This is the pre-peer reviewed version of the article published in final form
at
https://doi.org/10.1111/sapm.12566.

Dynamics for a quantum parliament

F. Bagarello

Dipartimento di Ingegneria,

Università di Palermo,

I-90128 Palermo, Italy

and I.N.F.N., Sezione di Catania

F. Gargano

Dipartimento di Ingegneria,

Università di Palermo, I - 90128 Palermo

###### Abstract

In this paper we propose a dynamical approach based on the Gorini-Kossakowski-
Sudarshan-Lindblad equation for a problem of decision making. More
specifically, we consider what was recently called a quantum parliament, asked
to approve or not a certain law, and we propose a model of the connections
between the various members of the parliament, proposing in particular some
special form of the interactions giving rise to a collaborative or non
collaborative behaviour.

##  I Introduction

As it is well established since many centuries, the role of mathematics is
relevant not only in pure, but also in applied sciences. Classical mechanics
is a prototypical example of this claim: we know very well that mathematics,
and mathematical modelling in particular, is essential if we want to describe
the motion of macroscopic systems, like pendulums, bullets, spinning tops and
so on. And mathematics is essential also in quantum mechanics, which is all
based on some basic facts in functional analysis, Hilbert spaces, and operator
theory.

Quantum mechanics is usually associated to the microscopic world, like atoms
and molecules, for instance. However, it is now well diffused the feeling that
quantum tools, and quantum ideas, can be relevant also for describing some
macroscopic systems. Nowadays there are hundreds of papers which explore these
connections, and several monographs: We only cite the latter here, where
several other references can also be found: [1]-[6].

In relation with our paper, quantum techniques have been applied to political
systems with different problems in mind: for instance, in [7, 8] the problem
of alliances was considered, whilst in [9] competition and cooperation were
analyzed in a ”coalition” equilibrium model. In other papers, [10]-[14], the
role of quantum protocols in a voting process has been discussed, while in
[15, 16] the attitude of people elected in one party to move to a different
one has been analyzed.

Recently, [17], the interest was more focused on the behaviour of two or three
groups of legislators, members of what the authors have nicely called a
quantum parliament. They need to vote for a given law, which should be
accepted or refused, and, while taking their decision, they follow their
leader’s suggestions. But not completely. In other words, they have a sort of
free will, which each legislator experience while producing its own decision.

In this paper we consider the same problem as in [17], but we adopt a
different, dynamical, strategy based on the use of the Lindblad operators. We
suppose that each legislator ${\cal L}_{j}$, $j=1,2,\ldots,N$, where $N$ is
the total number of the members of the parliament $\pi$, is described by a
quantum state, and we give a time dependence to the state by means of a master
equation simulating the interaction of ${\cal L}_{j}$ with the other members
of $\pi$, using both a suitable Hamiltonian and a Lindblad operator to model
the possible influences of a leader’s party and other effects. The result of
these interactions is reflected in the time evolution of the state, and
consequently in the final decision of ${\cal L}_{j}$. As in [17], we consider
in our treatment a sort of free will in $\pi$. This introduces some alea in
the final decision of each ${\cal L}_{j}$, which, for this reason, is not
obvious a priori, so that the acceptance of the law is not granted. Our
interest is mainly in the derivation of the outcome of the vote, using a
minimal set of working assumptions.

The paper is organized as follows: in Section II we introduce the problem and
discuss some of the essential aspects of the framework used all along the
paper. In Section III we discuss the dynamics of our quantum parliament,
considering different simplifying situations, from a single party with its
leader, to the case of three parties with different peculiarities. The case of
more leaders of the same party is also considered. Section IV contains our
conclusions and some plans for the future.

##  II Stating the problem

Let $\pi$ be the whole parliament. Following [17] we consider three groups of
legislators (or agents) in $\pi$, ${\mathcal{P}}_{\cal A}$,
${\mathcal{P}}_{\cal B}$ and ${\mathcal{P}}_{\cal M}$. These are Alice’s
party, made of $n_{\cal A}$ agents, Bob’s party, made of $n_{\cal B}$
legislators, and the mixed group, consisting in $n_{\cal M}$ legislators. We
have $n_{\cal A}+n_{\cal B}+n_{\cal M}=N$, the total number of the members of
$\pi$. We call $p_{\cal C}(j)$ the $j$-th member of the group ${\cal C}$,
where ${\cal C}={\cal A},{\cal B},{\cal M}$ and $j=1,2,\ldots,n_{\cal C}$. The
difference between the parties is as follows: suppose $\pi$ have to decide
whether to accept or refuse a law, $\Lambda$. Alice wants $\Lambda$ to be
accepted: she says yes to $\Lambda$. On the other side Bob, and its group
${\mathcal{P}}_{\cal B}$ can in principle vote ”yes” or ”no” depending on the
interaction with ${\mathcal{P}}_{{\cal A}}$. However, we will assume later
that Bob is against $\Lambda$. The case with no interaction is almost trivial,
since in this case ${\mathcal{P}}_{\cal B}$ is essentially a copy of
${\mathcal{P}}_{\cal A}$, and we could simply double the analysis given in
Section III.1 below where the influence of the leader, Alice or whoever, on
its party has been analyzed in detail. More interesting is the case in which
${\mathcal{P}}_{{\cal A}}$ and ${\mathcal{P}}_{\cal B}$ interact. And even
more interesting is the introduction of a third party, ${\mathcal{P}}_{\cal
M}$, which we suppose to have no leader and no a priori position to follow.

Since the only possible choices for $\Lambda$ are ”yes” and ”no”, it is
natural to imagine that each $p_{\cal C}(j)$ is described by a linear
combination of two orthogonal vectors, one representing the choice ”yes”, the
vector $e_{0}^{\cal C}(j)$, and the vector $e_{1}^{\cal C}(j)$, corresponding
to the choice ”no”. These two vectors form an orthonormal (o.n.) basis in the
Hilbert space $\mathcal{H}_{j}^{\cal C}=\mathbb{C}^{2}$, endowed with its
standard scalar product $\langle.,.\rangle_{j}$. The general vector of
$p_{\cal C}(j)$ can be written as

$\psi_{j}^{\cal C}=\alpha_{j}^{\cal C}e_{0}^{\cal C}(j)+\beta_{j}^{\cal
C}e_{1}^{\cal C}(j),$ (2.1)

with $|\alpha_{j}^{\cal C}|^{2}+|\beta_{j}^{\cal C}|^{2}=1$. Here, as before,
$j=1,2,\ldots,n_{\cal C}$ and ${\cal C}={\cal A},{\cal B},{\cal M}$.
$|\alpha_{j}^{\cal C}|^{2}$ and $|\beta_{j}^{\cal C}|^{2}$ can be seen
respectively as the probability of $p_{\cal C}(j)$ to vote ”yes” or ”no”. We
can naturally associate a composite Hilbert space to each of the parties in
$\pi$, made of copies of $\mathbb{C}^{2}$. In particular we put

$\mathcal{H}_{\cal A}=\otimes_{j=1}^{n_{\cal A}}\mathcal{H}_{j}^{\cal
A},\qquad\mathcal{H}_{\cal B}=\otimes_{j=1}^{n_{\cal B}}\mathcal{H}_{j}^{\cal
B},\qquad\mathcal{H}_{\cal M}=\otimes_{j=1}^{n_{\cal M}}\mathcal{H}_{j}^{\cal
M},$

and

$\mathcal{H}=\mathcal{H}_{\cal A}\otimes\mathcal{H}_{\cal
B}\otimes\mathcal{H}_{\cal M}.$ (2.2)

An o.n. basis of $\mathcal{H}$ is clearly consisting of tensor products of
states $e_{0}^{\cal C}(j)$, and $e_{1}^{\cal C}(j)$ for various $j$ and
different ${\cal C}$, that is of the vectors of the set

$\displaystyle\mathcal{E}=\\{e_{0}^{\cal A}(1),e_{1}^{\cal
A}(1),\ldots,e_{0}^{\cal A}(n_{\cal A}),e_{1}^{\cal A}(n_{\cal A}),e_{0}^{\cal
B}(j),e_{1}^{\cal B}(1),\dots,e_{0}^{\cal M}(n_{\cal M}),e_{1}^{\cal
M}(n_{\cal M})\\}.$ (2.3)

For instance, a vector describing a situation in which all the legislators of
${\mathcal{P}}_{\cal A}$ and ${\mathcal{P}}_{\cal M}$ vote ”yes”, while all
those of ${\mathcal{P}}_{\cal B}$ vote ”no” is

$(e_{0}^{\cal A}(1)\otimes\cdots\otimes e_{0}^{\cal A}(n_{\cal
A}))\otimes(e_{1}^{\cal B}(1)\otimes\cdots\otimes e_{1}^{\cal B}(n_{\cal
B}))\otimes(e_{0}^{\cal M}(1)\otimes\cdots\otimes e_{0}^{\cal M}(n_{\cal
M})).$

Of course the dimensionality of $\mathcal{H}$ increases with $N$. In fact we
have $\dim(\mathcal{H})=2^{N}$. An operator $X$ acting on, say,
$\mathcal{H}_{1}^{\cal A}$, is identified with the tensor product
$X\otimes\mathcal{I}_{2}\otimes\cdots\mathcal{I}_{2}$, i.e. the tensor product
of $X$ with $N-1$ copies of the identity operator $\mathcal{I}_{2}$, acting on
all the other Hilbert spaces. However, see Sections II and III, in our
applications we shall consider some simplifications that reduce significantly
the dimensionality of the Hilbert space. In particular, we will suppose that
all the members of a specific party are indistinguishable, i.e., they all feel
the same interactions and are described by the same parameters, with the same
values, so that it is reasonable to restrict to $n_{\cal A}=n_{\cal B}=n_{\cal
M}=1$, and the various $\mathcal{H}_{\cal A},\mathcal{H}_{\cal
B},\mathcal{H}_{\cal M}$ have dimension 2, while $\mathcal{H}$ has dimension
8. This is why, in the rest of the paper, we will often focus on a single
legislator of $\pi$, ${\cal L}$, or, at most, on a single representative
member for each party.

We consider now the vector

$\psi=\alpha\,e_{0}+\beta\,e_{1},$ (2.4)

with $|\alpha|^{2}+|\beta|^{2}=1$. This $\psi$ is, of course, analogous to the
one in (2.1) adopting a simplifying notation which is sufficient now since we
are only focusing on ${\cal L}$. We should also mention that, in the
following, sometimes we will use $|e_{j}\rangle$ rather than simply $e_{j}$,
$j=0,1$. This is useful when, see for instance $\rho_{\psi}$ below, rather
than to the vectors, we deal with density matrices. A possible parametrization
of the vector $\psi$, adopted in particular in [17], is the following:
$\alpha=\cos(\theta/2)$ and $\beta=\sin(\theta/2)\,e^{i\varphi}$, with
$\theta\in[0,\pi]$, and $\varphi\in[0,2\pi[$.

