# Quantum clock in the projection evolution formalism

A. Góźdź email<EMAIL_ADDRESS>ORCID: 0000-0003-4489-5136
Institute of Physics
Maria Curie–Skłodowska University in Lublin, Poland  M. Góźdź email:
<EMAIL_ADDRESS>ORCID: 0000-0003-4958-8880 Institute of Computer
Science
Maria Curie–Skłodowska University in Lublin, Poland

###### Abstract

Using the projection evolution (PEv) approach, time can be included in the
quantum mechanics as an observable. Having the time operator, it is possible
to explore the temporal structure of various quantum events. In the present
paper we discuss the possibility of constructing a quantum clock, which
advances in time during its quantum evolution, in each step having some
probability to localize itself on the time axis in the new position. We
propose a working two-state model as the simplest example of such a clock.

##  1 Introduction

Time is one of the most important feature of our physical world. Its
measurement is a fundamental procedure not only for physics. At every step of
technical development there are projects of investigation leading to
construction of better and better clocks which are able to measure extremaly
small time intervals [34, 33, 25, 7].

In the book [36] three types of time are discussed: time as a parameter,
dynamical time and time as a quantum observable. The most consistent with
quantum mechanics is the last concept, i.e., time as a quantum observable
considered on the same footing as the other position operators. In physics,
especially in the quantum regime, one requires more and more precise
measurements of time to better understand and predict evolution of such
systems. On the other hand, to measure very short time intervals one needs to
construct a clock which, in fact, is a specific quantum system working
according to quantum rules and, in addition, is often (weakly) coupled to the
measured system.

For many years time was treated in physics as a universal parameter
enumerating evolution of physical systems. The special and general relativity
change this notion substantially. In quantum mechanics, however, especially
after the publication of the Pauli theorem [39, 40] time is still considered
as an evolution parameter.

An overview of the role of time in quantum mechanics can be found, among
others, in [31, 45, 3, 41, 2, 42, 29, 9, 16, 27, 28, 1, 14, 43, 38, 5, 35, 19,
8, 37, 26, 13, 18, 11, 10, 6, 15, 4, 22, 23, 17, 30, 44, 24].

In this paper we treat time to be a component of a composite quantum
observable of the spacetime position. To be consistent, we use the projection
evolution model (PEv) which recent version can be found in [21, 20]. These
references present also more extended introduction to the problem and the
corresponding bibliography.

According to the PEv approach quantum time and space positions should be
considered on the same footing. However, one needs to realize that we have
direct access to the space position observable but only indirect access to the
time position observable on the time axis. The latter requires special
measuring devices called clocks. A proper clock definition is vital for
investigating the structure of the real spacetime.

##  2 Projection evolution

For readers’ convenience we summarize the main points of the PEv approach.

The main assumption of the PEv evolution model is the changes principle which
states that:

The evolution of a system is a random process caused by spontaneous changes in
the Universe. These spontaneous changes are primary processes in the Universe.

In this approach the changes in the quantum state space happen according to a
probability distribution which is dictated by the properties of the Universe
and its subsystems. A very important attribute of our Universe is the
spacetime itself, which should emerge from the quantum state space. In this
article we consider the flat spacetime.

Every single step of the evolution describes the state of a physical subsystem
(possibly the Universe as a whole) in this state space. It means that the
evolution is not driven by time, which is a part of the system’s description
at a given evolution step, but it is driven by an extra parameter $\tau$. This
parameter is not time; it belongs to a linearly ordered set with no aditional
structure required.

In the following we are using $\tau$ which can be represented by integers
$\mathbb{Z}$. This allows to order quantum events and to have the notion of
the predecessor and the successor in any set of physical events. As a
consequence one may expect the existence of a kind of pseudo-causality based
on the ordering of the quantum events, which leads to the causality principle
in the case of macroscopic physical systems. An additional, very important
feature of the PEv approach is that this idea does not need the spacetime as
the background, it is background free. The spacetime is “generated” by the
spacetime position observable.

In the projection evolution formalism we propose to use the generalized form
of the Lüders [32] type of the projection postulate as an evolution principle.
It allows to reproduce, as a special case, the standard unitary evolution
represented by relativistic and non-relativistic evolution equations.

In the following we introduce the evolution operators which can characterize a
given physical system. They are formally responsible for quantum evolution of
this object. In this paper we are using the evolution operators represented by
an orthogonal resolution of unity:

$\displaystyle\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})^{\dagger}=\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n}),$ $\displaystyle\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n}^{\prime})=\delta_{\nu_{n}\nu_{n}^{\prime}}\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n}),$ $\displaystyle\sum_{\nu_{n}\in{\cal
Q}_{n}}\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})=\hat{1\kern-4.5pt1}\,,$ (1)

where $\hat{1\kern-4.5pt1}$ denotes the unit operator and $\nu_{n}\in{\cal
Q}_{n}$ represents a set of quantum numbers describing quantum states.
Different alternatives representing random choices of quantum states are
described by different sets of quantum numbers $\nu_{n}$. Such type of
operators are able to describe a wide class of physical systems.

Assume now, that the vector $|\Phi(\tau_{n-1};\nu_{n-1})\rangle$ represents a
given quantum state at the evolution step $\tau_{n-1}$, where
$\nu_{n-1}\in{\cal Q}_{n-1}$.

The changes principle implies that there exists in the Universe a physical
mechanism, we call it the chooser, which chooses randomly the next state of
the system from the set of states determined by the projection postulates,

$|\Phi(\tau_{n};\nu_{n})\rangle=e^{i\alpha_{n}}\frac{\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})|\Phi(\tau_{n-1};\nu_{n-1})\rangle}{\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})\Phi(\tau_{n-1};\nu_{n-1})\right\|}\,,$ (2)

where the global phase $\alpha_{n}$ can be chosen arbitrarily and
$\left\|v\right\|$ denotes the norm of the vector $v$.

In other words, Eq.(2) determines the set of allowed states to which a
physical system can randomly evolve from the state
$|\Phi(\tau_{n-1};\nu_{n-1})\rangle$. To fully describe this stochastic
process, one needs to know the probability distribution for getting a given
state in the next step of the evolution. It is given by:

$\mathrm{pev}\kern-2.0pt\left(\nu_{n-1}\to\nu_{n}\right)=\langle\Phi(\tau_{n-1};\nu_{n-1})|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})|\Phi(\tau_{n-1};\nu_{n-1})\rangle\,.$ (3)

It is also useful to introduce a tool which facilitates the construction of
the evolution operators in terms of the projection operators. We have found
that required evolution operators can be obtained from some operators
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}$ which we call generators of the
projection evolution [21].

For a given evolution step $\tau$ the projection evolution generator
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau)$ is defined as a self-
adjoint operator which spectral decomposition gives the orthogonal resolution
of unity representing the required set of the evolution operators.

For example, assuming a discrete spectrum of an evolution generator
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$, the spectral theorem
gives the following relation between $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}$ and the evolution operators $\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}$: $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n})=\sum_{\nu}w_{\nu}\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu)$, where $w_{\nu}$ are the eigenvalues of
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}$. In the case of a continuous
spectrum one needs to use the integral form of the spectral theorem.

The evolution generators allow to use the already known quantum operators to
generate the appropriate evolution operators $\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n},\nu)$.

##  3 Quantum clock in the structureless flat spacetime

In particle, atomic, molecular and some other branches of physics the clock
can be considered to be independent of the physical system under
consideration. In this case one can treat time as the so-called external time.
However, it is only an approximation because the clock is, in fact, a part of
this system. In models in which time is a quantum observable, it should be
considered as a part of an observable describing position in spacetime. The
position in the quantum spacetime is, in turn, one of the attributes of
physical matter.

In this paper we consider the simplest, approximate model which leads to the
spacetime based on the $L^{2}(\mathbb{R}^{4},d^{4}x)$ quantum state space,
i.e., the Hilbert space of square integrable functions with the scalar product
(note the integration over time)
$\langle\psi_{2}|\psi_{1}\rangle=\int_{\mathbb{R}^{4}}d^{4}x\,\psi_{2}(x)^{\star}\,\psi_{1}(x)$
[21, 20].

In a fixed coordinate frame, a possible realization of the spacetime position
operator can be given by the four-vector operator
$\hat{x}^{\mu}:\hat{x}^{\mu}f(x)=x^{\mu}f(x)$, where $\mu=0,1,2,3$.

We understand the term four-vector as a four-component object which transforms
with respect to a given group of the spacetime transformations. In our case we
think about two groups: either the Galilean or the Lorentz group. It means
that using the same denotations we may consider either a non-relativistic or
relativistic four dimensional flat spacetime.

The canonically conjugated observable is the four-momentum operator
$\hat{p}_{\mu}=i\frac{\partial}{\partial x^{\mu}}$. Because time is a quantum
observable $\hat{t}\equiv\hat{x}^{0}$, the temporal momentum $\hat{p}_{0}$ is
the corresponding counterpart. By analogy to the space components of the
momentum operator one can think about $\hat{p}_{0}$ as an observable
representing the measure of motion in the temporal dimension. The sign of
$\hat{p}_{0}$ describes the arrow of time. The temporal momentum can be
measured using different equations of motions which relate it to the spatial
and other properties of the system. For example, the Schrödinger
equation$(\hat{p}_{0}-\hat{H})\psi(x,\zeta)=0$, where the variables $\zeta$
represent some additional degrees of freedom of the quantum system, relates
$\hat{p}_{0}$ to the Hamiltonian $\hat{H}$. One can say the same about the
Klein-Gordon, Dirac and other equations of motion.

In principle, we need some kind of “detectors” measuring positions in
spacetime. They can be represented by POV measures [12]. In the ideal case,
this measure is represented by a sharp observable given by the projection
operators
$M_{X}(x^{0},x^{1},x^{2},x^{3})=|x^{0},x^{1},x^{2},x^{3}\rangle\langle
x^{0},x^{1},x^{2},x^{3}|$, where the vectors $|x^{0},x^{1},x^{2},x^{3}\rangle$
are eigenstates of the position operators $\hat{x}^{\mu}$.

In such models we come across the (1+3)-D position measurement. There is no
problem with performing the 3D spatial measurement, however, we do not have
devices which allow to see the whole time axis, i.e., the past, present and
the future. It seems that our material world is, in most cases, rather well
localized in time and that only a narrow temporal window moving along the time
axis is available for us in the experiments. From this perspective a clock has
to be a part of the quantum system, well localized in time, possibly coupled
to other physical subsystems. Having a reference clock, one can construct
other types of clocks as devices synchronized with this clock. This procedure
allows to introduce the notion of the “external time”, however only for the
systems which are well decoupled from the reference clock.

We define a quantum reference clock as a kind of device, localized on the time
axis and moving with a fixed sign of the average temporal momentum
$\langle\hat{p}_{0}\rangle_{\text{clock state}}$ in spacetime. The sign is
conventional but it defines the arrow of time. The clock localizing itself in
a time interval (temporal window) shows us, usually indirectly by some kind of
an interface (measurement), its position in time. It means that the clock
consists of two subsystems: the proper clock, evolving from one localization
on the time axis to another, and the interface reading off the clock’s time
position.

In the PEv model [20] the average spacetime localization of a physical system
can change only with the change of the evolution parameter $\tau_{n}$. In the
following we denote by $\tau_{n}$, $n=1,2,3,\dots$ the evolution steps of the
proper clock, and by $\tau^{\prime}_{n}$, where
$\tau_{n}<\tau^{\prime}_{n}<\tau_{n+1}$, the evolution steps at which the
clock interface reads its temporal position. It implies that a given quantum
clock at the evolution step $\tau_{n}$ is represented by a set of clock states
$\Phi(\tau_{n};\nu_{n})$. The label $\nu_{n}$ represents a set of quantum
numbers describing both the temporal and other properties of observables
required for the construction of a quantum clock. For a good clock, the
variance of the time operator

$\hat{t}=\int_{\mathbb{R}}dx^{0}M_{T}(x^{0}),$ (4)

where

$M_{T}(x^{0})=\int_{\mathbb{R}^{3}}d\mathbf{x}M_{X}(x^{0},\mathbf{x})$ (5)

projects onto the subspaces of simultaneous events, should be small. More
precisely, the variance should fulfill the following inequality:

$\mathrm{var}\kern-2.0pt\left(\hat{t};\Phi(\tau_{n};\nu_{n})\right)=\langle\Phi(\tau_{n};\nu_{n})|(\hat{t}-\langle\Phi(\tau_{n};\nu_{n})|\hat{t}|\Phi(\tau_{n};\nu_{n})\rangle)^{2}|\Phi(\tau_{n};\nu_{n})\rangle\leq\mathrm{cr}\,,$
(6)

where $\mathrm{cr}$ is a small number and denotes the clock resolution. The
corresponding expectation value of the time operator within the clock states,
$t=\langle\Phi(\tau_{n};\nu_{n})|\hat{t}|\Phi(\tau_{n};\nu_{n})\rangle$, gives
the expected temporal position of the proper clock, i.e., it gives a parameter
$t$ representing the classical time. In the ideal case the clock states are
eigenstates of the time operator with the variance (6) being equal to 0.

