
  * [16] E. A. Nordgren, Composition operators on Hilbert spaces, in Hilbert Space Operators (Proc. Conf., Calif. State Univ., Long Beac Calif., 1977), pp. 3763, Lecture Notes in Math., 693, Springer, Berlin, (1978).
  * [17] A. Pełczyński, Projections in certain Banach spaces, Studia Mathematica 2, no. 19 (1960), 209-228.
  * [18] R. S. Phillips, On linear transformations, Transactions of the American Mathematical Society, 48(3)(1940), 516-541.
  * [19] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22.
  * [20] H.P. Rosenthal, On complemented and quasi-complemented subspaces of quotients of $C(S)$ for Stonian S. Proceedings of the National Academy of Sciences, 60(4), (1968), pp.1165-1169.
  * [21] W. Rudin, Functional analysis 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York (1991).
  * [22] J. V. Ryff, Subordinate $H^{p}$ functions, Duke J. Math. 33(1966)347-354.
  * [23] R. K. Singh, J. S. Manhas, Composition operators on function spaces, North-Holland, New York, 1993.
  * [24] A. R. Sourour, The isometries of $L_{p}(\Omega,X),$ Journal of Functional Analysis 30, no. 2 (1978): 276-285.
  * [25] H. Takagi, Compact weighted composition operators on $L_{,}p$ Proceedings of the American Mathematical Society 116, no. 2 (1992) 505-511.