We can now define a density matrix $\rho_{\psi}$ as an operator acting on
$\mathcal{H}=\mathbb{C}^{2}$ as follows:
$\rho_{\psi}f=\langle\psi,f\rangle\psi$, $\forall f\in\mathcal{H}$. In the
bra-ket language, $\rho_{\psi}$ is often written as
$\rho_{\psi}=|\psi\rangle\langle\psi|$. More explicitly, $\rho_{\psi}$ is the
two-by-two matrix

$\displaystyle\rho_{\psi}=\left(\begin{array}[]{cc}|\alpha|^{2}&\alpha\,\overline{\beta}\\\
\overline{\alpha}\,\beta&|\beta|^{2}\\\ \end{array}\right),$ (2.7)

which is manifestly self-adjoint and with unit trace. In particular, if ${\cal
L}$ is in a ”yes” or in a ”no” mood, $e_{0}$ or $e_{1}$, the related density
matrices are

$\displaystyle\rho_{0}=\left(\begin{array}[]{cc}1&0\\\ 0&0\\\
\end{array}\right),\qquad\rho_{1}=\left(\begin{array}[]{cc}0&0\\\ 0&1\\\
\end{array}\right),$ (2.12)

with obvious notation. In [17] the main idea was to measure the distance of
the density matrix of the various legislators with those corresponding to
Alice ($\rho_{0}$) and Bob ($\rho_{1}$). This is a way to check which the
final decision of each legislator is. Moreover, as already observed, a free
will parameter was introduced in [17] for ${\mathcal{P}}_{\cal A}$ and
${\mathcal{P}}_{\cal B}$. For instance, let $r_{\cal A}$ be this parameter for
${\mathcal{P}}_{\cal A}$. Then, it is not required that $\rho_{\psi}=\rho_{0}$
to conclude that ${\cal L}$ is going to vote ”yes”. It is sufficient that the
difference between $\rho_{\psi}$ and $\rho_{0}$ is less than $r_{\cal A}$ or,
more explicitly, that $d(\rho_{\psi},\rho_{0})\leq r_{\cal A}$. Here $d(.,.)$
is a distance between density matrices. Like any distance, $d(.,.)$ must
satisfy some constraints: it must be symmetric,
$d(\rho_{\psi},\rho_{\varphi})=d(\rho_{\varphi},\rho_{\psi})$, non negative,
$d(\rho_{\psi},\rho_{\varphi})\geq 0$, with in particular
$d(\rho_{\psi},\rho_{\varphi})=0$ if and only if $\rho_{\psi}=\rho_{\varphi}$,
and it must satisfy the triangular inequality:
$d(\rho_{\psi},\rho_{\varphi})\leq
d(\rho_{\psi},\rho_{\eta})+d(\rho_{\eta},\rho_{\varphi})$, for all density
matrices $\rho_{\psi}$, $\rho_{\eta}$ and $\rho_{\varphi}$. In [17] the
explicit expression for the distance was the following:

$d(\rho_{\psi},\rho_{\varphi})=\frac{1}{2}\,\text{Tr}|\rho_{\psi}-\rho_{\varphi}|,$

where $\text{Tr}|A|$ is the trace and $|A|=\sqrt{A^{\dagger}A}$, $A\in{\cal
M}_{2}$, the set of all two-by-two matrices. Here we just want to briefly
comment that, in our opinion, this choice of distance does not always perform
well, at least if we still want to give a full meaning to the previous
parametric representation for $\alpha$ and $\beta$. In fact, using this
parametrization, we get $d(\rho_{\psi},\rho_{0})=\sin(\theta/2)$ and
$d(\rho_{\psi},\rho_{1})=\cos(\theta/2)$, which are both independent of
$\varphi$. This is reasonable when we use density matrices, but not so much
when we adopt vectors, which is also a natural choice, used several times in
the literature in similar situations. In this latter case, the distance
$d_{n}(\psi,\phi)=\|\psi-\phi\|$, $\forall\psi,\phi\in\mathcal{H}$, could be a
good alternative, having no drawback of the kind shown for $d(.,.)$.

Our approach is based on the derivation of suitable mean values of some
observables describing the final decision of the legislator $\mathcal{L}$. In
particular, using the density matrices defined in (2.12), their mean values
are obtained through

$\displaystyle\langle\rho_{0}\rangle_{\psi}=\textrm{Tr}\left[\rho_{\psi}\rho_{0}\right],\qquad\langle\rho_{1}\rangle_{\psi}=\textrm{Tr}\left[\rho_{\psi}\rho_{1}\right].$
(2.13)

As we shall see, the above expressions can be straightforwardly extended to
take into account the presence of multiple members belonging to different
parties (see for instances (3.18) below). The meaning of (2.13) is quite
evident: $\langle\rho_{0}\rangle_{\psi}$ is the mean value of the operator
$|e_{0}\rangle\langle e_{0}|$ which represent the pure state corresponding to
”${\cal L}$ votes yes”, whereas $\langle\rho_{1}\rangle_{\psi}$ is the mean
value of $|e_{1}\rangle\langle e_{1}|$, the pure state representing, this
time, ”${\cal L}$ votes no”. Both these mean values are computed on the
density matrix $\rho_{\psi}$ describing ${\cal L}$. Using the properties
$\text{Tr}[\rho_{\psi}]=1$ and $\rho_{0}+\rho_{1}=1\\!\\!1$, one can easily
obtain $\langle\rho_{0}\rangle_{\psi}=1-\langle\rho_{1}\rangle_{\psi}$. Hence,
the time evolution of $\langle\rho_{0}\rangle_{\psi}$ and
$\langle\rho_{1}\rangle_{\psi}$ can be phenomenologically interpreted as a
measure of the legislator’s decision to vote ”yes” or ”no”.

##  III The dynamics of the system

Our main effort consists in proposing a plausible dynamics for the generic
member of the group $\cal C$. This means that we are supposing that the
member’s decision can change in time due, for instance, to the parties’
influence or to the presence of one or more leaders, not necessarily belonging
to different parties. The original vector (2.4) $\psi$ becomes now time-
dependent, $\psi(t)$, and this new vector still belongs to the same Hilbert
space $\mathcal{H}$ in (2.2): again we omit the label of the party since the
mechanisms we shall describe are essentially the same for each legislator, and
we are focusing on just one of them, ${\cal L}$.

It is well known that the dynamics of the wave function of a closed quantum
system is governed by the Schrödinger equation:

$\displaystyle i\,\frac{d}{dt}\psi(t)=H\psi(t),$ (3.1)

where $H=H^{\dagger}$ is the self-adjoint Hamiltonian operator containing all
mechanisms acting in the closed system. Given the initial condition $\psi(0)$
the evolution $\psi(t)$ is completely determined. Formally we have
$\psi(t)=e^{-iHt}\psi(0)$.

However our system can be seen as open, where the small closed subsystem is
made by the various members of the parties, while their reservoirs are nothing
but the parties themselves. This is because the parties can influence the
single legislator, while the opposite is quite less plausible. Stated
differently, Alice (resp. Bob) influences what each single $p_{\cal A}(j)$
(resp. $p_{\cal B}(j)$) decides, but $p_{\cal A}(j)$’s (resp. $p_{\cal
B}(j)$’s) opinion is not relevant for Alice (resp. Bob). This kind of one-
directional flow of influence suggests that the dynamics in (3.1) is not the
most appropriate, then. The reason is twofold: first of all, self-adjointess
of $H$ does not allow, by itself, to avoid identical strength of interactions
between, say, Alice and $p_{\cal A}(j)$: if $H=H^{\dagger}$, then if Alice
communicates with $p_{\cal A}(j)$ with a given strength, then $p_{\cal A}(j)$
communicates back with Alice with the same strength. Secondly, it has been
proved in recent years, see [6] for instance, that $H=H^{\dagger}$ is only
compatible, if $dim(\mathcal{H})<\infty$, as in our case, with periodic or
quasi-periodic dynamics. But such an oscillating dynamics, of course, is not
really what one expects in a decision-making process, where one imagines to
find some limiting value, corresponding to the final decision. Hence, we need
to include some non-Hermitian effect, that we mimic here trough the Gorini-
Kossakowski-Sudarshan-Lindblad (GKSL) equation, see for instance [18, 19] for
open quantum systems, and [20, 21, 22] for applications outside the quantum
realm.

Remark:– The GKSL equation below is not the only possibility to analyse the
time evolution of a given open quantum system. Other possibilities are also
well known, as a purely Hamiltonian approach in which the Hamiltonian includes
also terms of the reservoir, [5, 6, 19], or using the so-called
$(H,\rho)$-induced dynamics, [23].

In particular, considering the density operator
$\rho(t)=|\psi(t)\rangle\langle\psi(t)|$ whose matrix representation is (2.7),
but with time dependent parameters, the resulting differential equation for
the evolution of $\rho$ is the well known GKSL equation:

$\displaystyle\frac{d}{dt}\rho(t)=-i[H,\rho(t)]+\sum_{j=1}^{N}\left(L_{j}\rho(t)L_{j}^{\dagger}-\frac{1}{2}\left\\{L_{j}^{\dagger}L_{j},\rho(t)\right\\}\right),$
(3.2)

where the various $L_{j}$ are the Lindblad operators, generally taken
traceless and connected to the influence of the larger system (the parties) on
the small system (the various members of $\pi$), and where $\\{X,Y\\}=XY+YX$
is the anti-commutator between two generic operators $X$ and $Y$. In writing
this equation we have assumed that the dynamics of the system is Markovian,
and the Lindblad operators (which describe the way the Leader influences the
various members of his party) are independent of the current state of the
system. This can be simply understood under the assumption that the leader is
not influenced in any way by the reaction of the members. This requirement is
a natural way of interpreting an environment which is negligibly perturbed by
a small system. Of course this is a strong hypothesis that however is
plausible in a dynamics ruled by some leadership. We notice that when the
interaction with the larger system is not included, we recover the Von Neumann
equation:

$\displaystyle\frac{d}{dt}\rho(t)=-i[H,\rho(t)],$

which could be easily deduced from (3.1), since
$\rho(t)=|\psi(t)\rangle\langle\psi(t)|$.