In the projection evolution model [20] an important element is the
construction of the evolution operators for the subsequent steps of the
evolution $\tau_{n-1}\to\tau_{n-1}^{\prime}\to\tau_{n}$. It means that we have
to describe the following process: for the evolution step $\tau_{n-1}$ the
proper clock is localized at some time $t_{n-1}$, this position is read off by
the interface at $\tau_{n-1}^{\prime}$ and the proper clock is moving to the
next temporal position at $\tau_{n}$.

In this paper, we propose a set of unitary operations, driven by a random
variable $\xi\in\mathbb{R}_{+}$, transforming the evolution generators, not
states, from the previous to the next step of the evolution process:

$U(\xi,\gamma)=\exp(i(\xi\hat{p}_{0}-\gamma\hat{A}))\,,$ (7)

where $\gamma$ is a coupling constant. The temporal momentum operator
$\hat{p}_{0}$ and the self adjoint reconfiguring operator $\hat{A}$ commute,
i.e., $[\hat{p}_{0},\hat{A}]=0$. The temporal momentum operator $\hat{p}_{0}$
is responsible for the motion along the time axis. The values of the random
variable $\xi$ are related to the internal processes of the clock and to the
influence of the environment on the clock. This leads effectively to a
movement of the clock along the time axis. Direction of the clock motion along
the time axis is determined by the expectation value of the temporal momentum
operator $\hat{p}_{0}$, calculated in the clock states. The reconfiguring
operator $\hat{A}$ changes some internal states of the clock and allows to
define the required clock interface.

Let us denote by $\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$ the
evolution generator of the proper clock at the evolution step $\tau_{n}$. The
unitary transformations (7) form a two-parameter group,
$U(\xi+\xi^{\prime},(k+k^{\prime})\gamma)=U(\xi,k\gamma)U(\xi^{\prime},k^{\prime}\gamma)$,
where $k,k^{\prime}\in\mathbb{Z}$. This feature allows to relate the evolution
generator $\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$ to its
initial form $\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{0})$:

$\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n})=U(\xi_{n},\gamma)\,\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n-1})\,U(\xi_{n},\gamma)^{\dagger}=U(\beta_{n},n\gamma)\,\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})\,U(\beta_{n},n\gamma)^{\dagger}\,,$ (8)

where $\beta_{0}=0$, $\beta_{n}=\sum_{k=1}^{n}\xi_{k}$ and
$\xi_{1},\xi_{2},\dots\xi_{n}$ are subsequent values of the random variable
$\xi$.

To construct the evolution operators $\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\nu_{n})$ from the evolution generator
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$ one needs to find its
spectral decomposition. Assuming discrete spectrum of the generator
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$ one has to solve the
following eigenequation

$\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n})|\psi^{(n)}_{bc}\rangle=w^{(n)}_{b}|\psi^{(n)}_{bc}\rangle\,,$
(9)

where $c$ represents the possible degeneration of the spectrum. The
transformation property (8) implies that it is sufficient to solve the
eigenequation (9) only for $\tau_{0}$ to get all solutions for every
$\tau_{n}$. If $|\psi^{(0)}_{bc}\rangle\equiv|\psi_{bc}\rangle$ are
eigenvectors of $\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{0})$, the
states

$|\psi^{(n)}_{bc}\rangle=U(\beta_{n},n\gamma)|\psi_{bc}\rangle$ (10)

are eigenstates of $\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$.
The eigenvalues $w^{(n)}_{b}=w^{(0)}_{b}\equiv w_{b}$ are independent of $n$.

The distribution of $\xi$ should have a pronounced maximum very close to
zero111In this work we assume $\xi>0$ but the general condition is, that
$\xi\in\mathbb{R}$ has a distribution around zero, assymetric towards $\xi>0$
values. This will result in the net time flow in the positive direction of the
time axis. We will discuss this problem in a subsequent paper.. It implies,
that the next instant of classical time during the evolution should be very
close to the previous one. The probability distribution for $\xi$ is a
parameter of the clock. It seems that this distribution is strongly related to
both the construction of the clock, and the structure of spacetime.

The second component of the clock is an interface allowing to read the clock.
The interface is therefore a measuring device and its full evolution operator
has to contain the appropriate projection operators.

The projection evolution generators simplify the construction of the required
evolution operators for a quantum reference clock. This method will be used in
the next section.

###  3.1 The proper clock

For a given observer the quantum spacetime splits into time and 3D position
space. In the following, to present the clock idea we choose a non-covariant
description of a proper clock and its interface.

Because we treat time on the same footing as the other coordinates one expects
that every interaction $V_{int}(x-y)$, where
$x=(t_{x},\mathbf{x}),y=(t_{y},\mathbf{y})$ are spacetime points, depends not
only on the positions in space but also on positions in time [21]. One can say
the same about the effective potentials $V(t,\mathbf{x})$. According to
observations, in a wide range of its density, the physical matter is well
localized in time. This supports the parametric time as a good approximation.
On the other hand, what is even more important, this feature also suggests
that the interaction in the temporal dimension has a very short range. In the
normal density matter the temporal parts of interactions among many particles
lead to a temporal mean-field effective potential operator $V_{T}(\hat{t})$.
One can find similar effect in nuclear physics, where the mean-field approach
to a short range nuclear interaction is a good approximation.

In the following, we present a nonrelativistic proper clock described by a
modified Schrödinger type of quantum motion in the structureless flat
spacetime [21]. Relativistic versions of the clock can be constructed in a
similar way.

Following the discussion about the evolution generators described in [21], the
initial projection evolution generator $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})$ for a proper clock is postulated in the following
form:

$\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})=\hat{p}_{0}-\hat{H}-\frac{\hat{p}_{0}^{2}}{2m_{T}}-V_{T}(\hat{t}),$
(11)

where the temporal momentum operator is $\hat{p}_{0}=i\frac{\partial}{\partial
t}$, the Hamiltonian $\hat{H}$ acts on functions of the position variables
$\mathbf{x}\in\mathbb{R}^{3}$ and potentially on some intrinsic variables,
i.e., it does not depend on time. The signs in front of the temporal part of
the generator (11) are chosen to keep the coefficient $m_{T}$ positive. By
analogy to the spatial kinetic term in the Hamiltonian $\hat{H}$ the term
$m_{T}$ can be called “the temporal inertia”.

The proper clock is determined by the generator, which describes the physical
system localized in time, and the Hamiltonian $\hat{H}$ which allows to build
an appropriate interface in its state space. The proper clock is localized on
the time axis by the effective temporal potential operator $V_{T}(\hat{t})$.

Using the unitary operator (7) one gets the evolution generator for any
arbitrary evolution step $\tau_{n}$:

$\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n})=U(\beta_{n},n\gamma)\,\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})\,U(\beta_{n},n\gamma)^{\dagger}=\hat{p}_{0}-\hat{H}^{(n)}-\frac{\hat{p}_{0}^{2}}{2m_{T}}-V^{(n)}_{T}(\hat{t})\,,$
(12)

where the potential localizing the clock in time is given by

$V_{T}^{(n)}(\hat{t})=V_{T}\left(\hat{t}-\beta_{n}\hat{1\kern-4.5pt1}\right)\,.$
(13)

The modified Hamiltonian has the following form:

$\hat{H}^{(n)}=\exp\left(-in\gamma\,\hat{A}\right)\hat{H}\exp\left(+in\gamma\,\hat{A}\right)\,.$
(14)

According to the definition of the projection evolution generators we have to
construct the spectral decomposition of $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n})$. For this purpose one needs to look for the
eigenfunctions of $\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{0})$. In
our case we can find them by separating the variables,

$\psi_{\lambda\kappa\nu\mu}(t,\mathbf{x})=\chi_{\lambda\kappa}(t)\phi_{\nu\mu}(\mathbf{x})\,,$
(15)

where $\chi_{\lambda\kappa}(t):=e^{-im_{T}t}f_{\lambda\kappa}(t)$. The indices
$\kappa$ and $\mu$ indicate a possible degeneration of the solutions. In the
following we assume that the temporal functions $\chi_{\lambda\kappa}(t)$
represent the states which, on average, move in the positive direction of
time. This means that the temporal functions $\chi_{\lambda\kappa}(t)$ are the
states for which the expectation value of the temporal momentum operator,

$\langle\chi_{\lambda\kappa}|\hat{p}_{0}|\chi_{\lambda\kappa}\rangle=m_{T}+\int_{R}dt\,f_{\lambda\kappa}(t)^{\star}\hat{p}_{0}f_{\lambda\kappa}(t),$
(16)

is a positive number.

The functions $\phi_{\nu\mu}(\mathbf{x})$ denote the eigenvectors of the
Hamiltonian $\hat{H}$,

$\hat{H}\phi_{\nu\mu}(\mathbf{x})=E_{\nu}\phi_{\nu\mu}(\mathbf{x})\,,$ (17)

where $E_{\nu}$ are the eigenvalues of $\hat{H}$.

The function $f_{\lambda\kappa}(t)$ solves the equation

$\left(\frac{\hat{p}_{0}^{2}}{2m_{T}}+V_{T}(\hat{t})\right)f_{\lambda\kappa}(t)=\left(\frac{m_{T}}{2}-\epsilon_{\lambda}^{(T)}\right)f_{\lambda\kappa}(t),$
(18)

while together with the exponential function it constitutes the full solution
of the equation

$\left(\hat{p}_{0}-\frac{\hat{p}_{0}^{2}}{2m_{T}}-V_{T}(\hat{t})\right)\chi_{\lambda\kappa}(t)=\epsilon_{\lambda}^{(T)}\chi_{\lambda\kappa}(t)\,.$
(19)

This allows one to write the eigenequation for $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})$ as

$\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})\psi_{\lambda\kappa\nu\mu}(t,\mathbf{x})=w_{b(\lambda,\nu)}\psi_{\lambda\kappa\nu\mu}(t,\mathbf{x}),$
(20)

with $\psi_{\lambda\kappa\nu\mu}$ given by Eq. (15) and

$w_{b(\lambda,\nu)}=\epsilon^{(T)}_{\lambda}-E_{\nu}\,,$ (21)

where the function labelling the eigenvalues of $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{0})$ fulfil the condition:
$b(\lambda^{\prime},\nu^{\prime})=b(\lambda,\nu)\Leftrightarrow
w_{b(\lambda^{\prime},\nu^{\prime})}=w_{b(\lambda,\nu)}$.

Collecting the above partial solutions the eigenstates of
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{n})$ for a given evolution
step $\tau_{n}$ are given by the vectors

$\psi^{(n)}_{\lambda\kappa\nu\mu}(t,\mathbf{x})=U(\beta_{n},n\gamma)\psi_{\lambda\kappa\nu\mu}(t,\mathbf{x})=\chi^{(n)}_{\lambda\kappa}(t)\phi^{(n)}_{\nu\mu}(\mathbf{x})\,,$
(22)

where $\chi^{(n)}_{\lambda\kappa}(t):=\chi_{\lambda\kappa}(t-\beta_{n})$ and
$\phi^{(n)}_{\nu\mu}(\mathbf{x}):=\exp(-in\gamma\hat{A})\phi_{\nu\mu}(\mathbf{x})$.

The evolution operators generated by $\mathsf{W}\kern-1.0pt\hbox
to0.0pt{\hss$-$}(\tau_{n})$ are the projections onto eigenspaces of the
evolution generator,

$\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};w)=\sum_{\lambda\nu}\delta(w=\epsilon^{(T)}_{\lambda}-E_{\nu})\sum_{\kappa\mu}|\psi^{(n)}_{\lambda\kappa\nu\mu}\rangle\langle\psi^{(n)}_{\lambda\kappa\nu\mu}|=U(\beta_{n},n\gamma)\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{0};w)U(\beta_{n},n\gamma)^{\dagger}\,.$ (23)

The Kronecker’s type function is defined as:
$\delta(w=\epsilon^{(T)}_{\lambda}-E_{\nu})=1$ if
$w=\epsilon^{(T)}_{\lambda}-E_{\nu}$, otherwise it is equal to 0. The sums
will be substituted by some integrals in the case of a continuous spectrum.

###  3.2 The clock interface

To read the change of the internal state of the proper clock, we introduce the
interface evolution operators denoted by $\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\tau_{n}^{\prime},\nu)$, where
$\tau_{n}<\tau_{n}^{\prime}<\tau_{n+1}$ describes an intermediate event
between $\tau_{n}$ and $\tau_{n+1}$. They form an orthogonal resolution of
unity projecting onto eigenstates of the Hamiltonian $\hat{H}$ showing in
which eigenstate of $\hat{H}$ the clock actually is. Changes of the states of
the Hamiltonian along the time axis represent the clock ticks. The clock
interface should disturb neither the proper clock localization in time nor its
movement along the time axis but it should show in which eigenstate of the
Hamiltonian the proper clock is.