For the interpretation of our model it is useful to remind that the
interaction between the small system with the environment produces a mixture
of states from a generic pure state, [24]. In fact, adopting a standard
perturbative approach for small times and neglecting for a moment the action
of the Hamiltonian111We are assuming that its effect is negligible with
respect to the Lindbladian part, so that the Markovian process is highlighted.
$H$ in (3.2), the evolved density operator of a pure state
$\rho=|\psi\rangle\langle\psi|=\rho(0)$ in a small time step $dt$ can be
rewritten, to the leading order in $dt$, as

$\displaystyle\rho(dt)\approx\rho-\frac{1}{2}dt\sum_{j=1}^{N}\left(L^{\dagger}_{j}L_{j}\rho+\rho
L^{\dagger}_{j}L_{j}\right)+dt\sum_{j=1}^{N}L_{j}\rho
L^{\dagger}_{j}\approx\mathcal{A}\,\rho\,\mathcal{A}^{\dagger}+\sum_{j=1}^{N}\mathcal{B}_{j}\,\rho\,\mathcal{B}_{j}^{\dagger},$
(3.3)

where

$\displaystyle\mathcal{A}=1\\!\\!1-\frac{dt}{2}\sum_{j=1}^{N}L_{j}^{\dagger}L_{j},\qquad\mathcal{B}_{j}=\sqrt{dt}\,L_{j},\,j=1,\ldots,N.$
(3.4)

In other words, the evolved state is a mixture of the pure states defined by
$\mathcal{A}|\psi\rangle$ and by the various $\mathcal{B}_{j}|\psi\rangle$. In
particular we have

$\displaystyle\mathcal{A}\,\rho\,\mathcal{A}^{\dagger}=\mathcal{A}\,|\psi\rangle\langle\psi|\,\mathcal{A}^{\dagger}=p_{\mathcal{A}}\mathcal{\tilde{A}}\,|\psi\rangle\langle\psi|\mathcal{\tilde{A}}^{\dagger},$

with $\mathcal{\tilde{A}}=\frac{1}{\|\mathcal{A|\psi\rangle\|}}\mathcal{A}$.
Here
$p_{\mathcal{A}}={\|\mathcal{A}\,|\psi\rangle\|}^{2}\simeq\left(1-dt\sum_{j=1}^{N}\|L_{j}|\psi\rangle\|^{2}\right)$
can be seen as the probability that the vector $\psi$ evolves in
$\mathcal{\tilde{A}}\,|\psi\rangle$. This vector for $dt\rightarrow 0$ tends
to the initial vector $|\psi\rangle$, and follows the so called continuous
drift-type evolution, [24].

With similar computations we have

$\displaystyle\mathcal{B}_{j}\,\rho\,\mathcal{B}_{j}^{\dagger}=\mathcal{B}_{j}\,|\psi\rangle\langle\psi|\,\mathcal{B}_{j}^{\dagger}=p_{\mathcal{B}_{j}}\mathcal{\tilde{B}}_{j}\,|\psi\rangle\langle\psi|\mathcal{\tilde{B}}_{j}^{\dagger},\,\forall
j=1,\ldots,N,$ (3.5)

where
$\mathcal{\tilde{B}}_{j}=\frac{1}{\|\mathcal{B}_{j}|\psi\rangle\|}\mathcal{B}_{j}$
and
$p_{\mathcal{B}_{j}}={\|\mathcal{B}_{j}\,|\psi\rangle\|}^{2}=dt\|L_{j}|\psi\rangle\|^{2}$
is the probability that the vector $\psi$ evolves in
$\mathcal{\tilde{B}}_{j}\,|\psi\rangle$222Notice that due to (3.3), here we
simply have
$\mathcal{\tilde{B}}_{j}=\frac{1}{\|\mathcal{L}_{j}|\psi\rangle\|}\mathcal{L}_{j}$.
The process of evolving in such a state is called evolutionary jumps as for
$dt\rightarrow 0$ the vector $\mathcal{\tilde{B}}_{j}\,|\psi\rangle$ does not
tend to the original $|\psi\rangle$. This is the key process that produces
mixed states as a consequence of the interaction of the system with the large
environment. As it is well known, this process can be detected by looking at
the so-called purity, $\mathcal{P}=\text{Tr}(\rho^{2})$, and at the Von
Neumann entropy

$\displaystyle\mathcal{S}=-\text{Tr}(\rho\log\rho),$ (3.6)

that for a mixed state satisfy the inequalities
$\mathcal{P}<1,\,\mathcal{S}>0$, while $\mathcal{P}=1$ and $\mathcal{S}=0$ for
pure states.

###  III.1 A single party: the role of the leader

After this general introduction, we want to analyse next how the choice of a
single agent is influenced by its own party, and from its leader in
particular. Assuming that all the members of the party are indistinguishable,
it is natural to focus on a single member, ${\cal L}$, so to that the
dimension of the problem reduces, and the relevant Hilbert space is just
$\mathcal{H}=\mathbb{C}^{2}$. The vector representing ${\cal L}$’s choice is
simply $\psi=\alpha|e_{0}\rangle+\beta|e_{1}\rangle$, as we already discussed
before. The only effects we consider here are the free will (or its
uncertainty) of ${\cal L}$ and the leader’s influence.

In this case, the Hamiltonian governing the various processes occurring for
${\cal L}$ is assumed to be

$H=H_{f}+H_{v},$ $H_{f}=\omega\hat{a}^{\dagger}\hat{a},\qquad
H_{v}=\lambda(\hat{a}^{\dagger}+\hat{a}),$ (3.7)

where $\omega,\lambda$ are non negative parameters. Here we have introduced
the (fermionic) ladder operators $\hat{a}$ and $\hat{a}^{\dagger}$. These
operators are very well known and used originally in quantum mechanics, see
[25, 26] for instance, but then adopted also in other contexts, [5, 6]. For
our purposes, it is sufficient to say that these operators are defined on the
o.n. basis $\\{e_{0},e_{1}\\}$ of $\mathbb{C}^{2}$ as follows:

$\hat{a}e_{0}=0,\quad\hat{a}e_{1}=e_{0},\qquad\hat{a}^{\dagger}e_{0}=e_{1},\quad\hat{a}^{\dagger}e_{1}=0.$

They satisfy the canonical anti-commutation relations (CAR)
$\\{\hat{a},\hat{a}^{\dagger}\\}=\hat{a}\,\hat{a}^{\dagger}+\hat{a}^{\dagger}\,\hat{a}=\mathcal{I}_{2}$
and $\hat{a}^{2}=0$.

As widely discussed in the literature, see [5, 6] for an overview, the term
$H_{f}$ is responsible of an inertial mechanism which somehow forces ${\cal
L}$ to maintain its initial choice, whereas $H_{v}$ works in the opposite way
by inducing some change in ${\cal L}$ while forming its final decision. This
can be understood as follows: suppose that ${\cal L}$ is described, at $t=0$,
by the vector $\psi=e_{j}$. Then $\psi$ is an eigenstate of $H_{f}$, so that
$\psi$ is not modified when acting on it with $H_{f}$. On the other hand,
suppose that $\psi=e_{0}$. Then, using (3.7), we see that $H_{v}\psi=\lambda
e_{1}$, while, if $\psi=e_{1}$, we find that $H_{v}\psi=\lambda e_{0}$: the
action of $H_{v}$ on $\psi$ changes drastically the state of the system.
Notice that, so far, there is no reason for ${\cal L}$ to change his status
toward ”no” or ”yes”. In other words, there is no reason for ${\cal L}$ to
move from its original state $\psi$ to a new state $\psi_{new}$ which is
either $e_{1}$ or $e_{0}$ (or close to them). The way this task can be
achieved is by introducing in the model the two Lindblad operators

$L_{{\cal A},1}=\tau_{1}\hat{a},\qquad L_{{\cal
A},2}=\tau_{2}\hat{a}^{\dagger},$ (3.8)

which represent the action of ${\cal L}$’s leader on ${\cal L}$ itself. In
particular, under the action of $L_{{\cal A},1}$, ${\cal L}$ is influenced by
its leader to vote ”yes”, since it is forced to the vector $e_{0}$, whereas
$L_{{\cal A},2}$ does the opposite forcing to the vote ”no”, i.e., to the
vector $e_{1}$: the real parameters $\tau_{1}$ and $\tau_{2}$ fix the
strengths of these actions, and having $\tau_{1}$ and $\tau_{2}$ both non zero
can be seen as the simultaneous presence of two different leaders of the same
party proposing the two opposite final choice ”yes” and ”no” respectively. Of
course, the more influent is the leader, the higher the value of its parameter
$\tau_{j}$. Looking at the action of the operators on a pure state identified
by $|\psi\rangle=\alpha|e_{0}\rangle+\beta|e_{1}\rangle$, for a small time
$dt$, according to (3.3) the pure state becomes the mixed state

$\displaystyle\rho(dt)=p_{A}\mathcal{\tilde{A}}|\psi\rangle\langle\psi|\mathcal{\tilde{A}}^{\dagger}+p_{B_{1}}|e_{0}\rangle\langle
e_{0}|+p_{B_{2}}|e_{1}\rangle\langle e_{1}|$ (3.9)

with

$\mathcal{\tilde{A}}=\frac{1}{\sqrt{p_{A}}}\left(1\\!\\!1-\frac{dt}{2}\left(|\alpha\tau_{2}|^{2}|e_{0}\rangle\langle
e_{0}|+|\beta\tau_{1}|^{2}|e_{1}\rangle\langle e_{1}|\right)\right)$

and

$p_{A}=(1-p_{B_{1}}-p_{B_{2}}),\qquad p_{B_{1}}=dt|\beta\tau_{1}|^{2},\qquad
p_{B_{2}}=dt|\alpha\tau_{2}|^{2}.$

Hence there is a chance that the state is evolved in $|e_{0}\rangle$ (vote
”yes”) with probability $p_{B_{1}}$, in $|e_{1}\rangle$ (vote ”no”) with
probability $p_{B_{2}}$, otherwise it follows the continuous drift-type
evolution. It is expected that as time passes the mixture of the states
becomes a relevant phenomenon.

Some numerical simulations for different values of $\tau_{1}$ and $\tau_{2}$
are shown in Figures 1(a)-(b) and 2(a)-(b), where the mean values
$\langle\rho_{0}\rangle_{\psi}$, defined in (2.13), and the entropy $S$,
defined in (3.6), are shown. Considering the case $\tau_{1}\neq 0,\tau_{2}=0$,
that represents the situation where ${\cal L}$ is influenced by Alice to vote
”yes”, we reach an equilibrium that, depending also on the balance with the
other contribution in (3.2), tends faster to 1 as $\tau_{1}$ increases. We see
that the whole dynamics behaves as one would expect, given that the final
choice to vote ”yes” turns out to be highly probable333Even if the long time
value of $\langle\rho_{0}\rangle_{\psi}$ is not exactly one, it appears to be
almost one. This is a sort of uncertainty in our model, which replace the free
will in [17]..