Let us denote an orthonormal basis
$g_{s\nu\mu}(t,x)=e_{s}(t)\phi_{\nu\mu}(\mathbf{x})$ in the state space. We
obtain the required properties by assuming that

$\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\tau_{n}^{\prime},\nu)=\sum_{\mu}|\phi_{\nu\mu}\rangle\left(\sum_{s}|e_{s}\rangle\langle
e_{s}|\right)\langle\phi_{\nu\mu}|\equiv\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\nu)\,,$ (24)

where $\left(\sum_{s}|e_{s}\rangle\langle
e_{s}|\right)=\hat{1\kern-4.5pt1}_{T}$ is the unit operator in the time
domain. The projection operators (24) are independent of $\tau_{n}^{\prime}$,
i.e., we keep the same interface for every evolution step.

###  3.3 The clock

The evolution operators described above allow to construct different reference
clocks using different forms of the operators: $\hat{A},\hat{H}$ and
$V_{T}(\hat{t})$.

The clock starts from any arbitrary quantum state $|\Phi_{0}\rangle$. The
evolution operator $\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{0};u_{0})$ prepares the initial state of the clock

$\Phi(\tau_{0};u_{0};t,\mathbf{x})=\frac{\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{0};u_{0})\Phi_{0}(t,\mathbf{x})}{\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{0};u_{0})\Phi_{0}\right\|}\,,$ (25)

where $u_{0}$ represents, according to the PEv formalism, a randomly chosen
eigenvalue $w_{b(\lambda,\nu)}$ of the evolution generator
$\mathsf{W}\kern-1.0pt\hbox to0.0pt{\hss$-$}(\tau_{0})$.

The state (25) is read by the interface

$\Phi(\tau_{0}^{\prime};\sigma_{0},u_{0};t,\mathbf{x)}=\frac{\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{0})\Phi(\tau_{0};u_{0};t,\mathbf{x})}{\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{0})\Phi(\tau_{0};u_{0})\right\|}\,,$ (26)

where $\sigma_{0}$ represents a randomly chosen eigenvalue $E_{\sigma_{0}}$ of
the Hamiltonian $\hat{H}$. The simplest choice of the initial state is any
eigenstate of the evolution generator (15),

$\Phi_{0}(t,\mathbf{x})=\psi_{\lambda_{0}\kappa_{0}\nu_{0}\mu_{0}}(t,\mathbf{x})\equiv\chi_{\lambda_{0}\kappa_{0}}(t)\phi_{\nu_{0}\mu_{0}}(\mathbf{x})\,.$
(27)

Then
$\Phi(\tau_{0}^{\prime};\sigma_{0},u_{0};t,\mathbf{x})=\Phi_{0}(t,\mathbf{x})$,
where $\sigma_{0}=\nu_{0}$ and $u_{0}=\epsilon^{T}_{\lambda_{0}}-E_{\nu_{0}}$.

To simplify the notation let us denote by $(\sigma,u)_{n}$ the sequence of
quantum numbers which gives a possible evolution path

$(\sigma,u)_{n}=(\sigma_{n},u_{n},\sigma_{n-1},u_{n-1},\dots,\sigma_{0},u_{0})=(\sigma_{n},u_{n},(\sigma,u)_{n-1}).$

The subsequent cycles of the clock are described by the following recurrence
relations

$\Phi(\tau_{n};u_{n},(\sigma,u)_{n-1};t,x)=\frac{\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1};t,x)}{\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1})\right\|}\,,$
(28)

and

$\displaystyle\Phi(\tau_{n}^{\prime};(\sigma,u)_{n};t,x)$ $\displaystyle=$
$\displaystyle\frac{\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\Phi(\tau_{n};u_{n},(\sigma,u)_{n-1};t,x)}{\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\Phi(\tau_{n};u_{n},(\sigma,u)_{n-1})\right\|}$
(29) $\displaystyle=$ $\displaystyle\frac{\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1};t,x)}{\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1})\right\|}\,,$

where the probabilities of choosing next states are given by (3), i.e., by the
denominators of (28) and (29), respectively.

The action of the evolution operator $\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})$ on any function
$\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1};t,x)$ can be written as

$\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1};t,x)=\mathcal{G}((\sigma,u)_{n};\lambda\kappa\mu^{\prime})\chi^{(n)}_{\lambda\kappa}(t)\phi_{\sigma_{n}\mu^{\prime}}(\mathbf{x})\,,$
(30)

where the expansion coefficients, taking into account (23) and (24), read

$\mathcal{G}((\sigma,u)_{n};\lambda\kappa\mu^{\prime})=\sum_{\nu\mu}\delta(u_{n}=\epsilon^{(T)}_{\lambda}-E_{\nu})\,\langle\phi_{\sigma_{n}\mu^{\prime}}|\phi^{(n)}_{\nu\mu}\rangle\,\langle\psi^{(n)}_{\lambda\kappa\nu\mu}|\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1})\rangle\,.$
(31)

Because the functions
$\chi^{(n)}_{\lambda\kappa}(t)\phi_{\sigma_{n}\mu^{\prime}}(\mathbf{x})$ and
$\chi^{(n)}_{\lambda^{\prime}\kappa^{\prime}}(t)\phi_{\sigma_{n}^{\prime}\mu^{\prime\prime}}(\mathbf{x})$
are orthonormal, the coefficient $N(\tau_{n};(\sigma,u)_{n})$ which normalizes
Eq. (30) can be expressed by the coefficients (31)

$N(\tau_{n};(\sigma,u)_{n})^{2}=\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1})\right\|^{2}=\sum_{\lambda\kappa\mu^{\prime}}|\mathcal{G}((\sigma,u)_{n};\lambda\kappa\mu^{\prime})|^{2}\,.$
(32)

Using the coefficients $\mathcal{G}$ the clock state (29) can be rewriten as

$\Phi(\tau_{n}^{\prime};(\sigma,u)_{n};t,x)=N(\tau_{n};(\sigma,u)_{n})^{-1}\sum_{\lambda\kappa\mu^{\prime}}\mathcal{G}((\sigma,u)_{n};\lambda\kappa\mu^{\prime})\chi^{(n)}_{\lambda\kappa}(t)\phi_{\sigma_{n}\mu^{\prime}}(\mathbf{x})\,.$
(33)

The definition (31) and Eqs. (32) and (33) lead to the recurrence relation for
the $\mathcal{G}$-coefficients

$\displaystyle\mathcal{G}((\sigma,u)_{n};\lambda\kappa\mu^{\prime})=N(\tau_{n-1};(\sigma,u)_{n-1})\sum_{\nu_{1}\mu_{1}}\delta(u_{n}=\epsilon^{(T)}_{\lambda}-E_{\nu_{1}})\langle\phi_{\sigma_{n}\mu^{\prime}}|\phi^{(n)}_{\nu_{1}\mu_{1}}\rangle$
$\displaystyle\sum_{\lambda_{2}\kappa_{2}\mu_{2}^{\prime}}\mathcal{G}((\sigma,u)_{n-1};\lambda_{2}\kappa_{2}\mu_{2}^{\prime})\langle\chi^{(n)}_{\lambda\kappa}|\chi^{(n-1)}_{\lambda_{2}\kappa_{2}}\rangle\langle\phi^{(n)}_{\nu_{1}\mu_{1}}|\phi_{\sigma_{n-1}\mu_{2}^{\prime}}\rangle.$
(34)

Having the $\mathcal{G}$-coefficients, one can calculate the transition
probability between the subsequent readings off the interface. The square of
the normalization factor represents this probability:

$\displaystyle\mathrm{pev}\kern-2.0pt\left(\tau_{n-1}^{\prime}\to\tau_{n}\to\tau_{n}^{\prime}\right)$
$\displaystyle=|\langle\Phi(\tau_{n}^{\prime};(\sigma,u)_{n})|\Phi(\tau_{n};u_{n},(\sigma,u)_{n-1})\rangle\langle\Phi(\tau_{n};u_{n},(\sigma,u)_{n-1})|\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1})\rangle|^{2}$
$\displaystyle=\left\|\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};u_{n})\Phi(\tau_{n-1}^{\prime};(\sigma,u)_{n-1})\right\|^{2}=N(\tau_{n};(\sigma,u)_{n})^{2}\,.$
(35)

In our model, the clock shifts subsequently on the time axis about the random
variable $\xi$. However, our knowledge about this shift is supplied only by
the clock interferface. It is important to calculate the expectation value of
the time operator within the states corresponding to reading off the
interface. For this purpose, let us denote by $t_{n}$ the average value of the
time operator in the state (29), i.e.,
$t_{n}=\langle\Phi(\tau_{n}^{\prime};(\sigma,u)_{n})|\hat{t}|\Phi(\tau_{n}^{\prime};(\sigma,u)_{n})\rangle$.
Using (29) and the following matrix elements:

$\displaystyle\langle\chi^{(n)}_{\lambda_{1}\kappa_{1}}|\hat{t}|\chi^{(n)}_{\lambda_{2}\kappa_{2}}\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}dt\,f_{\lambda_{1}\kappa_{1}}(t-\beta_{n})^{\star}tf_{\lambda_{2}\kappa_{2}}(t-\beta_{n})$
(36) $\displaystyle=$
$\displaystyle\delta_{\lambda_{1}\lambda_{2}}\delta_{\kappa_{1}\kappa_{2}}\beta_{n}+\int_{\mathbb{R}}dt\,f_{\lambda_{1}\kappa_{1}}(t)^{\star}\,t\,f_{\lambda_{2}\kappa_{2}}(t)$
$\displaystyle=$
$\displaystyle\delta_{\lambda_{1}\lambda_{2}}\delta_{\kappa_{1}\kappa_{2}}\beta_{n}+\langle\chi_{\lambda_{1}\kappa_{1}}|\hat{t}|\chi_{\lambda_{2}\kappa_{2}}\rangle,$

the required expectation value reads

$\displaystyle
t_{n}=\beta_{n}+\left(\frac{1}{N(\tau_{n};(\sigma,u)_{n})}\right)^{2}\Bigg{[}\sum_{\lambda_{1}\kappa_{1}}\sum_{\lambda_{2}\kappa_{2}}$
$\displaystyle\sum_{\mu^{\prime}}\mathcal{G}((\sigma,u)_{n};\lambda_{1}\kappa_{1},\mu^{\prime})^{\star}\mathcal{G}((\sigma,u)_{n};\lambda_{2}\kappa_{2},\mu^{\prime})\Bigg{]}\langle\chi_{\lambda_{1}\kappa_{1}}|\hat{t}|\chi_{\lambda_{2}\kappa_{2}}\rangle\,.$
(37)

The ideal clock should show the value $t_{n}=\beta_{n}$, i.e., the place where
the clock is localized on the time axis. The interface, however, is also a
quantum device and acts randomly. There is therefore a finite probability that
the proper clock moves to the next position on the time axis, but the
interface does not change its state, i.e., the “hand of the clock” does not
advance forward. This implies that for a good clock the second term in (3.3)
should be always close to zero.

##  4 A two-state clock

As an example, we build a schematic clock for which there is no summation over
$\lambda,\mu,\kappa$ and $\nu$ in Eq. (23).

This is often the case when both (17) and (18) have discrete nondegenerate
spectra, which allows to rewrite the evolution operators (23) in a simpler
form:

$\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\lambda\nu)=|\psi^{(n)}_{\lambda\nu}\rangle\langle\psi^{(n)}_{\lambda\nu}|.$
(38)

We express the clock Hamiltonian describing the clock interface by making use
of the spectral theorem,

$\hat{H}=\sum_{\nu=0}^{\infty}E_{\nu}|\phi_{\nu}\rangle\langle\phi_{\nu}|\,.$
(39)

The reconfiguration operator is taken in the following form:

$\hat{A}=|\phi_{0}\rangle\langle\phi_{1}|+|\phi_{1}\rangle\langle\phi_{0}|\,.$
(40)

This means that our clock interface is oscillating between two states,
$|\phi_{0}\rangle$ and $|\phi_{1}\rangle$, and the corresponding evolution
operators, reading the actual state of the interface, are given by

$\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma)=|\phi_{\sigma}\rangle\langle\phi_{\sigma}|,\quad\sigma=0,1.$
(41)

The state of the proper clock and its interface at the evolution step
$\tau^{\prime}_{n-1}$ is
$|\Phi(\tau^{\prime}_{n-1};(\sigma,\lambda,\nu)_{n-1})\rangle$. To get to the
next step of the evolution one needs to calculate

$\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\lambda_{n},\nu_{n})|\Phi(\tau^{\prime}_{n-1};(\sigma,\lambda,\nu)_{n-1})\rangle=\langle\psi^{(n)}_{\lambda_{n}\nu_{n}}|\Phi(\tau^{\prime}_{n-1};(\sigma,\lambda,\nu)_{n-1})\rangle|\psi^{(n)}_{\lambda_{n}\nu_{n}}\rangle$
(42)

and

$\displaystyle\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}_{I}(\sigma_{n})\mathsf{E}\kern-1.0pt\hbox
to0.0pt{\hss$|$}(\tau_{n};\lambda_{n},\nu_{n})|\Phi(\tau^{\prime}_{n-1};(\sigma,\lambda,\nu)_{n-1})\rangle=$
$\displaystyle\langle\phi_{\sigma_{n}}|\phi^{(n)}_{\nu_{n}}\rangle\langle\psi^{(n)}_{\lambda_{n}\nu_{n}}|\Phi(\tau^{\prime}_{n-1};(\sigma,\lambda,\nu)_{n-1})\rangle|\chi^{(n)}_{\lambda_{n}}\rangle|\phi_{\sigma_{n}}\rangle\,.$
(43)