We notice that, for moderate low values of $\tau_{1}$, see the cases
$\tau_{1}=0.1$ and $\tau_{1}=0.5$, strong amplitude oscillations are visible
in the early-mid phase and tend to be damped for later times. They are
consequences of the member’s indecision mechanism due to the Hamiltonian term
$H_{v}$ which, as observed in other contexts ([5, 6]), is the main responsible
of the oscillatory behaviour in the mean values of the density or number
operators. We should consider that, in a real situation, there could be also
(few) members of ${\mathcal{P}}_{{\cal A}}$ reaching a different final choice
and therefore voting ”no”. Hence, from a pure quantum interpretation, the
final state is a mixture of states most of them representing the vote ”yes”
and few others the vote ”no”. The evolution of the entropy $S$ can be seen in
this sense as a possible measure of this mixture. In Figures 1(b) we observe
that $S(t)$ is always strictly positive, indicative of the presence of a
mixture of states, and showing the presence of a main peak in time followed by
a rapid decreasing up to some equilibrium value. This value is reached more
rapidly for increasing values of $\tau_{1}$. We can justify this behaviour by
imagining that in a first phase the influence of $\cal{L}$’s leader due to the
Lindblad operator $L_{{\cal A},1}$ strongly modifies the member’s state of
mind, also because of the presence of $H_{v}$. However, the higher is the
strength of $L_{{\cal A},1}$, i.e. the value of $\tau_{1}$, the more rapidly
this condition moves toward an equilibrium. The fact that the final
equilibrium value increases for lower $\tau_{1}$ can be interpreted as the
lower influence of $\cal{L}$’s leader and hence to a richer mixture of states
related in principle to the presence of more members of the party voting ”no”
(which, however, remain a minority with respect to those voting ”yes”)444This
interpretation is based on the assumption that all the members of the various
parties are indistinguishable, so that each member of ${\mathcal{P}}_{{\cal
A}}$ behaves as ${\cal L}$..

Adding the Lindblad operator $L_{{\cal A},2}$ creates a richer dynamics, as
shown in Figures 2(a)-(b). The presence of two Lindblad operators, inducing
opposite effects, can be seen as a situation in which two different leaders of
the same party influence the final choice of ${\cal L}$. As expected when
$\tau_{1}>\tau_{2}$ the final mean value $\langle\rho_{0}\rangle_{\psi}$ is
closer to 1 rather than to 0, and it approaches 1 more and more as the
difference $\tau_{1}-\tau_{2}$ increases. A perfect equilibrium is reached
when $\tau_{1}=\tau_{2}$, and in this case the final value of
$\langle\rho_{0}\rangle_{\psi}$ is equal to 0.5. Finally, for
$\tau_{2}>\tau_{1}$, $\langle\rho_{0}\rangle_{\psi}$ is closer to 0,
regardless its initial value. The various mechanisms that could lead to this
dynamics can be straightforwardly deduced by the previous discussion made on
the case $\tau_{2}=0$. Concerning the time evolution of the entropy $S$ we can
observe that the case $\tau_{1}=\tau_{2}$ can be considered critical in the
sense that the equilibrium value is $\log(2)\approx 0.693$ which is the
maximum admissible value for $S$ (we recall that $S$ is bounded by the value
$\log{(d)}$, being $d$ the dimension of the Hilbert space in this case). This
case represents a situation of uncertainty in which no clear final choice is
achieved, and the state is a perfect mixture of $\rho_{0}$ and $\rho_{1}$. The
other cases show also that adding a second Lindblad operators induces a
stronger mixture as compared to the case where $\tau_{2}=0$: two competing
leaders of a single party create more uncertainty!

Figure 1: (a) The time evolution of the mean value
$\langle\rho_{0}\rangle_{\psi}(t)$ for different values of $\tau_{1}$ and with
$\tau_{2}=0$. Other parameters: $\omega=1,\lambda=0.25$. The initial condition
is $|\psi\rangle=\sqrt{0.7}|e_{0}\rangle+\sqrt{0.3}|e_{1}\rangle$. In the
small inset the time evolution for early times. (b) The time evolution of the
entropy $\mathcal{S}(t)$ for the same parameters and initial condition.

Figure 2: (a) The time evolution of the mean value
$\langle\rho_{0}\rangle_{\psi}(t)$ for different values of $\tau_{2}$ and with
$\tau_{1}=0.5$. Other parameters: $\omega=1,\lambda=0.25$. The initial
condition is $|\psi\rangle=\sqrt{0.7}|e_{0}\rangle+\sqrt{0.3}|e_{1}\rangle$.
(b) The time evolution of the entropy $\mathcal{S}(t)$ for the same parameters
and initial condition.

###  III.2 Two parties: the dynamics of alliance

In this section we want to model a situation in which the members of the first
party $\mathcal{P}_{\cal A}$ interact with the members of the second party
$\mathcal{P}_{\cal B}$, assuming however that only the members of
$\mathcal{P}_{\cal A}$ receive specific indications on the option they should
vote (specifically the option ”yes”). It is like if Bob’s influence on his own
party is very low, if not completely negligible: this is what is usually
called lack of leadership. We will consider the case in which Bob influences
the various $p_{\cal B}(j)$ later on. Our goal here is to derive the proper
model and operators to describe: i) a collaborative-like attitude of the
parties, that is the second members of $\mathcal{P}_{\cal B}$ are inclined to
vote for the same option as the members of $\mathcal{P}_{\cal A}$; ii) a
conflictual dynamics in which the two parties move in different directions
(one votes ”yes” while the other votes ”no”).

More in details, we suppose here that only the Lindblad operator $L_{{\cal
A},1}$ in (3.8) acts by forcing the members of $\mathcal{P}_{\cal A}$ to vote
”yes”, and that the behaviour of the members of $\mathcal{P}_{\cal B}$ is only
dictated by their own interactions with $\mathcal{P}_{\cal A}$. It follows
that the Lindblad operator is simply $L_{{\cal A},1}=\tau_{1}\hat{a}_{1}$,
whereas the Hamiltonian ruling the interactions between the members can be
assumed to be

$H=H_{f}+H_{v}+H_{c}+H_{nc},$

where $H_{f},H_{v}$ here extend those given in the previous section, while
$H_{c},\,H_{nc}$ are operators describing the new collaborative or conflictual
(non collaborative) dynamics. To be specific,

$\displaystyle H_{f}$ $\displaystyle=$
$\displaystyle\omega_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}+\omega_{2}\hat{a}_{2}^{\dagger}\hat{a}_{2},\,$
(3.10) $\displaystyle H_{v}$ $\displaystyle=$
$\displaystyle\lambda_{1}\left(\hat{a}_{1}^{\dagger}+\hat{a}_{1}\right)+\lambda_{2}\left(\hat{a}_{2}^{\dagger}+\hat{a}_{2}\right),\,$
(3.11)

with $\omega_{1,2}\geq 0,\lambda_{1,2}\geq 0$, while the new operators are

$\displaystyle
H_{c}=\gamma_{c}\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1}\right),\qquad
H_{nc}=\gamma_{nc}\left(\hat{a}_{1}^{\dagger}\hat{a}^{\dagger}_{2}+\hat{a}_{2}\hat{a}_{1}\right).$
(3.12)

Here our interest is focused on just two legislators, ${\cal L}_{1}$ and
${\cal L}_{2}$, as representants of Alice’s and Bob’s parties, assuming as in
Section III.1 that all the members of a given party share a similar attitude
towards $\Lambda$. The operators $\hat{a}_{j}$ and $\hat{a}_{j}^{\dagger}$
obey the following two-dimensional CAR:

$\\{\hat{a}_{j},\hat{a}_{k}^{\dagger}\\}=\delta_{j,k}\mathcal{I}_{4},\qquad\\{\hat{a}_{j},\hat{a}_{k}\\}=0,$
(3.13)

$j,k=1,2$. Here $\mathcal{I}_{4}$ is the identity operator on
$\mathbb{C}^{4}$.

The motivation that leads to these terms in the Hamiltonian is based on the
way in which the Lindblad operator acts on the system. In fact, in view of
what we have seen before, $L_{{\cal A},1}$ drives a generic vector
$|\tilde{\psi}\rangle=\sum_{j,k=0,1}\alpha_{j,k}|e_{j,k}\rangle$ into a new
vector where the components proportional to $\alpha_{1,0}$ and $\alpha_{1,1}$
tend to disappear555In fact, the action of $L_{{\cal A},1}$ makes these terms
in $|\tilde{\psi}\rangle$ disappear. However, the simultaneous effect of $H$,
partly restores them..

Given that, it is interesting to describe how, at least heuristically, the
combined action of $H$ and $L_{{\cal A},1}$ works on the members of
$\mathcal{P}_{\mathcal{B}}$ supposing that the initial configuration is
represented by the pure state $|\psi(0)\rangle$ (with its related density
operator $\rho=|\psi(0)\rangle\langle\psi(0)|$) given by

$|\psi(0)\rangle=\sum_{j,k=0,1}\alpha_{j,k}|e_{j,k}\rangle,\,\textrm{ with
}\sum_{j,k=0,1}|\alpha_{j,k}|^{2}=1.$

Since the behaviour of the members of $\mathcal{P}_{\mathcal{B}}$ is not
directly modified by $L_{{\cal A},1}$, and observing that $H_{f},H_{v}$ are
not responsible for the interaction, we start describing naively what happens
if we focus for small times on a branch which is firstly affected by the
action of $H_{c}$ and then by $L_{{\cal A},1}$. In particular, we have

$\displaystyle
H_{c}|\psi(0)\rangle=\gamma_{c}\left(\alpha_{1,0}|e_{0,1}\rangle+\alpha_{0,1}|e_{1,0}\rangle\right),$
(3.14)

since all the other terms of $|\psi(0)\rangle$ are annihilated by the action
of $H_{c}$. Hence, after a small time $dt$, we obtain the new vector (up to a
suitable normalization)

$\displaystyle|\psi(dt)\rangle\approx|\psi(0)\rangle-
idt\,\gamma_{c}\left(\alpha_{1,0}|e_{0,1}\rangle+\alpha_{0,1}|e_{1,0}\rangle\right).$
(3.15)

This vector is different from $|\psi(0)\rangle$ if $\alpha_{1,0}$ or
$\alpha_{0,1}$ are non zero. Then, following the same ideas discussed in the
previous sections, and considering the subsequent action of $L_{{\cal A},1}$
only, the obtained state is induced to jump, with some non zero probability,
to a state expressed by the density operator

$\mathcal{\tilde{B}}_{1}\,|\psi(dt)\rangle\langle\psi(dt)|\mathcal{\tilde{B}}_{1}^{\dagger},$

where
$\mathcal{\tilde{B}}_{1}=\frac{1}{\|\hat{a}_{1}|\psi\rangle\|}\hat{a}_{1}$
(see (3.5)). It is clear that, in view of (3.15), the state following this
jump can only be of the form

$\displaystyle|\psi_{\textrm{jump}}\rangle=\sum_{k=0,1}\tilde{\alpha}_{1,k}|e_{0,k}\rangle\,-i\,dt\gamma_{c}\alpha_{0,1}|e_{0,0}\rangle,$
(3.16)

where $\tilde{\alpha}_{1,k}=\alpha_{1,k}/||\hat{a}_{1}|\psi\rangle||$ and
where as usual the proper normalization should be inserted. It is now clear
that the second populations can have an excitement of the 0 level due to the
appearance of the term $-i\,dt\gamma_{c}\alpha_{0,1}|e_{0,0}\rangle$: the two
parties are driven to the same decision. This perturbative analysis reflects
our numerical results, as we will show next.