Because in the formulas (28) and (29) the common multiplication factors in the
nominators and denominators can be simplified, and overall phases are
unimportant, the resulting clock states, after simplification of notation, are
represented by the vectors

$\displaystyle|\Phi(\tau_{n};(\sigma,\lambda,\nu)_{n})\rangle=|\Phi(\tau_{n};\lambda_{n}\nu_{n})\rangle\equiv|\psi^{(n)}_{\lambda_{n}\nu_{n}}\rangle,$
(44)
$\displaystyle|\Phi(\tau^{\prime}_{n};(\sigma,\lambda,\nu)_{n})\rangle=|\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})\rangle\equiv|\chi^{(n)}_{\lambda_{n}}\rangle|\phi_{\sigma_{n}}\rangle\,.$
(45)

The clock states (44) and (45) allow to calculate the transition probability
between the states representing two subsequent readings from the clock
interface. The probability of passing the evolution path

$(\tau^{\prime},\sigma,\lambda,\nu)_{n-1}\to(\tau_{n},\lambda_{n},\nu_{n},(\tau,\sigma,\lambda,\nu)_{n-1})\to(\tau^{\prime},\sigma,\lambda,\nu)_{n}$

according to the expression (3.3) reads

$\mathrm{pev}^{\prime}=|\langle\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})|\Phi(\tau_{n};\lambda_{n},\nu_{n})\rangle\langle\Phi(\tau_{n};\lambda_{n},\nu_{n})|\Phi(\tau^{\prime}_{n-1};\sigma_{n-1},\lambda_{n-1})\rangle|^{2}\,.$
(46)

Using relations (42) and (43), one can cast (46) in the form

$\mathrm{pev}^{\prime}=|\langle\chi_{\lambda_{n}}^{(n)}|\chi_{\lambda_{n-1}}^{(n-1)}\rangle|^{2}|\langle\phi_{\sigma_{n}}|\phi_{\nu_{n}}^{(n)}\rangle\langle\phi_{\nu_{n}}^{(n)}|\phi_{\sigma_{n-1}}\rangle|^{2}.$
(47)

For the reconfiguration operator (40), the $\hat{A}$-dependent part of the
$U(\xi,\gamma)$ operator (7) is given by

$\exp(-i\gamma\hat{A})=\hat{1}-i\sin\gamma\hat{A}+\left(\cos\gamma-1\right)\hat{A}^{2},$
(48)

where $\hat{1}$ is the unit operator in the full state space, while $\hat{A}$
acts in its two-dimensional subspace only. It follows that the scalar product
reads

$\langle\phi_{\sigma_{n}}|\phi_{\nu_{n}}^{(n)}\rangle=\langle\phi_{\sigma_{n}}|\exp(-i\gamma\hat{A})|\phi_{\nu_{n}}\rangle=\delta_{\sigma_{n}\nu_{n}}\cos\gamma-i(\delta_{\sigma_{n}1}\delta_{\nu_{n}0}+\delta_{\sigma_{n}0}\delta_{\nu_{n}1})\sin\gamma,$
(49)

which allows to calculate the transition amplitudes:

$\displaystyle\langle\phi_{1}|\phi_{1}^{(n)}\rangle\langle\phi_{1}^{(n)}|\phi_{0}\rangle=\frac{i}{2}\sin(2\gamma),$
(50)
$\displaystyle\langle\phi_{1}|\phi_{0}^{(n)}\rangle\langle\phi_{0}^{(n)}|\phi_{0}\rangle=-\frac{i}{2}\sin(2\gamma),$
(51)
$\displaystyle\langle\phi_{0}|\phi_{1}^{(n)}\rangle\langle\phi_{1}^{(n)}|\phi_{0}\rangle=\sin^{2}\gamma,$
(52)
$\displaystyle\langle\phi_{0}|\phi_{0}^{(n)}\rangle\langle\phi_{0}^{(n)}|\phi_{0}\rangle=\cos^{2}\gamma,$
(53)
$\displaystyle\langle\phi_{0}|\phi_{1}^{(n)}\rangle\langle\phi_{1}^{(n)}|\phi_{1}\rangle=-\frac{i}{2}\sin(2\gamma),$
(54)
$\displaystyle\langle\phi_{0}|\phi_{0}^{(n)}\rangle\langle\phi_{0}^{(n)}|\phi_{1}\rangle=\frac{i}{2}\sin(2\gamma),$
(55)
$\displaystyle\langle\phi_{1}|\phi_{1}^{(n)}\rangle\langle\phi_{1}^{(n)}|\phi_{1}\rangle=\cos^{2}\gamma,$
(56)
$\displaystyle\langle\phi_{1}|\phi_{0}^{(n)}\rangle\langle\phi_{0}^{(n)}|\phi_{1}\rangle=\sin^{2}\gamma.$
(57)

To register the time passing is equivalent for the clock to change the state
from $\sigma_{n}=1$ to $\sigma_{n}=0$ or from $0$ to $1$, i.e., the clock
“clicks”. Using Eq. (47) and the appropriate amplitudes (50)–(57) gives the
probabilities that the clock “clicks” in the form

$\displaystyle
p(\lambda_{n}):=\mathrm{Prob}(\sigma_{n-1}=1\to\sigma_{n}=0)=\mathrm{Prob}(\sigma_{n-1}=0\to\sigma_{n}=1)$
$\displaystyle=\frac{1}{2}\sin^{2}(2\gamma)|\langle\chi_{\lambda_{n}}^{(n)}|\chi_{\lambda_{n-1}}^{(n-1)}\rangle|^{2}\leq\frac{1}{2}\,.$
(58)

The probability of the opposite event, i.e., that the clock changes its state
without the “click” is $1-p(\lambda_{n})$.

The probability that the clock “clicks” exactly in the $\ell$-th step is given
by

$p(\lambda_{\ell})\prod_{k=1}^{\ell-1}(1-p(\lambda_{k}))\,.$ (59)

The probability amplitude
$\langle\chi_{\lambda_{n}}^{(n)}|\chi_{\lambda_{n-1}}^{(n-1)}\rangle$ can be
expressed as

$\langle\chi_{\lambda_{n}}^{(n)}|\chi_{\lambda_{n-1}}^{(n-1)}\rangle=e^{-im_{T}\xi_{n}}\int_{R}dt\,f_{\lambda_{n}}(t-\xi_{n})^{\star}f_{\lambda_{n-1}}(t)\,.$
(60)

The elementary step of movement along the time axis should be extremely small,
i.e., the expectation value of $\langle\xi_{k}\rangle\approx 0$, for every
step $k$ of the clock evolution. This implies that
$|\langle\chi_{\lambda_{n}}^{(n)}|\chi_{\lambda_{n-1}}^{(n-1)}\rangle|^{2}=1$
with a very good approximation and one can write $p(\lambda_{\ell})\approx
p=\frac{1}{2}\sin^{2}(2\gamma)$.

Let us calculate the average value of the time operator $\hat{t}$. It is given
by

$\langle\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})|\hat{t}|\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})\rangle=\beta_{n}+\langle\chi_{\lambda}|\hat{t}|\chi_{\lambda}\rangle.$
(61)

Since $\chi_{\lambda}=e^{-i\leavevmode\nobreak\ m_{T}t}f_{\lambda}(t)$, we
have

$\langle\chi_{\lambda}|\hat{t}|\chi_{\lambda}\rangle=\int_{\mathbb{R}}dt\
t|f_{\lambda}(t)|^{2}.$ (62)

A good clock is characterized by the term (62) being as small as possible. If
$f_{\lambda}(-t)=e^{i\alpha}f_{\lambda}(t)$ this term vanishes. It follows,
that after $n$ steps of evolution, the localization of the clock on the time
axis is close to $\beta_{n}$.

In a similar way one may obtain the average value of the temporal momentum
operator,

$\langle\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})|\hat{p}_{0}|\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})\rangle=m_{T}+\int_{\mathbb{R}}dtf_{\lambda}(t)^{\star}\hat{p_{0}}f_{\lambda}(t).$
(63)

Since $\hat{p}_{0}=i\frac{\partial}{\partial t}$, the condition
$f_{\lambda}(-t)=e^{i\alpha}f_{\lambda}(t)$ implies that the second term in
Eq. (63) vanishes, which results in a constant arrow of time:

$\langle\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})|\hat{p}_{0}|\Phi(\tau^{\prime}_{n};\sigma_{n},\lambda_{n})\rangle=m_{T}>0.$
(64)

To estimate how long, on average, do we have to wait until a “click” appears,
we calculate the variance of $\ell$. The normalized probability that a single
“click” takes place at any of the steps, during an $\ell$-step evolution, is
given by

$\mathrm{Prob}(\mathrm{click},\ell)=\frac{\ell
p(1-p)^{\ell-1}}{\sum_{\ell=1}^{\infty}\ell p(1-p)^{\ell-1}}=\ell
p^{2}(1-p)^{\ell-1}.$ (65)

This results in the average $\ell$,

$\langle\ell\rangle=\sum_{\ell=1}^{\infty}\ell\
\mathrm{Prob}(\mathrm{click},\ell)=\sum_{\ell=1}^{\infty}\ell
p^{2}(1-p)^{\ell-1}=\frac{2-p}{p},$ (66)

and the average $\ell^{2}$,

$\langle\ell^{2}\rangle=\sum_{\ell=1}^{\infty}\ell^{2}\
\mathrm{Prob}(\mathrm{click},\ell)=\sum_{\ell=1}^{\infty}\ell^{2}p^{2}(1-p)^{\ell-1}=\frac{p^{2}-6p+6}{p^{2}}.$
(67)

Thus the variance $\mathrm{var}\kern-2.0pt\left(\ell\right)$ is given by

$\mathrm{var}\kern-2.0pt\left(\ell\right)=\langle\ell^{2}\rangle-\langle\ell\rangle^{2}=\frac{2-2p}{p^{2}}.$
(68)

Figure 1:  The probability (65) of a single click during $\ell$ steps of the
clock’s evolution. The values of $p$ are $0.1$, $0.25$, and $0.5$. Figure 2:
The variance (68) of the number of steps $\ell$ as a function of $p$.

The probability Eq. (65) is presented on Fig. 1 as a function of the number of
steps $\ell$. We notice that for higher probabilities $p$ the number of
required evolution steps $\ell$ decreases. The exact position of the maxima is
given by the relation

$\ell_{\mathrm{max}}=-\frac{1}{\ln(1-p)},$ (69)

which for $p=0.1$ gives $\ell_{\mathrm{max}}=9.49$, for $p=0.25$ is
$\ell_{\mathrm{max}}=3.48$, and for $p=0.5$ drops to
$\ell_{\mathrm{max}}=1.44$.

The variance of $\ell$ is presented on Fig. 2. For the ideal case of $p=0.5$,
the average $\ell$ from Eq. (66) is $\langle\ell\rangle=3$ while the standard
deviation reads $\sigma=\sqrt{\mathrm{var}(\ell)}=2$. This gives the average
number of the required evolution steps $3\pm 2$.

##  5 Final remarks

One may ask, how to compare the discussed quantum clock with the currently
used atomic clocks? The time unit measured by an atomic clock is derived from
the frequency of the photon emitted by an atom during its de-excitation. From
the perspective of our model, the stochastic process of the atomic de-
excitation plays the role of the proper clock, while the interface can be
found in the clock’s system which detects the photon.

The stability of the chosen atomic transitions is so high, that modern atomic
clocks [34, 33, 25, 7] achieve systematic uncertainty on the level of
$10^{-18}$s, where this number is measured with respect to an external
laboratory clock. Since each evolution step advances the quantum clock on the
time axis by $\xi$ and we are required to wait between 1 and 5 steps to
register the click, the value of $\xi$ must be small enough to assure the
already achieved precission. If we allow $\xi$ to be a random variable, with
different values in each step of the clock’s evolution, the average
$\langle\xi\rangle$ should be small. In that case one may take the statistics
from as many as needed evolution steps, which will lower the value of
$\mathrm{var}(\ell)$ and increase the clock’s accuracy to the required level.

We notice also that every clock, even theoreticaly considered as a quantum
system, is influenced by spacetime and physical fields. Observed changes in
the clock can give information about the temporal structure of these objects.
This is an open problem for future investigations.

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# Dynamical Representation of Frames in Tensor Product of Hardy Spaces

###### Abstract

Dynamical Sampling of frames and tensor products are important topics in
harmonic analysis. This paper combines the concepts of dynamical sampling of
frames and the Carleson condition in the tensor product of Hardy spaces.
Initially we discuss the preservation of the frame property under the tensor
product on the Hilbert spaces. Then we discuss the iterative representation of
frames in tensor product of Hardy spaces. The key ingredient of this paper is
the so-called Carleson condition on the sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$ in
the open unit disc $\mathbb{D}_{1}\otimes\mathbb{D}_{2}$. Our proof is
motivated by the result of Shapiro and Shields.

Nabin Kumar Sahu∗ |  Vishesh Kumar∗
---|---
<EMAIL_ADDRESS>|<EMAIL_ADDRESS>

∗Dhirubhai Ambani Institute of Information and Communication Technology,
Gandhinagar, India

**footnotetext: Vishesh Kumar and Nabin Kumar Sahu are co-first authors with
equal contribution.

Keywords: Frames, tensor product, dynamical sampling, the Carleson condition,
Hardy space.

MSC: 42C15, 46B15

##  1 Introduction

Dynamical sampling of frames is a technique used in signal processing that
deals with recovering a signal from samples taken in space and time. This
technique aims to accurately represent the evolution process of the signal
over time by selecting frames that best capture its dynamic behavior. By
capturing the dynamic behavior of the signal over time and space, dynamical
sampling of frames can help improve the quality of signal recovery in
scenarios where an evolution process generates the signal. It is a relatively
new topic in applied harmonic analysis but has already garnered considerable
interest among researchers and practitioners alike [1, 3, 2, 6, 7, 15, 16,
19].

Aldroubi et al. [1] initiated the study of dynamical sampling problem and the
frame properties of sequences in the form of
$\\{T^{n}\varphi\\}_{n=0}^{\infty},$ where $T:\mathcal{H}\to\mathcal{H}$
belongs to certain classes of linear operators and $\varphi\in\mathcal{H}$,
where $\mathcal{H}$ is a separable Hilbert space. The classical dynamical
sampling problem is as follows: Consider a measurement set
$Y=\\{x(i),Tx(i),T^{2}x(i),.....,T^{m_{i}}x(i):\leavevmode\nobreak\
i\in\Omega\\}$, where $T$ is a bounded operator on $l^{2}(I)$, $\Omega$ is an
index set. The classical dynamical sampling problem is to find the necessary
and sufficient conditions on $A$, $\Omega$ and $m_{i}$ such that any $f\in
l^{2}(I)$ can be recovered efficiently from the measurement set $Y$.

In 2017, Christensen and Hasannasab [9] classified the frames which is
generated by a linear operator not necessarily bounded. They have also
discussed dynamically generated dual frames and they found that the dynamical
frames are unstable under the classical perturbation conditions (Paley-Wiener
type conditions). Again in 2019, Christensen and Hasannasab [10] proved that
every frame which is norm-bounded below can be represented as a finite union
of sequences $\\{(T_{j})^{n}\phi_{j}\\}_{n=0}^{\infty}$ for some bounded
operators $T_{j}$ and some elements $\phi_{j}$ in the underlying Hilbert
space. Recently, Ashbrock and Powell [4] proved that every redundant finite
frame for $\mathbb{F}^{d}$, where $\mathbb{F}=\mathbb{R}$ or $\mathbb{C},$ has
infinitely many dynamical dual frames, and introduced a low complexity error
diffusion quantization algorithm based on dynamical dual frames.

Hardy Hilbert space $H^{2}(\mathbb{D})$ is the space of all analytic functions
defined on the unit disk $\mathbb{D}$ with square summable coefficients. That
is
$H^{2}(\mathbb{D})=\Big{\\{}f:\mathbb{D}\rightarrow\mathbb{C}:\leavevmode\nobreak\
\leavevmode\nobreak\ f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
\sum_{n=0}^{\infty}|a_{n}^{2}|<\infty\Big{\\}}$. The space $H^{2}(\mathbb{D})$
is equipped with the inner product $\displaystyle\langle
f(z),g(z)\rangle=\sum_{n=0}^{\infty}a_{n}\overline{b_{n}}$, where
$f(z)=\displaystyle\sum_{n=0}^{\infty}a_{n}z^{n}$ and
$g(z)=\displaystyle\sum_{n=0}^{\infty}b_{n}z^{n}$ . The theory of frames has
been well investigated in the space $H^{2}(\mathbb{D})$, and for that one may
refer [8, 18, 20] and the references there in. Carleson condition [12] is the
most important ingredient to construct dynamically generated frames in a
separable Hilbert space. We recall that a sequence
$\displaystyle\\{\alpha_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ satisfies the
Carleson condition if

$\displaystyle\inf_{n\in\mathbb{N}}\displaystyle\Pi_{k\neq
n}\frac{|\alpha_{k}-\alpha_{n}|}{|1-\overline{\alpha_{k}}\alpha_{n}|}>0.$

We quote the main result from [12].

###### Theorem 1

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal basis
$\displaystyle\\{e_{k}\\}_{k=1}^{\infty}$. Let
$\displaystyle\\{\alpha_{k}\\}_{k=1}^{\infty}$ be a sequence in the unit disk
$\mathbb{D}$ in the complex plane. Assume that
$\displaystyle\big{\\{}\sqrt{1-|\alpha_{k}|^{2}}\big{\\}}_{k=1}^{\infty}\in
l^{2}(\mathbb{N})$. Let $T:\mathcal{H}\rightarrow\mathcal{H}$ be a bounded
linear operator such that $Te_{k}=\alpha_{k}e_{k}$. Then the sequence
$\displaystyle\\{T^{n}h\\}_{n=0}^{\infty}=\Big{\\{}\sum_{k=1}^{\infty}\alpha_{k}^{n}\sqrt{1-|\alpha_{k}|^{2}}e_{k}\Big{\\}}$
is a frame for the Hilbert space $\mathcal{H}$ if and only if
$\displaystyle\\{\alpha_{k}\\}_{k=1}^{\infty}$ satisfies the Carleson
condition.

In [5], it is proved that the tensor product of two sequences is frame if and
only if each part of this product is a frame. In [14], it is also proved that
if $\\{x_{n}\\}_{n\in I}$ and $\\{y_{m}\\}_{m\in I}$ be a frame for
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, respectively. Then $\\{x_{n}\otimes
y_{m}\\}_{n,m\in I}$ is a frame for $\mathcal{H}_{1}\otimes\mathcal{H}_{2}.$

The above work persuade us to carry out investigation on dynamically generated
frames on tensor product of two Hilbert spaces. The main contribution of this
report is Section 3, where the Carleson condition has been newly formulated.
The relationship between the Carleson condition and the frame properties of an
iterated sequence has been established.

##  2 Iterative representation of sequences in tensor product of two Hilbert
spaces

In this section we consider linearly independent sequences
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ in Hilbert space
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ and explore the existence of a
representation
$\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}$
with a bounded operator $T_{1}\otimes
T_{2}:span\\{f_{k}\\}_{k=1}^{\infty}\otimes span\\{g_{k}\\}_{k=1}^{\infty}\to
span\\{f_{k}\\}_{k=1}^{\infty}\otimes span\\{g_{k}\\}_{k=1}^{\infty}.$ This
generalization is motivated by the results for frames proved in [12, 11]. In
general the existence of a representation of a sequence
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ of the form
$\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}$
for a bounded operator $T_{1}\otimes T_{2}$ is not nearly related with frame
properties of the given sequence
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}.$

###### Example 1

Consider a linearly independent frame
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ for an infinite
dimensional Hilbert space $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ and the
associated representation
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}=\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}$
in terms of a linear operator $T_{1}\otimes
T_{2}:\operatorname{span}\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}\rightarrow\operatorname{span}\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$.
Assume that $\inf_{k\in\mathbb{N}}||f_{k}\otimes
g_{k}||=\inf_{k\in\mathbb{N}}||f_{k}||||g_{k}||>0$. Consider the sequence
$\\{\phi_{k}\\}_{k=1}^{\infty}\otimes\\{\psi_{k}\\}_{k=1}^{\infty}\subset\mathcal{H}_{1}\otimes\mathcal{H}_{2}$
given by $\phi_{k}\otimes\psi_{k}=2^{k}f_{k}\otimes 2^{k}g_{k}$
$,k\in\mathbb{N},$ which leads to a frame like expansion and satisfies the
lower frame condition but fails the upper one. For any $k\in\mathbb{N},$

$\displaystyle\phi_{k+1}\otimes\psi_{k+1}$
$\displaystyle=2^{k+1}f_{k+1}\otimes 2^{k+1}g_{k+1}$
$\displaystyle=2^{k+1}T_{1}f_{k}\otimes 2^{k+1}T_{2}g_{k}$
$\displaystyle=4T_{1}(2^{k}f_{k})\otimes T_{2}(2^{k}g_{k})$
$\displaystyle=4T_{1}(\phi_{k})\otimes T_{2}(\psi_{k})$
$\displaystyle=4(T_{1}\otimes T_{2})(\phi_{k}\otimes\psi_{k}).$

This shows that
$\\{\phi_{k}\\}_{k=1}^{\infty}\otimes\\{\psi_{k}\\}_{k=1}^{\infty}$ has the
representation
$\\{W_{1}^{n}\phi_{1}\\}_{n=0}^{\infty}\otimes\\{W_{2}^{n}\psi_{1}\\}_{n=0}^{\infty}$,
where $W_{1}=2T_{1},W_{2}=2T_{2}.$ In particular we can say that the frame
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ is represented
by a bounded operator if and only if the non-Bessel sequence
$\\{\phi_{k}\\}_{k=1}^{\infty}\otimes\\{\psi_{k}\\}_{k=1}^{\infty}$ is
represented by a bounded operator.

It is important to note a sequence
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ that has nice
frame properties and a nice representation
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}=\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}$
do not necessarily correlate with one another, as illustrated by the following
example. Although the family of functions
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ is not a frame
and cannot provide a frame-like expansion, but it has a representation
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}=\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}$
for a bounded isometric operator $T_{1}\otimes T_{2}.$

###### Example 2

Consider the sequence
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ given by
$f_{k}\otimes g_{k}=(e_{k}+e_{k+1})\otimes(d_{k}+d_{k+1}),k\in\mathbb{N}$,
where $\\{e_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}$ is an
orthonormal basis for the Hilbert space
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. We find that
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ is a Bessel
sequence but not a frame, despite the fact that
$\overline{span}\\{f_{k}\\}_{k=1}^{\infty}\otimes\overline{span}\\{g_{k}\\}_{k=1}^{\infty}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}.$
Since $\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}$ is
linearly independent, we define the operator $T_{1}\otimes
T_{2}:span\\{f_{k}\\}_{k=1}^{\infty}\otimes span\\{g_{k}\\}_{k=1}^{\infty}\to
span\\{f_{k}\\}_{k=1}^{\infty}\otimes span\\{g_{k}\\}_{k=1}^{\infty}$, by
$T_{1}f_{k}\otimes T_{2}g_{k}=f_{k+1}\otimes g_{k+1},$ and we have
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}=\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}.$
Then, for any $a_{k},c_{k}\in\mathbb{C},$ and any $N\in\mathbb{N},$

$\displaystyle\footnotesize\left|\left|T_{1}\sum_{k=1}^{N}c_{k}f_{k}\otimes
T_{2}\sum_{k=1}^{N}a_{k}g_{k}\right|\right|^{2}$
$\displaystyle=\left|\left|(T_{1}\otimes
T_{2})\left(\sum_{k=1}^{N}c_{k}f_{k}\otimes\sum_{k=1}^{N}a_{k}g_{k}\right)\right|\right|^{2}$
$\displaystyle=\left|\left|\sum_{k=1}^{N}c_{k}(e_{k+1}+e_{k+2})\otimes
a_{k}(d_{k+1}+d_{k+2})\right|\right|^{2}$
$\displaystyle=\left|\left|\sum_{k=1}^{N}c_{k}(e_{k}+e_{k+1})\otimes
a_{k}(d_{k}+d_{k+1})\right|\right|^{2}$
$\displaystyle=\left|\left|\sum_{k=1}^{N}c_{k}f_{k}\otimes\sum_{k=1}^{N}a_{k}g_{k}\right|\right|^{2}$
$\displaystyle=\left|\left|\sum_{k=1}^{N}c_{k}f_{k}\right|\right|^{2}\left|\left|\sum_{k=1}^{N}a_{k}g_{k}\right|\right|^{2}$

It follows that $T_{1}\otimes T_{2}$ has an extension to an isometric operator
$T_{1}\otimes
T_{2}:\mathcal{H}_{1}\otimes\mathcal{H}_{2}\to\mathcal{H}_{1}\otimes\mathcal{H}_{2}.$

Given any sequence
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}\subset\mathcal{H}_{1}\otimes\mathcal{H}_{2},$
then the synthesis operator is defined as

$V:\mathcal{D}(V)\to\mathcal{H}_{1}\otimes\mathcal{H}_{2},V(\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}):=\sum_{k=1}^{\infty}c_{k}f_{k}\otimes\sum_{k=1}^{\infty}d_{k}g_{k},$

where the domain $\mathcal{D}(V)$ consists the set of all scalar-valued
sequences $\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}$ for
which
$\displaystyle\sum_{k=1}^{\infty}c_{k}f_{k}\otimes\sum_{k=1}^{\infty}d_{k}g_{k}$
is convergent.Here we do not restrict our attention to sequences
$\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}$ belonging to
$l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$.