With similar computations, we can derive the approximated state obtained by
the action of $H_{nc}$ first, and the jump induced by $L_{{\cal A},1}$ after.
In this case we get

$\displaystyle|\psi_{\textrm{jump}}\rangle=\sum_{k=0,1}\tilde{\alpha}_{1,k}|e_{0,k}\rangle\,-i\,dt\gamma_{nc}\alpha_{0,0}|e_{0,1}\rangle,$
(3.17)

which, when compared to (3.16), leads to an excitement of $|e_{0,1}\rangle$,
that is the state representing the two opposite members’ decision.

Numerical results confirming this kind of behavior are shown in Figure 3 for
the cooperative attitude, and in Figure 4 for the non cooperative one. Figures
contain the time evolutions of the legislator’s intention to vote ”yes” for
both parties, expressed by the mean values

$\displaystyle\langle\rho_{0}\rangle_{\psi}^{(1)}=\textrm{Tr}\left[\rho_{\psi}(\rho_{0}\otimes\mathcal{I}_{2})\right],\qquad\langle\rho_{0}\rangle_{\psi}^{(2)}=\textrm{Tr}\left[\rho_{\psi}(\mathcal{I}_{2}\otimes\rho_{0})\right],$
(3.18)

where $\rho_{0}=\begin{pmatrix}1&0\\\
0&0\end{pmatrix},\,\mathcal{I}_{2}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$
and of the entropy $\mathcal{S}(t)$.

As we can see in Figures 3(a)-(b), for the cooperative case, increasing the
value of $\gamma_{c}$ leads to a continuous growth of the asymptotic value of
$\langle\rho_{0}\rangle_{\psi}^{(2)}(t)$, showing that the action of $H_{c}$
is responsible of the common attitude of the members of $\mathcal{P}_{\cal A}$
and $\mathcal{P}_{\cal B}$. We also notice that, again for increasing
$\gamma_{c}$, also $\langle\rho_{0}\rangle_{\psi}^{(1)}(t)$ slightly increases
its asymptotic value: we can speculate on this effect by supposing that the
members of $\mathcal{P}_{\cal A}$ reinforce their attitude in voting yes when
they interact and influence the members of $\mathcal{P}_{\cal B}$ in doing the
same. This is also supported by (3.16) according to which there is a jump
proportional to $\gamma_{c}$ toward the state $|e_{0,0}\rangle$, that is all
members vote ”yes”. Concerning the measure of the entropy $\mathcal{S}(t)$,
Figure 3(c), the initial phase (up to $t\approx 50$) is characterized by an
overall growth of ${\cal S}(t)$, and then by a decreasing behaviour and
convergence towards an asymptotic value which in general decreases faster for
increasing $\gamma_{c}$. The peaks in $\mathcal{S}(t)$ are higher for larger
values of $\gamma_{c}$. This could be interpreted by the fact that the
cooperative dynamics induced by $H_{c}$ creates, together with the action of
$L_{{\cal A},1}$, a rapid mixture of states representing the same final
decision taken by the parties, and the rapid decay can be seen as an immediate
settlement to the asymptotic value.

The non-cooperative case is shown in Figures 4, where the various time
evolutions are shown by changing the key parameter $\gamma_{nc}$ which tunes
the strength of $H_{nc}$. As expected, and predicted by our perturbative
approach, increasing the effect of $H_{nc}$ leads to two opposite final
choices taken by the members of $\cal{P}_{\cal{A}}$ and $\cal{P}_{\cal{B}}$,
even if the behaviour of these latter is not as sharp as that of
${\mathcal{P}}_{\cal A}$. It is interesting to note that also the entropy
${\cal S}(t)$ attains its equilibrium to larger values than those obtained in
the cooperative case; this can be explained by the fact that the action of the
Lindbladian operator $L_{{\cal A},1}$ together with the action of $H$, induces
a strong mixture of states increased by the non collaborative effect.

Figure 3: (a) The time evolution of the mean value
$\langle\rho_{0}\rangle_{\psi}^{(1)}(t)$ for different values of $\gamma_{c}$
(cooperative attitudes). Other parameters:
$\omega_{1}=\omega_{2}=1,\,\lambda_{1}=\lambda_{2}=0.25,\,\gamma_{nc}=0,\tau_{1}=0.5$.
Initial state is such that
$\langle\rho_{0}\rangle_{\psi}^{(1)}=0.6,\langle\rho_{0}\rangle_{\psi}^{(2)}=0.4$.
(b) The time evolution of the mean value
$\langle\rho_{0}\rangle_{\psi}^{(2)}(t)$ for the same parameters and initial
condition. (c) The time evolution of the entropy $\mathcal{S}(t)$ for the same
parameters and initial condition.

Figure 4:  The time evolution of the expected value
$\langle\rho_{0}\rangle_{\psi}^{(1)}(t)$ for different values of $\gamma_{nc}$
(non cooperative attitudes). Other parameters:
$\omega_{1}=\omega_{2}=1,\,\lambda_{1}=\lambda_{2}=0.25,\,\gamma_{c}=0,\tau_{1}=0.5$.
Initial state is such that
$\langle\rho_{0}\rangle_{\psi}^{(1)}=0.6,\langle\rho_{0}\rangle_{\psi}^{(2)}=0.4$.
(b) The time evolution of the expected value
$\langle\rho_{0}\rangle_{\psi}^{(2)}(t)$ for the same parameters and initial
condition. (c) The time evolution of the entropy $\mathcal{S}(t)$ for the same
parameters and initial condition.

###  III.3 The effects of Bob and ${\mathcal{P}}_{\cal M}$

We conclude the analysis of our quantum parliament $\pi$ by briefly
considering the inclusion in our model of a third party ${\mathcal{P}}_{{\cal
M}}$ that does not follow a leader’s influence, and the case in which Bob has
some (weak) influence on ${\mathcal{P}}_{\cal B}$. We shall perform two
different kind of experiments. In the first one, we suppose that only
${\mathcal{P}}_{\cal A}$ is influenced by its leader, whereas the members of
${\mathcal{P}}_{\cal B}$ and ${\mathcal{P}}_{\cal M}$ are left to interact
with the members of ${\mathcal{P}}_{\cal A}$ but they both have no strong
leader to follow. In the second experiment we suppose that
${\mathcal{P}}_{\cal B}$ has a leader, Bob, weak compared to Alice. The
presence of a third party in our model is interesting because allows us to
consider non-trivial cubic terms in the Hamiltonian of $\pi$, see (3.22)
below.

For the first experiment, we use the same hypothesis of the previous sections,
assuming that the only Lindblad operator is $L_{{\cal
A},1}=\tau_{1}\hat{a}_{1}$. In this case, adding a third agent to our system,
the Hilbert space of the micro-system becomes $\mathcal{H}=\mathbb{C}^{8}$,
and the vector representing the member’s choice is given by
$|\psi\rangle=\sum_{j,k,l=0,1}\alpha_{j,k,l}|e_{j,k,l}\rangle$ with
$\sum_{j,k,l=0,1}|\alpha_{j,k,l}|^{2}=1$. Concerning the Hamiltonian ruling
the behaviour of the three parties, it is natural to consider the following
one:

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{f}+H_{v}+H_{i},$ (3.19)

where, similarly to (3.10)-(3.11) we have

$\displaystyle H_{f}$ $\displaystyle=$
$\displaystyle\sum_{k=1,2,3}\omega_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k},$
(3.20) $\displaystyle H_{v}$ $\displaystyle=$
$\displaystyle\sum_{k=1,2,3}\lambda_{k}\left(\hat{a}_{k}^{\dagger}+\hat{a}_{k}\right)$
(3.21)

with $\omega_{1,2,3}\geq 0,\lambda_{1,2,3}\geq 0$, and where the interaction
term $H_{int}$ is assumed to be

$\displaystyle
H_{int}=\gamma_{1}\,\hat{a}_{1}^{\dagger}\hat{a}_{2}\hat{a}_{3}+\gamma_{2}\,\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{3}+\gamma_{3}\,\hat{a}_{1}^{\dagger}\hat{a}_{2}\hat{a}_{3}^{\dagger}+\gamma_{4}\,\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{3}^{\dagger}+\text{h.c.}$
(3.22)

which contains all the possible cubic terms related to the ways the three
parties could mutually interact, and where $\gamma_{k}\geq 0,\,k=1,2,3,4$.
Here h.c. stands for hermitian conjugate. This is needed if we require
$H=H^{\dagger}$. The CAR for the operators involved extend those in (3.13) to
three dimensions. To clarify the effect of each term in $H_{int}$, in our
numerical simulations we always assume that only one of the parameters
$\gamma_{k}$ is different from zero. Of course, one can easily relax this
assumption to create more complex dynamics considering various terms in
$H_{int}$ acting simultaneously.

Some numerical results for different cases of $H_{int}$ are shown in Figures
5, where the 4 possible triple interactions are considered. Here we plot the
functions

$\begin{cases}&\langle\rho_{0}\rangle_{\psi}^{(1)}=\textrm{Tr}\left[\rho_{\psi}(\rho_{0}\otimes\mathcal{I}_{2}\otimes\mathcal{I}_{2})\right],\\\
&\langle\rho_{0}\rangle_{\psi}^{(2)}=\textrm{Tr}\left[\rho_{\psi}(\mathcal{I}_{2}\otimes\rho_{0}\otimes\mathcal{I}_{2})\right],\\\
&\langle\rho_{0}\rangle_{\psi}^{(3)}=\textrm{Tr}\left[\rho_{\psi}(\mathcal{I}_{2}\otimes\mathcal{I}_{2}\otimes\rho_{0})\right],\end{cases}$
(3.23)

which extend those in (3.18), and are a measure of the members’ will to vote
”yes”.