Consider the right-shift operators $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$,
which acts on arbitrary scalar sequences $\\{c_{k}\\}_{k=1}^{\infty}$ and
$\\{d_{k}\\}_{k=1}^{\infty}$ respectively by
$\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}=\\{0,c_{1},c_{2},c_{3},.......\\}$
and
$\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty}=\\{0,d_{1},d_{2},d_{3},......\\}$.Now
define an operator
$\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty}=\\{0,c_{1},c_{2},....\\}\otimes\\{0,d_{1},d_{2},....\\}$.
A vector space $W$ of scalar-valued sequences
$\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}$ is said to be
invariant under the defined operator $\mathcal{T}_{1}\otimes\mathcal{T}_{2}$
if $(\mathcal{T}_{1}\otimes\mathcal{T}_{2})(W)\subseteq W$.

The following result generalizes one of the important results in [11, 12] for
any arbitrary sequence (not necessarily frame).

###### Theorem 2

Consider a sequence
$\\{f_{k}\\}_{k=1}^{\infty}\otimes\\{g_{k}\\}_{k=1}^{\infty}\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}$
which has representation
$\\{T_{1}^{n}f_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{n}g_{1}\\}_{n=0}^{\infty}$
for a linear operator $T_{1}\otimes
T_{2}:span\\{f_{k}\\}_{k=1}^{\infty}\otimes span\\{g_{k}\\}_{k=1}^{\infty}\to
span\\{f_{k}\\}_{k=1}^{\infty}\otimes span\\{g_{k}\\}_{k=1}^{\infty}.$ Then
the following results hold.

  1. 1.

If $T_{1}\otimes T_{2}$ is bounded, then the domain $\mathcal{D}(V)$ and the
kernel $N(V)$ of the synthesis operator are invariant under the defined
operator $\mathcal{T}_{1}\otimes\mathcal{T}_{2}$, and
$\left\\{\frac{||f_{k+1}\otimes g_{k+1}||}{||f_{k}\otimes
g_{k}||}\right\\}_{k=1}^{\infty}\in l^{\infty}$ where $f_{k}\otimes g_{k}\neq
0,\forall k\in\mathbb{N}.$

  2. 2.

$T_{1}\otimes T_{2}$ is bounded on $span\\{f_{k}\\}_{k=1}^{\infty}\otimes
span\\{g_{k}\\}_{k=1}^{\infty}$ if and only if there exists a positive
constant $K$ such that
$\displaystyle\left|\left|V(\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty})\right|\right|\leq
K\left|\left|V(\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty})\right|\right|$
for all finite sequences
$\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}.$

Proof: We know if $T_{1}$ and $T_{2}$ is bounded then $T_{1}\otimes T_{2}$ is
bounded, let $\widetilde{T_{1}}\otimes\widetilde{T_{2}}$ is a unique extension
of the bounded linear operator $T_{1}\otimes T_{2}$ on
$\overline{span}\\{f_{k}\\}_{k=1}^{\infty}\otimes\overline{span}\\{g_{k}\\}_{k=1}^{\infty}.$

(1) Assume $T_{1}\otimes T_{2}$ is bounded and consider sequence
$\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}\in\mathcal{D}(V).$
In order to show that
$\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty}\in\mathcal{D}(V),$
i.e., that
$\displaystyle\sum_{k=1}^{\infty}c_{k}f_{k+1}\otimes\sum_{k=1}^{\infty}d_{k}g_{k+1}$
is convergent. Consider any $M,N\in\mathbb{N}$ where $N>M.$ Then

$\displaystyle\displaystyle\left|\left|\sum_{k=1}^{N}c_{k}f_{k+1}\otimes\sum_{k=1}^{N}d_{k}g_{k+1}-\sum_{k=1}^{M}c_{k}f_{k+1}\otimes\sum_{k=1}^{M}d_{k}g_{k+1}\right|\right|$
$\displaystyle=\left|\left|\sum_{k=M+1}^{N}c_{k}f_{k+1}\otimes\sum_{k=M+1}^{N}d_{k}g_{k+1}\right|\right|$
$\displaystyle=\left|\left|T_{1}\sum_{k=M+1}^{N}c_{k}f_{k}\otimes
T_{2}\sum_{k=M+1}^{N}d_{k}g_{k}\right|\right|$
$\displaystyle=\left|\left|(T_{1}\otimes
T_{2})\left(\sum_{k=M+1}^{N}c_{k}f_{k}\otimes\sum_{k=M+1}^{N}d_{k}g_{k}\right)\right|\right|$
$\displaystyle\leq\left|\left|T_{1}\otimes
T_{2}\right|\right|\left|\left|\sum_{k=M+1}^{N}c_{k}f_{k}\right|\right|\left|\left|\sum_{k=M+1}^{N}d_{k}g_{k}\right|\right|$
$\displaystyle\longrightarrow 0$

as $M,N\to\infty.$ Thus
$\displaystyle\sum_{k=1}^{\infty}c_{k}f_{k+1}\otimes\sum_{k=1}^{\infty}d_{k}g_{k+1}$
is convergent, i.e, $\mathcal{D}(V)$ is indeed invariant under the operator
$\mathcal{T}_{1}\otimes\mathcal{T}_{2}$.

To prove the invariance of $\mathcal{N}(V),$ assume that
$\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}\in N(V).$ The
series
$\displaystyle\sum_{k=1}^{\infty}c_{k}f_{k+1}\otimes\sum_{k=1}^{\infty}d_{k}g_{k+1}$
converges by what is already proved, and furthermore

$\displaystyle\sum_{k=1}^{\infty}c_{k}f_{k+1}\otimes\sum_{k=1}^{\infty}d_{k}g_{k+1}$
$\displaystyle=\sum_{k=1}^{\infty}c_{k}T_{1}f_{k}\otimes\sum_{k=1}^{\infty}d_{k}T_{2}g_{k}$
$\displaystyle=\left(\widetilde{T_{1}}\otimes\widetilde{T_{2}}\right)\left(\sum_{k=1}^{\infty}c_{k}f_{k}\otimes\sum_{k=1}^{\infty}d_{k}g_{k}\right)$
$\displaystyle=0;$

this shows that
$\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty}\in\mathcal{N}_{V}.$

Finally, for every $k\in\mathbb{N},$

$\displaystyle\left|\left|f_{k+1}\otimes g_{k+1}\right|\right|$
$\displaystyle=\left|\left|T_{1}f_{k}\otimes T_{2}g_{k}\right|\right|$
$\displaystyle=\left|\left|(T_{1}\otimes T_{2})(f_{k}\otimes
g_{k})\right|\right|$ $\displaystyle\leq\left|\left|T_{1}\otimes
T_{2}\right|\right|\left|\left|f_{k}\otimes g_{k}\right|\right|,$

and thus $\left\\{\frac{||f_{k+1}\otimes g_{k+1}||}{||f_{k}\otimes
g_{k}||}\right\\}_{k=1}^{\infty}\in l^{\infty}.$

(2) Assume firstly that $T_{1}\otimes T_{2}$ is bounded for every
$\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty}\in\mathcal{D}(V)$.
By (1) that
$\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty}\in\mathcal{D}(V);$
furthermore,

$\displaystyle\left|\left|V(\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty})\right|\right|$
$\displaystyle=\left|\left|\sum_{k=1}^{\infty}c_{k}f_{k+1}\otimes\sum_{k=1}^{\infty}d_{k}g_{k+1}\right|\right|$
$\displaystyle=\left|\left|\sum_{k=1}^{\infty}c_{k}\widetilde{T_{1}}f_{k}\otimes\sum_{k=1}^{\infty}d_{k}\widetilde{T_{2}}g_{k}\right|\right|$
$\displaystyle=\left|\left|(\widetilde{T_{1}}\otimes\widetilde{T_{2}})\left(\sum_{k=1}^{\infty}c_{k}f_{k}\otimes\sum_{k=1}^{\infty}d_{k}g_{k}\right)\right|\right|$
$\displaystyle\leq\left|\left|\widetilde{T_{1}}\otimes\widetilde{T_{2}}\right|\right|\left|\left|V(\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty})\right|\right|.$

Conversely, assume that there is a constant $K>0$ so that
$\left|\left|V(\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty})\right|\right|\leq
K\left|\left|V(\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty})\right|\right|$
for all finite sequences.

Take an arbitrary $f\otimes g\in span\\{f_{k}\\}_{k=1}^{\infty}\otimes
span\\{g_{k}\\}_{k=1}^{\infty}$.i.e., $\displaystyle f\otimes
g=\sum_{k=1}^{N}c_{k}f_{k}\otimes\sum_{k=1}^{N}d_{k}g_{k}$ for some
$N\in\mathbb{N}$ and some
$c_{1},c_{2},.......,c_{N},d_{1},d_{2},......,d_{N}\in\mathbb{C};$ letting
$c_{k}=0,d_{k}=0$ for $k>N$ we have that

$\displaystyle\left|\left|T_{1}f\otimes T_{2}g\right|\right|$
$\displaystyle=\left|\left|\sum_{k=1}^{\infty}c_{k}f_{k+1}\otimes\sum_{k=1}^{\infty}d_{k}g_{k+1}\right|\right|$
$\displaystyle=\left|\left|V(\mathcal{T}_{1}\\{c_{k}\\}_{k=1}^{\infty}\otimes\mathcal{T}_{2}\\{d_{k}\\}_{k=1}^{\infty})\right|\right|$
$\displaystyle\leq
K\left|\left|V(\\{c_{k}\\}_{k=1}^{\infty}\otimes\\{d_{k}\\}_{k=1}^{\infty})\right|\right|$
$\displaystyle\leq K\left|\left|f\otimes g\right|\right|,$

i.e., $T_{1}\otimes T_{2}$ is bounded.

##  3 Frames and the Carleson condition in the tensor product of Hardy Spaces

In this section, we consider a class of frames that can be represented via
bounded operators $T_{1}\otimes T_{2}$. We extended the construction that
first appeared in Corollary 3.17 in [1] in terms of the tensor product. Our
purpose is to extend the result given by Shapiro and Shields [17] in the
tensor product of Hardy spaces. First, we will discuss the Carleson condition
on sequences $\\{\lambda_{k}\\}_{k=1}^{\infty}$ and
$\\{\gamma_{l}\\}_{l=1}^{\infty}$ in the open unit discs $\mathbb{D}_{1}$ and
$\mathbb{D}_{2}$ respectively.

###  3.1 The Carleson Condition and tensor product

Let $\mathbb{D}_{1}$ and $\mathbb{D}_{2}$ denote the open unit discs in the
complex plane. The Hardy space $H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2})$ is defined by

$H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2})=\biggl{\\{}f\otimes
g:\mathbb{D}_{1}\otimes\mathbb{D}_{2}\to\mathbb{C}\big{|}f(z)\otimes
g(\omega)=\sum_{n=0}^{\infty}a_{n}z^{n}\otimes\sum_{m=0}^{\infty}b_{m}\omega^{m},$
$\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\in
l^{2}(\mathbb{N}_{0})\otimes l^{2}(\mathbb{N}_{0})\biggr{\\}}.$

The Hardy space $H_{1}^{2}(\mathbb{D}_{1})\otimes H_{2}^{2}(\mathbb{D}_{2})$
is a Hilbert space; given $w_{1},w_{2}\in H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2})$ where $w_{1}=f_{1}\otimes g_{1},w_{2}=f_{2}\otimes
g_{2}$ and $f_{1},f_{2}\in H_{1}^{2}(\mathbb{D}_{1})$ and $g_{1},g_{2}\in
H_{2}^{2}(\mathbb{D}_{2})$

$w_{1}=\sum_{n=0}^{\infty}a_{n}z^{n}\otimes\sum_{m=0}^{\infty}b_{m}\omega^{m},w_{2}=\sum_{n=0}^{\infty}c_{n}z^{n}\otimes\sum_{m=0}^{\infty}d_{m}\omega^{m}$

The inner product is defined by,

$\displaystyle\displaystyle\langle w_{1},w_{2}\rangle$
$\displaystyle=\langle\sum_{n=0}^{\infty}a_{n}z^{n}\otimes\sum_{m=0}^{\infty}b_{m}\omega^{m},\sum_{n=0}^{\infty}c_{n}z^{n}\otimes\sum_{m=0}^{\infty}d_{m}\omega^{m}\rangle$
$\displaystyle=\langle\sum_{n=0}^{\infty}a_{n}z^{n},\sum_{n=0}^{\infty}c_{n}z^{n}\rangle\langle\sum_{m=0}^{\infty}b_{m}\omega^{m},\sum_{m=0}^{\infty}d_{m}\omega^{m}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\bar{c_{n}}\cdot\sum_{m=0}^{\infty}b_{m}\bar{d_{m}}$

Note that $\\{z^{n}\\}_{n=0}^{\infty}\otimes\\{\omega^{m}\\}_{m=0}^{\infty}$
is an orthonormal basis for $H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2})$; denoting the canonical basis for
$l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$ by
$\\{\delta_{n}\\}_{n=1}^{\infty}\otimes\\{\sigma_{m}\\}_{m=1}^{\infty},$ the
operator $\theta:H_{1}^{2}(\mathbb{D}_{1})\otimes H_{2}^{2}(\mathbb{D}_{2})\to
l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$ defined by
$\theta(z^{n}\otimes\omega^{m})=\delta_{n+1}\otimes\sigma_{n+1}$ for
$n,m=0,1,2,3,.......$ is a unitary operator from
$H_{1}^{2}(\mathbb{D}_{1})\otimes H_{2}^{2}(\mathbb{D}_{2})$ onto
$l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$ and its preserve the norm and
inner product of vectors.

###### Definition 3.1

A sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}_{1},\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{2}$
then $\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$
satisfies the Carleson condition if

$\displaystyle\inf_{n,m\in\mathbb{N}}\prod_{(k,l)\neq(n,m)}\frac{\mid\lambda_{k}\gamma_{l}-\lambda_{n}\gamma_{m}\mid}{\mid
1-\overline{\lambda_{k}\gamma_{l}}\lambda_{n}\gamma_{m}\mid}>0.$ (1)

For a given sequence
$\Lambda=\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{1}\otimes\mathbb{D}_{2},$
define the sequence-valued operator $\varphi_{\Lambda}$ by

$\varphi_{\Lambda}(f\otimes
g)=\\{f(\lambda_{k})\sqrt{1-|\lambda_{k}|^{2}}\\}_{k=1}^{\infty}\otimes\\{g(\gamma_{l})\sqrt{1-|\gamma_{l}|}\\}_{l=1}^{\infty},f\otimes
g\in H_{1}^{2}(\mathbb{D}_{1})\otimes H_{2}^{2}(\mathbb{D}_{2}).$ (2)

Since the sequence in equation (2) does not necessarily belong to
$l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N}).$ The following result is
motivated by the result given by Shapiro and Shields in [17] and the Theorem
9.1 of [13].

###### Theorem 3

A sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{1}\otimes\mathbb{D}_{2}$
satisfies the Carleson condition.i.e.,
$\displaystyle\prod_{k,l=1,(l\neq
m)}^{\infty}\frac{\mid\lambda_{k}\gamma_{l}-\lambda_{n}\gamma_{m}\mid}{\mid
1-\overline{\lambda_{k}\gamma_{l}}\lambda_{n}\gamma_{m}\mid}>0$ if and only if
the interpolation problem
$f(\lambda_{k})g(\gamma_{l})=z_{k}\omega_{l},f\otimes g$ are bounded analytic
in $\mathbb{D}_{1}\otimes\mathbb{D}_{2}$ is solvable for arbitrary
$z_{k}\omega_{l}$. i.e., $\varphi_{\Lambda}(H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2}))=l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$ and in
affirmative case, $\varphi_{\Lambda}$ is bounded.

###### Theorem 4

Let $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}_{1}$ and
$\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{2}$ be a sequence of
distinct numbers and
$\\{\lambda_{k}\\}_{k=1}^{\infty}\cap\\{\gamma_{l}\\}_{l=1}^{\infty}=\phi$
then
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{1}\otimes\mathbb{D}_{2}$
is also sequence of distinct numbers. If $\exists\;c\in(0,1)$ such that

$\dfrac{1-\mid\lambda_{k}\gamma_{l+1}\mid}{1-\mid\lambda_{k}\gamma_{l}\mid}\leq
c<1,\forall k,l\in\mathbb{N}$ (3)

then $\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\lambda_{l}\\}_{l=1}^{\infty}$
satisfies the carleson condition. If
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$ is
positive and increasing then condition (3) is also necessary for
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$ to
satisfy the Carleson condition.

Proof : Condition (3) implies that

$1-\mid\lambda_{k}\gamma_{l+1}\mid\leq c(1-\lambda_{k}\gamma_{l})$
$1-\mid\lambda_{k}\gamma_{l}\mid\leq
c^{l-m}(1-\lambda_{n}\gamma_{m}),l>m,k\geq n\;\text{and}\;k,m=1,2,3....$ (4)

In particular
$\displaystyle\sum_{k}\sum_{l}(1-\mid\lambda_{k}\gamma_{l}\mid)<\infty$

it follows from (4) that for $l>m$

$\mid\lambda_{k}\gamma_{l}\mid-\mid\lambda_{n}\gamma_{m}\mid\geq(1-c^{l-m})(1-\mid\lambda_{n}\gamma_{m}\mid)$

and

$\displaystyle 1-\mid\lambda_{k}\gamma_{l}\lambda_{n}\gamma_{m}\mid$
$\displaystyle=1-\mid\lambda_{k}\gamma_{l}\mid+\mid\lambda_{k}\gamma_{l}\mid(1-\mid\lambda_{n}\gamma_{m}\mid)$
$\displaystyle\leq(1+c^{l-m})(1-\mid\lambda_{n}\gamma_{m}\mid)$

Hence by the lemma

$\displaystyle\left|\dfrac{\lambda_{n}\gamma_{m}-\lambda_{k}\gamma_{l}}{1-\overline{\lambda_{k}\gamma_{l}}\lambda_{n}\gamma_{m}}\right|$
$\displaystyle\geq\dfrac{\mid\lambda_{k}\gamma_{l}\mid-\mid\lambda_{n}\gamma_{m}\mid}{1-\mid\lambda_{k}\gamma_{l}\lambda_{n}\gamma_{m}\mid}$
$\displaystyle\geq\dfrac{1-c^{l-m}}{1+c^{l-m}},\;\;\;\;\;\;l>m$

for $l<m$ this inequality takes the form

$\left|\dfrac{\lambda_{m}\gamma_{m}-\lambda_{k}\gamma_{l}}{1-\overline{\lambda_{k}\gamma_{l}}\lambda_{n}\gamma_{m}}\right|\geq\dfrac{1-c^{m-l}}{1+c^{m-l}}$

Consequently,

$\prod_{k,l=1,(l\neq
m)}^{\infty}\left|\dfrac{\lambda_{n}\gamma_{m}-\lambda_{k}\gamma_{l}}{1-\overline{\lambda_{k}\gamma_{l}}\lambda_{n}\gamma_{m}}\right|\geq\prod_{N=1}^{\infty}\left(\dfrac{1-c^{N}}{1+c^{N}}\right)^{2}>0$

which shows that
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$
satisfies the carleson condition. Now suppose
$0\leq\lambda_{1}\gamma_{1}<\lambda_{1}\gamma_{2}<.........<\lambda_{2}\gamma_{1}<\lambda_{2}\gamma_{2}<......$
and

$\prod_{k,l=1,(l\neq
m)}^{\infty}\left|\dfrac{\lambda_{n}\gamma_{m}-\lambda_{k}\gamma_{l}}{1-\lambda_{k}\gamma_{l}\lambda_{n}\gamma_{m}}\right|\geq\delta>0$

Then,

$\lambda_{k}\gamma_{l+1}-\lambda_{k}\gamma_{l}\geq\delta(1-\lambda_{k}\gamma_{l}\lambda_{k}\gamma_{l+1}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k,l=1,2,.....$

So that,

$1-\lambda_{k}\gamma_{l+1}\leq
1-\dfrac{\delta+\lambda_{k}\gamma_{l}}{1+\delta\lambda_{k}\gamma_{l}}\leq(1-\delta)(1-\lambda_{k}\gamma_{l})$

Thus $\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$
satisfies (3) as claimed.

###  3.2 Frame Properties and the Carleson Condition in terms of tensor
product

In this section, we have extended the result given Theorem 3.7 in [12], which
yields the construction of a class of operators $T_{1}\otimes
T_{2}:l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})\to l^{2}(\mathbb{N})\otimes
l^{2}(\mathbb{N})$ for which
$\\{T_{1}^{n}h_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{m}h_{2}\\}_{m=0}^{\infty}$
is a frame for $l^{2}(\mathbb{(}N))\otimes l^{2}(\mathbb{(}N))$ for certain
sequences $h_{1}\otimes h_{2}\in l^{2}(\mathbb{(}N))\otimes
l^{2}(\mathbb{(}N)).$

Consider a sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}_{1}$
and $\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{2}$, where
$\mathbb{D}_{1},\mathbb{D}_{2}$ are unit discs in arbitrary complex planes and
assume that
$\displaystyle\\{\sqrt{1-\mid\lambda_{k}\mid^{2}}\\}_{k=1}^{\infty}\in
l^{2}(\mathbb{N})$ and
$\\{\sqrt{1-\mid\gamma_{l}\mid^{2}}\\}_{l=1}^{\infty}\in l^{2}(\mathbb{N})$.
Given any Hilbert spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, choose an
orthonormal basis $\\{e_{k}\\}_{k=1}^{\infty}$ and
$\\{d_{l}\\}_{l=1}^{\infty}$ and consider the bounded linear operators
$T_{1}:\mathcal{H}_{1}\to\mathcal{H}_{1},T_{2}:\mathcal{H}_{2}\to\mathcal{H}_{2}$
for which $T_{1}e_{k}=\lambda_{k}e_{k},T_{2}d_{l}=\gamma_{l}d_{l}.$ Let
$\displaystyle
h_{1}=\sum_{k=1}^{\infty}\sqrt{1-\mid\lambda_{k}\mid^{2}}e_{k},h_{2}=\sum_{l=1}^{\infty}\sqrt{1-\mid\gamma_{l}\mid^{2}}d_{l}$
and consider the iterated systems

$\\{T_{1}^{n}h_{1}\\}_{n=0}^{\infty}=\\{\sum_{k=1}^{\infty}\lambda_{k}^{n}\sqrt{1-\mid\lambda_{k}\mid^{2}}e_{k}\\}_{n=0}^{\infty}$
(5)

$\\{T_{2}^{m}h_{2}\\}_{m=0}^{\infty}=\\{\sum_{l=1}^{\infty}\gamma_{l}^{m}\sqrt{1-\mid\gamma_{l}\mid^{2}}d_{l}\\}_{m=0}^{\infty}$
(6)

We will now state the following result.

###### Theorem 5

Let
$\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}_{1},\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{2}$
and assume that $\\{\sqrt{1-\mid\lambda_{k}\mid^{2}}\\}_{k=1}^{\infty}\in
l^{2}(\mathbb{N}),\\\ \\{\sqrt{1-\mid\gamma_{l}\mid^{2}}\\}_{l=1}^{\infty}\in
l^{2}(\mathbb{N})$ Then the sequence
$\\{T_{1}^{n}h_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{m}h_{2}\\}_{m=0}^{\infty}$
defined by $(5)$ and $(6)$ is a frame for
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ if and only if
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$
satisfies the Carleson condition.

Proof : Define the synthesis operator $V:l^{2}(\mathbb{N}_{0})\otimes
l^{2}(\mathbb{N}_{0})\to\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ by
$V\\{\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{0}^{\infty}\\}=\displaystyle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2}$.
The sequence
$\\{T_{1}^{n}h_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{m}h_{2}\\}_{m=0}^{\infty}$
is a frame for $\\{\mathcal{H}_{1}\\}\otimes\\{\mathcal{H}_{2}\\}$ if and only
if the operator $V$ is well defined and surjective.

First assume that
$\\{T_{1}^{n}h_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{m}h_{2}\\}_{m=0}^{\infty}$
is a frame for $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. Take an arbitrary
sequence $\\{c_{j}\\}_{j=1}^{\infty}\otimes\\{f_{r}\\}_{r=1}^{\infty}\in
l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$ The surjectivity of $V$ implies
that there exists
$\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\in
l^{2}(\mathbb{N}_{0})\otimes l^{2}(\mathbb{N}_{0})$ such that
$\displaystyle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2}=\sum_{j=1}^{\infty}c_{j}e_{j}\otimes\sum_{r=1}^{\infty}f_{r}d_{r}.$
It follows that for each $k,l\in\mathbb{N},$

$\begin{split}c_{k}f_{l}&=\langle\sum_{j=1}^{\infty}c_{j}e_{j}\otimes\sum_{r=1}^{\infty}f_{r}d_{r},e_{k}\otimes
d_{l}\rangle\\\
&=\langle\sum_{j=1}^{\infty}c_{j}e_{j},e_{k}\rangle\langle\sum_{r=1}^{\infty}f_{r}d_{r},d_{l}\rangle\\\
&=\sum_{n=0}^{\infty}a_{n}\langle
T_{1}^{n}h_{1},e_{k}\rangle\sum_{m=0}^{\infty}b_{m}\langle
T_{2}^{m}h_{2},d_{l}\rangle\\\
&=\sum_{n=0}^{\infty}a_{n}\lambda_{k}^{n}\sqrt{1-|\lambda_{k}|^{2}}\cdot\sum_{m=0}^{\infty}b_{m}\gamma_{l}^{m}\sqrt{1-|\gamma_{l}|^{2}},\end{split}$
(7)

where
$c_{k}=\displaystyle\langle\sum_{j=1}^{\infty}c_{j}e_{j},e_{k}\rangle=\sum_{n=0}^{\infty}a_{n}\left<T_{1}^{n}h_{1},e_{k}\right>=\sum_{n=0}^{\infty}a_{n}\lambda_{k}^{n}\sqrt{1-\mid\lambda_{k}\mid^{2}}$
and similarly
$\displaystyle
f_{l}=\sum_{m=0}^{\infty}b_{m}\gamma_{l}^{m}\sqrt{1-\mid\gamma_{l}\mid^{2}}.$

Defining $f\in H_{1}^{2}(\mathbb{D}_{1})$ and $g\in H_{2}^{2}(\mathbb{D}_{2})$
by

$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},g(\omega)=\sum_{m=0}^{\infty}b_{m}\omega^{m}.$

Equation $(7)$ turns into

$f(\lambda_{k})\sqrt{1-|\lambda_{k}|^{2}}\cdot
g(\gamma_{l})\sqrt{1-|\gamma_{l}|^{2}}=c_{k}f_{l}.$

Formulated in terms of the operator $\varphi_{\Lambda}$ in $(2)$, this means
that $l^{2}(\mathbb{N})\otimes
l^{2}(\mathbb{N})\subseteq\varphi_{\Lambda}(H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2}))$.