We notice that most of the results can be understood by following the same
analysis, based on the perturbative approach, proposed in the previous section
for the single party and the two-parties cases. Hence, for instance, the case
$H_{int}=\gamma_{1}\hat{a}_{1}^{\dagger}\hat{a}_{2}\hat{a}_{3}+\text{h.c}$
corresponds to a dynamics in which all the members of the parties tend to vote
”yes” (panel (a)), whereas the case
$H_{int}=\gamma_{4}\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{3}^{\dagger}+\text{h.c}$
implies that the members of $\cal{P}_{\cal{A}}$ vote ”yes”, whereas the others
vote ”no” (panel (d)). They suggest a globally collaborative and a partly non
collaborative behaviour. Other cases in the other panels follow
straightforwardly.

Remark:– It is worth observing that the perturbative approach considered
before for one or two parties, is much less clear in this case, with three
different parties. The presence of too many agents makes the global dynamics
much more complicated, and the perturbation expansion in, e.g., (3.9) is less
explicative of the full dynamics, especially when the strength of $L_{{\cal
A},1}$ is lower than that of $H$, that is $\tau_{1}$ is smaller as compared to
the other parameters of the model.

The second set of experiments is performed by adding a second Lindblad
operator that forces the members of ${\mathcal{P}}_{\cal B}$ to vote ”no”,
opposite to the choice of ${\mathcal{P}}_{\cal A}$. In particular the Lindblad
operator governing this mechanism is

$\displaystyle L_{{\cal B},1}=\kappa\hat{a}^{\dagger}_{2},$ (3.24)

with $\kappa>0$. According to the analysis performed in the previous section,
this term forces the generic member of ${\mathcal{P}}_{\cal B}$ to jump in a
state representing the vote ”no”. We also suppose that the Hamiltonian is same
defined in (3.19). Numerical results are shown in Figures 6, for $\kappa=0.1$,
lower than $\tau_{1}=0.5$ which fixes the strength of $L_{{\cal A},1}$, and
the other parameters are as in the previous experiments. The results are in
agreement with what expected, and governed mainly by the interaction between
the parties ${\mathcal{P}}_{{\cal A}}$ and ${\mathcal{P}}_{{\cal M}}$. In all
case the members of ${\mathcal{P}}_{\cal B}$ are forced to vote ”no”, given
that $\langle\rho_{0}\rangle_{\psi}^{(2)}$ reaches asymptotic values that are
always below $0.5$, and most often even below 0.2. The case $\gamma_{1}>0$ and
$\gamma_{2}>0$, panels (a)-(b), depict a complete cooperation between the
party ${\mathcal{P}}_{{\cal A}}$ and ${\mathcal{P}}_{{\cal M}}$, in agreement
with the previous experiment. The non cooperative attitude for
${\mathcal{P}}_{{\cal A}}$ and ${\mathcal{P}}_{{\cal M}}$ is instead obtained
in the case $\gamma_{3}>0$ and $\gamma_{4}>0$, panels (c)-(d).

Figure 5: (a) The time evolution of the expected value
$\langle\rho_{0}\rangle_{\psi}^{(1,2,3)}(t)$ for the three parties model. Non
zero parameters are:
$\omega_{1}=\omega_{2}=\omega_{3}=0.1,\,\lambda_{1}=\lambda_{2}=\lambda_{3}=0.025,\,\gamma_{1}=1,\tau_{1}=0.5$.
Initial state is such that
$\langle\rho_{0}\rangle_{\psi}^{(1)}(0)=0.7,\langle\rho_{0}\rangle_{\psi}^{(2)}(0)=0.6,\langle\rho_{0}\rangle_{\psi}^{(3)}(0)=0.5$.
In the inset the time evolution of the entropy $S$. (b) Same plots as in (a)
with $\gamma_{2}=1,\gamma_{1}=0$, and the same other parameters and initial
conditions. (c) Same plots as in (a) with $\gamma_{3}=1,\gamma_{1}=0$, and the
same other parameters and initial conditions. (d) Same plots as in (a) with
$\gamma_{4}=1,\gamma_{1}=0$, and the same other parameters and initial
conditions.

Figure 6: (a) The time evolution of the expected value
$\langle\rho_{0}\rangle_{\psi}^{(1,2,3)}(t)$ for the three parties model with
the add of a second Lindblad operator $\kappa\hat{a}^{\dagger}_{2}$. Non zero
parameters are: $\kappa=0.1$,
$\omega_{1}=\omega_{2}=\omega_{3}=0.1,\,\lambda_{1}=\lambda_{2}=\lambda_{3}=0.025,\,\gamma_{1}=1,\tau_{1}=0.5$.
Initial state is such that
$\langle\rho_{0}\rangle_{\psi}^{(1)}(0)=0.7,\langle\rho_{0}\rangle_{\psi}^{(2)}(0)=0.6,\langle\rho_{0}\rangle_{\psi}^{(3)}(0)=0.5$.
In the inset the time evolution of the entropy $S$. (b) Same plots as in (a)
with $\gamma_{2}=1,\gamma_{1}=0$, and the same other parameters and initial
conditions. (c) Same plots as in (a) with $\gamma_{3}=1,\gamma_{1}=0$, and the
same other parameters and initial conditions. (d) Same plots as in (a) with
$\gamma_{4}=1,\gamma_{1}=0$, and the same other parameters and initial
conditions.

##  IV Conclusions

We have proposed a dynamical approach based on the GKLS equation for the
analysis of the time evolution of a quantum parliament, whose members are
asked to approve or not a certain law. In particular, we have analysed in some
details those terms in the model which produce a collaborative and a non
collaborative behaviour, to discriminate between the two. Our approach is
deduced form the idea that a small-system is influenced by the external
environment, and each state of the system, representing the decision of a
generic member of a party, evolves with the aforementioned GKLS equation. In
particular, the various members (the small-system) can be influenced by some
leaders’ influence (the environment), with the possibility that the members of
different parties can interact between them (in a collaborative/ non
collaborative way). We have supposed that the interactions between members are
described by a Hermitian Hamiltonian, containing reversible effects, whereas
the influence of the leaders is a unidirectional effect described by Lindblad
operators. With this approach we avoid the use a non-Hermitian methods, like
in [27, 28] for different macrosystems, allowing for the standard assumptions
to derive the dynamics with operators in quantum mechanics. It is clear that
the models proposed here can be adapted to other systems, of the kind
discussed in the past, and a comparison between the efficiency of the various
approaches is surely interesting and worth to be carried out. These are part
of our future plans.

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# Hilbert-type operators acting between weighted Fock spaces

Jianjun Jin  School of Mathematics Sciences, Hefei University of Technology,
Xuancheng Campus, Xuancheng 242000, P.R.China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Shuan Tang  School of Mathematics Sciences, Guizhou Normal
University, Guiyang 550001, P.R.China<EMAIL_ADDRESS>and  Xiaogao Feng
College of Mathematics and Information, China West Normal University, Nanchong
637009, P.R.China<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce and study several new Hilbert-type operators acting
between the weighted Fock spaces. We provide some sufficient and necessary
conditions for the boundedness and compactness of certain Hilbert-type
operators from one weighted Fock space to another.

###### Key words and phrases:

Hilbert-type operators; weighted Fock spaces; boundedness of operator;
compactness of operator; generalized Carleson measure

###### 2010 Mathematics Subject Classification:

47B35; 30H20

The first author supported by National Natural Science Foundation of China
(Grant Nos. 11501157). The second author was supported by National Natural
Science Foundation of China (Grant Nos. 12061022) and the foundation of
Guizhou Provincial Science and Technology Department (Grant Nos. [2017]7337
and [2017]5726). The third author was supported by National Natural Science
Foundation of China (Grant Nos. 11701459).

Data Availability Statement: the datasets generated during and/or analysed
during the current study are available from the corresponding author on
reasonable request.

##  1\. Introduction and main results

In this paper we will use $C,C_{1},C_{2},\cdots$ to denote universal positive
constants that might change from one line to another. For two positive numbers
$A,B$, we write $A\preceq B$, or $A\succeq B$, if there exists a positive
constant $C$ independent of $A$ and $B$ such that $A\leq CB$, or $A\geq CB$,
respectively. We will write $A\asymp B$ if both $A\preceq B$ and $A\succeq B$.

Let $\mathbb{C}$ be the complex plane, $\Omega$ be an open domain in
$\mathbb{C}$. Let $\mathcal{H}(\Omega)$ be the class of all holomorphic
functions on $\Omega$. In particular, $\mathcal{H}(\mathbb{C})$ is the class
of all entire functions.

The Fock space, denoted by $\mathcal{F}^{2}$, is defined as

$\mathcal{F}^{2}=\\{f\in\mathcal{H}(\mathbb{C}):\|f\|_{2}^{2}:=\frac{1}{\pi}\int_{\mathbb{C}}|f(z)e^{-\frac{1}{2}|z|^{2}}|^{2}dA(z)<\infty\\},$

where $dA$ is the Lebesgue area measure, see [23].

For an entire function $f=\sum_{n=0}^{\infty}a_{n}z^{n}$. A calculation yields
that

$\|f\|_{2}^{2}=2\int_{0}^{\infty}(\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n})e^{-r^{2}}rdr=\sum_{n=0}^{\infty}|a_{n}|^{2}n!.$

Thus we have $f\in\mathcal{F}^{2}$ if and only if
$\sum_{n=0}^{\infty}|a_{n}|^{2}n!<\infty.$

For $\theta>0$, $\alpha\in\mathbb{R}$, we introduce the weighted Fock space,
which is denoted by $\mathcal{F}^{2}_{\theta,\alpha}$ and defined as

$\mathcal{F}^{2}_{\theta,\alpha}=\\{f=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{H}(\mathbb{C}):\|f\|_{\theta,\alpha}^{2}:=\sum_{n=0}^{\infty}(n+\theta)^{\alpha}|a_{n}|^{2}n!<\infty\\}.$

######  Remark 1.1.

When $\alpha=0$, $\mathcal{F}^{2}_{\theta,\alpha}$ becomes the Fock space
$\mathcal{F}^{2}$. When $\alpha\in\mathbb{N}$, we know from [4] that an entire
function $f(z)$ belongs to the Fock-Sobolev space $F^{2,\alpha}$ on
$\mathbb{C}$, if and only if $z^{\alpha}f(z)\in\mathcal{F}^{2}$. Then it is
easy to see that $\mathcal{F}^{2}_{\theta,\alpha}$ coincides with the Fock-
Sobolev space $F^{2,\alpha}$ on $\mathbb{C}$ for $\alpha\in\mathbb{N}$.