On the other hand, take an arbitrary $f\otimes g\in
H_{1}^{2}(\mathbb{D}_{1})\otimes H_{2}^{2}(\mathbb{D}_{2})$ and choose
$\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\in
l^{2}(\mathbb{N}_{0})\otimes l^{2}(\mathbb{N}_{0})$ such that, $\displaystyle
f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},g(\omega)=\sum_{m=0}^{\infty}b_{m}\omega^{m}$
for every $k,l\in\mathbb{N}$ we have

$\displaystyle\langle
V\\{\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\\},e_{k}\otimes
d_{l}\rangle$
$\displaystyle=\langle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2},e_{k}\otimes
d_{l}\rangle$
$\displaystyle=\langle\sum_{n=0}^{\infty}a_{n}\sum_{j=1}^{\infty}\lambda_{j}^{n}\sqrt{1-|\lambda_{j}|^{2}}e_{j}\otimes\sum_{m=0}^{\infty}b_{m}\sum_{r=1}^{\infty}\gamma_{r}^{m}\sqrt{1-|\gamma_{r}|^{2}}d_{r}$
$\displaystyle\hskip 142.26378pt,e_{k}\otimes d_{l}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\lambda_{k}^{n}\sqrt{1-|\lambda_{k}|^{2}}\cdot\sum_{m=0}^{\infty}b_{m}\gamma_{l}^{m}\sqrt{1-|\gamma_{l}|^{2}}$

Therefore,

$\varphi_{\Lambda}(f\otimes g)=\\{\langle
V\\{\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\\},e_{k}\otimes
d_{l}\rangle\\}_{k,l=1}^{\infty}\in l^{2}(\mathbb{N})\otimes
l^{2}(\mathbb{N})$

$\implies\varphi_{\Lambda}(H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2}))\subseteq l^{2}(\mathbb{N})\otimes
l^{2}(\mathbb{N})$

Hence,

$\varphi_{\Lambda}(H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2}))=l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N}),$

which by [Theorem 3] implies that
$\\{\lambda_{k}\\}_{k=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}$
satisfies the Carleson condition.

Conversely, assume that the sequence
$\\{\lambda_{j}\\}_{j=1}^{\infty}\otimes\\{\gamma_{l}\\}_{l=1}^{\infty}\subset\mathbb{D}_{1}\otimes\mathbb{D}_{2}$
satisfies the Carleson condition. We first show that
$\displaystyle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2}$
is convergent for all
$\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\in
l^{2}(\mathbb{N}_{0})\otimes l^{2}(\mathbb{N}_{0})$. Consider the
corresponding $f\in H_{1}^{2}(\mathbb{D}_{1})$ and $g\in
H_{2}^{2}(\mathbb{D}_{2})$ determined by $\displaystyle
f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},g(\omega)=\sum_{m=0}^{\infty}b_{m}\omega^{m}$.
By [Theorem 3] we know that
$\varphi_{\Lambda}(H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2}))=l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$. So
$\displaystyle\\{f(\lambda_{k})\sqrt{1-|\lambda_{k}|^{2}}\\}_{k=1}^{\infty}\otimes\\{g(\gamma_{l})\sqrt{1-|\gamma_{l}|^{2}}\\}_{l=1}^{\infty}\in
l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$. Now for $N,M\in\mathbb{N}$,
consider the truncated sequences
$\displaystyle\\{a_{n}\\}_{n=0}^{N},\\{b_{m}\\}_{m=0}^{M}$ and the associated
functions $\displaystyle f_{N}\in H_{1}^{2}(\mathbb{D}_{1}),g_{M}\in
H_{2}^{2}(\mathbb{D}_{2})$ ,respectively. Given by $\displaystyle
f_{N}(z)=\sum_{n=0}^{N}a_{n}z^{n},g_{M}(\omega)=\sum_{m=0}^{M}b_{m}\omega^{m}.$
Again
$\\{f(\lambda_{k})\sqrt{1-|\lambda_{k}|^{2}}\\}_{k=1}^{\infty}\otimes\\{g(\gamma_{l})\sqrt{1-|\gamma_{l}|^{2}}\\}_{l=1}^{\infty}\in
l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$, and since
$\varphi_{\Lambda}:H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2})\to l^{2}(\mathbb{N})\otimes l^{2}(\mathbb{N})$ is
bounded by [Theorem 3], there is a constant $C>0$ such that

$\displaystyle||\varphi_{\Lambda}(f\otimes g)-\varphi_{\Lambda}(f_{N}\otimes
g_{M})||^{2}$ $\displaystyle\leq C||f\otimes g-f_{N}\otimes g_{M}||^{2}$
$\displaystyle=C||\sum_{n=0}^{\infty}a_{n}z^{n}\sum_{m=0}^{\infty}b_{m}\omega^{m}-\sum_{n=0}^{N}a_{n}z^{n}\sum_{m=0}^{M}b_{m}\omega^{m}||^{2}$
$\displaystyle=C||\left(\sum_{n=0}^{N}a_{n}z^{n}+\sum_{n=N+1}^{\infty}a_{n}z^{n}\right)\left(\sum_{m=0}^{M}b_{m}\omega^{m}+\sum_{m=M+1}^{\infty}b_{m}\omega^{m}\right)$
$\displaystyle\;\;\;\;\;\;\hskip
56.9055pt-\sum_{n=0}^{N}a_{n}z^{n}\sum_{m=0}^{M}b_{m}\omega^{m}||^{2}$
$\displaystyle=C||\sum_{n=0}^{N}a_{n}z^{n}\sum_{m=0}^{M}b_{m}\omega^{m}+\sum_{n=0}^{N}a_{n}z^{n}\sum_{m=M+1}^{\infty}b_{m}\omega^{m}$
$\displaystyle\hskip
56.9055pt+\sum_{n=N+1}^{\infty}a_{n}z^{n}\sum_{m=0}^{M}b_{m}\omega^{m}+\sum_{n=N+1}^{\infty}a_{n}z^{n}\sum_{m=M+1}^{\infty}b_{m}\omega^{m}$
$\displaystyle\hskip
85.35826pt-\sum_{n=0}^{N}a_{n}z^{n}\sum_{m=0}^{M}b_{m}\omega^{m}||^{2}$
$\displaystyle=C||\sum_{n=0}^{N}a_{n}z^{n}\sum_{m=M+1}^{\infty}b_{m}\omega^{m}+\sum_{n=N+1}^{\infty}a_{n}z^{n}\sum_{m=0}^{M}b_{m}\omega^{m}$
$\displaystyle\hskip
56.9055pt+\sum_{n=N+1}^{\infty}a_{n}z^{n}\sum_{m=M+1}^{\infty}b_{m}\omega^{m}||^{2}$
$\displaystyle=C\sum_{n=0}^{N}|a_{n}|^{2}\sum_{m=M+1}^{\infty}|b_{m}|^{2}+C\sum_{n=N+1}^{\infty}|a_{n}|^{2}\sum_{m=0}^{M}|b_{m}|^{2}$
$\displaystyle\hskip
85.35826pt+C\sum_{n=N+1}^{\infty}|a_{n}|^{2}\sum_{m=M+1}^{\infty}|b_{m}|^{2}\to
0,N,M\to\infty$

It follows that

$\displaystyle\sum_{n=0}^{N}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{M}b_{m}T_{2}^{m}h_{2}$
$\displaystyle=\sum_{n=0}^{N}a_{n}\sum_{k=1}^{\infty}\sqrt{1-\mid\lambda_{k}\mid^{2}}\lambda_{k}^{n}e_{k}\otimes\sum_{m=0}^{M}b_{m}\sum_{l=1}^{\infty}\sqrt{1-\mid\gamma_{l}\mid^{2}}\gamma_{l}^{n}d_{l}$
$\displaystyle=\sum_{k=1}^{\infty}\sqrt{1-\mid\lambda_{k}\mid^{k}}\sum_{n=0}^{N}a_{n}\lambda_{k}^{n}e_{k}\cdot\sum_{l=1}^{\infty}\sqrt{1-\mid\gamma_{l}\mid^{2}}\sum_{m=0}^{M}b_{m}\gamma_{l}^{m}d_{l}$
$\displaystyle=\sum_{k=1}^{\infty}\sqrt{1-\mid\lambda_{k}\mid^{2}}f_{N}(\lambda_{k})e_{k}\cdot\sum_{l=1}^{\infty}\sqrt{1-\mid\lambda_{l}\mid^{2}}g_{M}(\gamma_{l})d_{l}$
$\displaystyle\to\sum_{k=1}^{\infty}f(\lambda_{k})\sqrt{1-\mid\lambda_{k}\mid^{2}}e_{k}\sum_{l=1}^{\infty}g(\gamma_{l})\sqrt{1-\mid\gamma_{l}\mid^{2}}d_{l}\;\;\;\;\text{as
}N,M\to\infty$

This proves that
$\displaystyle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2}$
is convergent, and thus $V$ is well defined from $l^{2}(\mathbb{N}_{0})\otimes
l^{2}(\mathbb{N}_{0})$ into $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. In order
to prove that
$\\{T_{1}^{n}h_{1}\\}_{n=0}^{\infty}\otimes\\{T_{2}^{m}h_{2}\\}_{m=0}^{\infty}$
is a frame, it is enough to show that the synthesis operator
$V:l^{2}(\mathbb{N}_{0})\otimes
l^{2}(\mathbb{N}_{0})\to\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ is surjective.
Let $x\otimes y\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}$.By [Theorem 3] there
is an $f\otimes g\in H_{1}^{2}(\mathbb{D}_{1})\otimes
H_{2}^{2}(\mathbb{D}_{2})$ such that
$f(\lambda_{k})\sqrt{1-\mid\lambda_{k}\mid^{2}}\otimes
g(\gamma_{l})\sqrt{1-\mid\gamma_{l}\mid^{2}}=\langle x\otimes y,e_{k}\otimes
d_{l}\rangle$ for all $k,l\in\mathbb{N}$. Choose
$\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}\in
l^{2}(\mathbb{N}_{0})\otimes l^{2}(\mathbb{N}_{0})$, such that $\displaystyle
f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},g(\omega)=\sum_{m=0}^{\infty}b_{m}\omega^{m}$.
Then for each $k,l\in\mathbb{N},$

$\displaystyle\langle
V(\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty}),e_{k}\otimes
d_{l}\rangle$
$\displaystyle=\langle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1}\otimes\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2},e_{k}\otimes
d_{l}\rangle$
$\displaystyle=\langle\sum_{n=0}^{\infty}a_{n}T_{1}^{n}h_{1},e_{k}\rangle\langle\sum_{m=0}^{\infty}b_{m}T_{2}^{m}h_{2},d_{l}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\langle\sum_{j=1}^{\infty}\lambda_{j}^{n}\sqrt{1-\mid\lambda_{j}\mid^{2}}e_{j},e_{k}\rangle$
$\displaystyle\hskip
56.9055pt\cdot\sum_{m=0}^{\infty}b_{m}\langle\sum_{r=1}^{\infty}\gamma_{r}^{m}\sqrt{1-\mid\gamma_{r}\mid^{2}}d_{r},d_{l}\rangle$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\lambda_{k}^{n}\sqrt{1-\mid\lambda_{k}\mid^{2}}\sum_{m=0}^{\infty}b_{m}\gamma_{l}^{m}\sqrt{1-\mid\gamma_{l}\mid^{2}}$
$\displaystyle=f(\lambda_{k})\sqrt{1-\mid\lambda_{k}\mid^{2}}\cdot
g(\gamma_{l})\sqrt{1-\mid\gamma_{l}\mid^{2}}$ $\displaystyle=\langle
x,e_{k}\rangle\langle y,d_{l}\rangle.$

Therefore
$V(\\{a_{n}\\}_{n=0}^{\infty}\otimes\\{b_{m}\\}_{m=0}^{\infty})=x\otimes y$
and thus $V$ is surjective , as desired.

Acknowledgement:  This work was supported by the Science and Engineering
Research Board, DST, Govt. of India [CRG/2020/003170].
Declaration:  I confirm that all authors listed on the title page have
contributed equally to the work, have read the manuscript, and agree to its
submission.
Conflict of Interest:  The authors declare that there is no conflict of
interest.
Data Availability: Data availability or data sharing is not applicable to this
article as no data sets were generated or analyzed during the current
investigation.

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