######  Remark 1.2.

We notice that, if the coefficients of a power series
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ satisfy the condition
$\sum_{n=0}^{\infty}(n+\theta)^{\alpha}|a_{n}|^{2}n!<\infty$, then $f(z)$ is
an entire function.

######  Remark 1.3.

If $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{F}^{2}_{\theta,\alpha}$,
then, by Cauchy-Schwartz inequality, we see that there is a positive constant
$C$, depends only on $\theta,\alpha$, such that

$|f(z)|\leq\left[\sum_{n=0}^{\infty}(n+\theta)^{\alpha}|a_{n}|^{2}n!\right]^{\frac{1}{2}}\left[\sum_{n=0}^{\infty}(n+\theta)^{-\alpha}\frac{|z|^{2n}}{n!}\right]^{\frac{1}{2}}\leq
e^{C|z|^{2}}\|f\|_{\theta,\alpha},$

for all $z\in\mathbb{C}$.

It should be pointed out that the weighted Fock space
$\mathcal{F}^{2}_{\theta,\alpha}$ can be considered as a special case of the
weighted Hardy spaces, which have been detailed studied in [18]. We recall the
definition of the weighted Hardy spaces. Let
$\beta=\\{\beta_{n}\\}_{n=0}^{\infty}$ be a sequence of positive numbers. The
weighted Hardy space $\mathcal{H}^{2}(\beta)$ is the class of formal complex
power series $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ such that

$\sum_{n=0}^{\infty}|a_{n}|^{2}\beta_{n}^{2}<\infty.$

From [18], we know that, if $\beta_{n+1}/\beta_{n}\rightarrow 1$, when
$n\rightarrow\infty,$ $\mathcal{H}^{2}(\beta)$ is a Hilbert space of
holomorphic functions on the unit disk $\mathbb{D}$ with the inner product

$\langle f,g\rangle=\sum_{n=0}^{\infty}a_{n}\overline{b}_{n}\beta_{n}^{2},$

where $f=\sum_{n=0}^{\infty}a_{n}z^{n}$ and $g=\sum_{n=0}^{\infty}b_{n}z^{n}$.
In particular, if we take $\beta_{n}=1$ for all
$n\in\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$, then $\mathcal{H}^{2}(\beta)$
reduces the classic Hardy space $H^{2}(\mathbb{D})$, which is a Hilbert space
of holomorphic functions on the unit disk $\mathbb{D}$ with the inner product
$\langle f,g\rangle=\sum_{n=0}^{\infty}a_{n}\overline{b}_{n},$ see [22].

######  Remark 1.4.

Define the inner product on $\mathcal{F}_{\theta,\alpha}^{2}$ as

$\langle
f,g\rangle=\sum_{n=0}^{\infty}(n+\theta)^{\alpha}a_{n}\overline{b}_{n}n!,$

where $f=\sum_{n=0}^{\infty}a_{n}z^{n}$ and $g=\sum_{n=0}^{\infty}b_{n}z^{n}.$
Consider the operator $W$ on $\mathcal{F}_{\theta,\alpha}^{2}$, defined as

$Wf(z)=\sum_{n=0}^{\infty}[(n+\theta)^{\frac{\alpha}{2}}\sqrt{n!}]a_{n}z^{n},\quad\forall
f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{F}_{\theta,\alpha}^{2}$

Then we conclude that the operator $W$ on $\mathcal{F}_{\theta,\alpha}^{2}$ is
an isometric isomorphism from $\mathcal{F}_{\theta,\alpha}^{2}$ to the Hardy
space $H^{2}(\mathbb{D})$. Consequently, we see that
$\mathcal{F}_{\theta,\alpha}^{2}$ is a Hilbert space with the above inner
product. For any nonnegative integer $n$, let
$e_{n}(z)=(n!)^{-\frac{1}{2}}(n+\theta)^{-\frac{\alpha}{2}}z^{n}.$ Then we see
that the set $\\{e_{n}\\}$ is an orthonormal basis of
$\mathcal{F}_{\theta,\alpha}^{2}$. By the standard theory of reproducing
kernel Hilbert spaces(see Theorem 2.4 in [15]), we get that the reproducing
kernel $K_{\theta,\alpha}(z,w)$ for $\mathcal{F}_{\theta,\alpha}^{2}$ is

$K_{\theta,\alpha}(z,w)=\sum_{n=0}^{\infty}e_{n}(z)\overline{e_{n}(w)}=\sum_{n=0}^{\infty}(n+\theta)^{-\alpha}\frac{(z\overline{w})^{n}}{n!}.$

In particular, when $\alpha=0$, we have $K_{\theta,0}(z,w)=e^{z\overline{w}}$;
when $\theta\in\mathbb{N},\alpha=-1$, we have
$K_{\theta,-1}(z,w)=e^{z\overline{w}}(z\overline{w}+\alpha).$

For $f=\sum_{k=0}^{\infty}a_{k}z^{k}\in\mathcal{H}(\Omega)$, $\Omega$ is an
open domain in $\mathbb{C}$. The Hilbert operator $\mathbf{H}$, which is an
operator on spaces of holomorphic functions by its action on the Taylor
coefficients, is defined as,

$\mathbf{H}(f)(z):=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_{k}}{k+n+1}\right)z^{n}.$

During the last two decades, the Hilbert operator $\mathbf{H}$ and its
generalizations defined on various spaces of holomorphic functions on the unit
disk $\mathbb{D}$ in $\mathbb{C}$ have been much investigated. See, for
example, [1-14], [17], [20], [21].

For $\lambda>0$, $f=\sum_{k=0}^{\infty}a_{k}z^{k}\in\mathcal{H}(\mathbb{C})$,
we define the following Hilbert-type operator $\mathbf{H}_{\lambda}$ as

$\mathbf{H}_{\lambda}(f)(z):=\sum_{n=0}^{\infty}\left[\frac{1}{\sqrt{n!}}\sum_{k=0}^{\infty}\frac{a_{k}\sqrt{k!}}{(k+n)^{\lambda}+1}\right]z^{n}.$

In this paper, we first study the boundedness of $\mathbf{H}_{\lambda}$ acting
from one weighted Fock space to another, and obtain that

######  Theorem 1.5.

Let $\lambda>0,-1<\alpha,\beta<1$. Then, for any $\theta>0$,
$\mathbf{H}_{\lambda}$ is bounded from $\mathcal{F}_{\theta,\alpha}^{2}$ to
$\mathcal{F}_{\theta,\beta}^{2}$ if and only if $\lambda\geq
1+\frac{1}{2}(\beta-\alpha)$ .

Next we consider the case $\lambda=1$. We see from Theorem 1.5 that
$\mathbf{H}_{1}$ is not bounded from $\mathcal{F}_{\theta,\alpha}^{2}$ to
$\mathcal{F}_{\theta,\beta}^{2}$ if $\alpha<\beta$. We notice that

$\frac{1}{k+n+1}=\int_{0}^{1}t^{k+n}dt,\>k,n\in\mathbb{N}_{0}.$

To make $\mathbf{H}_{1}$ bounded from $\mathcal{F}_{\theta,\alpha}^{2}$ to
$\mathcal{F}_{\theta,\beta}^{2}$ when $\alpha<\beta$. For
$f=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{H}(\mathbb{C})$, let $\mu$ be a
positive Bore measure on $[0,1)$, we introduce the Hilbert-type operator
$\mathbf{H}_{\mu}$, which is defined as

$\mathbf{H}_{\mu}(f)(z):=\sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{n!}}\sum_{k=0}^{\infty}\mu[k+n]a_{k}\sqrt{k!}\right)z^{n}.$

Where

(1.1) $\mu{[n]}=\int_{0}^{1}t^{n}d\mu(t),\>n\in\mathbb{N}_{0}.$

We then study the problem of characterizing measures $\mu$ such that
$\mathbf{H}_{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is bounded and prove that

######  Theorem 1.6.

Let $-1<\alpha,\beta<1$, $\mu$ be a finite positive Bore measure on $[0,1)$
and $\mathbf{H}_{\mu}$ be as above. Then, for any $\theta>0$,
$\mathbf{H}_{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is bounded if and only if $\mu$ is a $[1+\frac{1}{2}(\beta-\alpha)]$-Carleson
measure on $[0,1)$.

Here, for $s>0$, a positive Borel measure $\mu$ on $[0,1)$, we say $\mu$ is an
$s$-Carleson measure if there is a constant $C_{1}>0$ such that

$\mu([t,1))\leq C_{1}(1-t)^{s}$

holds for all $t\in[0,1)$.

Also, we characterize measures $\mu$ such that
$\mathbf{H}_{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is compact and shall show that

######  Theorem 1.7.

Let $-1<\alpha,\beta<1$, $\mu$ be a finite positive Bore measure on $[0,1)$
and $\mathbf{H}_{\mu}$ be as above. Then, for any $\theta>0$,
$\mathbf{H}_{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is compact if and only if $\mu$ is a vanishing
$[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on $[0,1)$.

Here, an $s$-Carleson measure $\mu$ on $[0,1)$ is said to be a vanishing
$s$-Carleson measure, if it satisfies further that

$\lim_{t\rightarrow 1^{-}}\frac{\mu([t,1))}{(1-t)^{s}}=0.$

The paper is organized as follows. The proofs of Theorem 1.5, 1.6 and 1.7 will
be given in Section 3, 4 and 5, respectively. Some lemmas will be proved in
Section 2. Several remarks will be finally presented in the last section.

##  2\. Some lemmas

In this section, we establish some lemmas, which will be used in the proofs of
Theorem 1.5, 1.6 and 1.7 .

######  Lemma 2.1.

For $\theta>0,-1<\alpha,\beta<1$, we define

$w_{\alpha,\beta}^{[1]}(n):=\sum_{k=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(n+\theta)^{\frac{1-\beta}{2}}}{(k+\theta)^{\frac{1+\alpha}{2}}},\>n\in\mathbb{N}_{0},$

and

$w_{\alpha,\beta}^{[2]}(k):=\sum_{n=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(k+\theta)^{\frac{1+\alpha}{2}}}{(n+\theta)^{\frac{1-\beta}{2}}},\>k\in\mathbb{N}_{0}.$

Then

(2.1) $w_{\alpha,\beta}^{[1]}(n)\leq
B(\frac{1+\beta}{2},\frac{1-\alpha}{2})(n+\theta)^{-\beta},n\in\mathbb{N}_{0};$
(2.2) $w_{\alpha,\beta}^{[2]}(k)\leq
B(\frac{1+\beta}{2},\frac{1-\alpha}{2})(k+\theta)^{\alpha},k\in\mathbb{N}_{0}.$

Here $B(\cdot,\cdot)$ is the Beta function, defined as

$B(u,v)=\int_{0}^{\infty}\frac{t^{u-1}}{(1+t)^{u+v}}\,dt,\>u>0,v>0.$

It is known that

$B(u,v)=\int_{0}^{1}t^{u-1}(1-t)^{v-1}\,dt=\frac{\Gamma(u)\Gamma{(v)}}{\Gamma(u+v)}.$

and $B(u,v)=B(v,u)$, where $\Gamma(x)$ is the Gamma function, defined as

$\Gamma(x)=\int_{0}^{\infty}e^{-t}t^{x-1}\,dt,\>x>0.$

For more detailed introduction to the Beta function and Gamma function, see
[19] .

###### Proof of Lemma 2.1 .

Since $-1<\alpha,\beta<1$, we see that

$\displaystyle
w_{\alpha,\beta}^{[1]}(n)\leq\int_{0}^{\infty}\frac{1}{(x+n+\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(n+\theta)^{\frac{1-\beta}{2}}}{x^{\frac{1+\alpha}{2}}}\,dx=B(\frac{1+\beta}{2},\frac{1-\alpha}{2})(n+\theta)^{-\beta}.$

Similarly, we can obtain that

$w_{\alpha,\beta}^{[2]}(k)\leq
B(\frac{1+\beta}{2},\frac{1-\alpha}{2})(k+\theta)^{\alpha}.$

Lemma 2.1 is proved. ∎

######  Lemma 2.2.

Let $\theta>0,-1<\alpha,\beta<1$. Let $\mu$ be a positive Borel measure on
$[0,1)$ and $\mu[n]$ is defined as in (1.1) for $n\in\mathbb{N}_{0}$. If $\mu$
is a $[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on $[0,1)$, then

(2.3) $\mu[n]\preceq\frac{1}{(n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}$

holds for all $n\in\mathbb{N}_{0}$. Furthermore, if $\mu$ is a vanishing
$[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on $[0,1)$, then

(2.4)
$\mu[n]=o\left(\frac{1}{(n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\right),\quad
n\rightarrow\infty.$

###### Proof.

For $n\in\mathbb{N}$, we get from integration by parts that

$\displaystyle\mu[n]=\int_{0}^{1}t^{n}d\mu(t)$ $\displaystyle=$
$\displaystyle\mu([0,1))-n\int_{0}^{1}t^{n-1}\mu([0,t))dt$ $\displaystyle=$
$\displaystyle n\int_{0}^{1}t^{n-1}\mu([t,1))dt.$

If $\mu$ is a $[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on $[0,1)$,
then we see that there is a constant $C_{3}>0$ such that

$\mu([t,1))\leq C_{3}(1-t)^{1+\frac{1}{2}(\beta-\alpha)}$

holds for all $t\in[0,1)$. It follows that

$\displaystyle\mu[n]$ $\displaystyle\leq$ $\displaystyle
C_{3}n\int_{0}^{1}t^{n-1}(1-t)^{1+\frac{1}{2}(\beta-\alpha)}dt$
$\displaystyle=$ $\displaystyle
C_{3}\frac{n\Gamma(n)\Gamma(2+\frac{1}{2}(\beta-\alpha))}{\Gamma(n+2+\frac{1}{2}(\beta-\alpha))}.$

By using the fact that

$\Gamma(x)=\sqrt{2\pi}x^{x-\frac{1}{2}}e^{-x}[1+r(x)],\,|r(x)|\leq
e^{\frac{1}{12x}}-1,\,x>0,$

we obtain that

$\frac{n\Gamma(n)\Gamma(2+\frac{1}{2}(\beta-\alpha))}{\Gamma(n+2+\frac{1}{2}(\beta-\alpha))}\asymp\frac{1}{n^{1+\frac{1}{2}(\beta-\alpha)}}.$

Consequently, it is easy to see that

$\mu[n]\preceq\frac{1}{(n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}$

holds for all $n\in\mathbb{N}_{0}$.

By minor modifications of above arguments, we can similarly show that (2.4)
holds if $\mu$ is a vanishing $[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure
on $[0,1)$. Lemma 2.2 is proved. ∎

######  Lemma 2.3.

Let $\theta>0,-1<\alpha,\beta<1$. For
$f=\sum_{k=0}^{\infty}a_{k}z^{k}\in\mathcal{H}(\mathbb{C})$, we define

$\mathbf{\check{H}}_{\theta}(f)(z):=\sum_{n=0}^{\infty}\left[\frac{1}{\sqrt{n!}}\sum_{k=0}^{\infty}\frac{a_{k}\sqrt{k!}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\right]z^{n}.$

Then $\mathbf{\check{H}}_{\theta}$ is bounded from
$\mathcal{F}^{2}_{\theta,\alpha}$ to $\mathcal{F}^{2}_{\theta,\beta}.$

###### Proof.

For $f=\sum_{k=0}^{\infty}{a_{k}}z^{k}\in\mathcal{F}_{\theta,\alpha}^{2}$,
$n\in\mathbb{N}_{0}$, by Cauchy’s inequality, we have

$\displaystyle\left|\sum_{k=0}^{\infty}\frac{a_{k}\sqrt{k!}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\right|\leq\sum_{k=0}^{\infty}\frac{|a_{k}|\sqrt{k!}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\left\\{[\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}]^{\frac{1}{2}}\cdot\frac{(k+\theta)^{\frac{1+\alpha}{4}}}{(n+\theta)^{\frac{1-\beta}{4}}}\cdot|a_{k}|\sqrt{k!}\right\\}$
$\displaystyle\quad\quad\times\left\\{[\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}]^{\frac{1}{2}}\cdot\frac{(n+\theta)^{\frac{1-\beta}{4}}}{(k+\theta)^{\frac{1+\alpha}{4}}}\right\\}$
$\displaystyle\leq$
$\displaystyle\left[\sum_{k=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(k+\theta)^{\frac{1+\alpha}{2}}}{(n+\theta)^{\frac{1-\beta}{2}}}\cdot|a_{k}|^{2}k!\right]^{\frac{1}{2}}$
$\displaystyle\quad\quad\times\left[\sum_{k=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(n+\theta)^{\frac{1-\beta}{2}}}{(k+\theta)^{\frac{1+\alpha}{2}}}\right]^{\frac{1}{2}}.$

Then, in view of Lemma 2.1, we get that

$\displaystyle\left|\sum_{k=0}^{\infty}\frac{a_{k}\sqrt{k!}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\right|$
$\displaystyle\leq$
$\displaystyle[w_{\alpha,\beta}^{[1]}(n)]^{\frac{1}{2}}\left[\sum_{k=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(k+\theta)^{\frac{1+\alpha}{2}}}{(n+\theta)^{\frac{1-\beta}{2}}}\cdot|a_{k}|^{2}k!\right]^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle[B(\frac{1+\beta}{2},\frac{1-\alpha}{2})]^{\frac{1}{2}}(n+\theta)^{-\frac{\beta}{2}}$
$\displaystyle\quad\quad\quad\quad\times\left[\sum_{k=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(k+\theta)^{\frac{1+\alpha}{2}}}{(n+\theta)^{\frac{1-\beta}{2}}}\cdot|a_{k}|^{2}k!\right]^{\frac{1}{2}}.$

Consequently, it follows from (2.2) that

$\displaystyle\|\mathbf{\check{H}}_{\theta}f\|_{\theta,\beta}^{2}=\sum_{n=0}^{\infty}(n+\theta)^{\beta}\left|\sum_{k=0}^{\infty}\frac{a_{k}\sqrt{k!}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\right|^{2}$
$\displaystyle\leq
B(\frac{1+\beta}{2},\frac{1-\alpha}{2})\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{1}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\cdot\frac{(k+\theta)^{\frac{1+\alpha}{2}}}{(n+\theta)^{\frac{1-\beta}{2}}}\cdot|a_{k}|^{2}k!$
$\displaystyle=B(\frac{1+\beta}{2},\frac{1-\alpha}{2})\sum_{k=0}^{\infty}w_{\alpha,\beta}^{[2]}(k)|a_{k}|^{2}k!\leq[B(\frac{1+\beta}{2},\frac{1-\alpha}{2})]^{2}\|f\|_{\theta,\alpha}^{2}.$

This means that
$\mathbf{\check{H}}_{\theta}:\mathcal{F}^{2}_{\theta,\alpha}\rightarrow\mathcal{F}^{2}_{\theta,\beta}$
is bounded and Lemma 2.3 is proved. ∎

##  3\. Proof of Theorem 1.5

We first show the ”if” part. For
$f=\sum_{k=0}^{\infty}{a_{k}}z^{k}\in\mathcal{F}_{\theta,\alpha}^{2}$,
$n\in\mathbb{N}_{0}$, when $\lambda\geq 1+\frac{1}{2}(\beta-\alpha)$, we have

$\left|\sum_{k=0}^{\infty}\frac{a_{k}\sqrt{k!}}{(k+n)^{\lambda}+1}\right|\leq\sum_{k=0}^{\infty}\frac{|a_{k}|\sqrt{k!}}{(k+n)^{1+\frac{1}{2}(\beta-\alpha)}+1}\preceq\sum_{k=0}^{\infty}\frac{|a_{k}|\sqrt{k!}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}.$

Consequently, by Lemma 2.3, we obtain that
$\mathbf{{H}}_{\lambda}:\mathcal{F}^{2}_{\theta,\alpha}\rightarrow\mathcal{F}^{2}_{\theta,\beta}$
is bounded. This proves the ”if” part of Theorem 1.5.

Next, we shall prove the ”only if” part. We will show that, if
$\lambda<1+\frac{1}{2}(\beta-\alpha)$, then, for any $\theta>0$,
$\mathbf{H}_{\lambda}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is not bounded.

Actually, let $\varepsilon>0$ and set
$f_{\varepsilon}=\sum_{k=0}^{\infty}a_{k}z^{k}$ with

$a_{0}=0,\,a_{k}=\sqrt{\varepsilon\theta^{\varepsilon}}(k+\theta)^{-\frac{\alpha+1+\varepsilon}{2}}\frac{1}{\sqrt{k!}},\,k\in\mathbb{N}.$

It is easy to see that

$\|f_{\varepsilon}\|_{\theta,\alpha}^{2}=\varepsilon\theta^{\varepsilon}\sum_{k=1}^{\infty}(n+\theta)^{-1-\varepsilon}\leq\varepsilon\theta^{\varepsilon}\int_{0}^{\infty}(x+\theta)^{-1-\varepsilon}dx=1.$

Then we have

$\displaystyle\|\mathbf{H}_{\lambda}f_{\varepsilon}\|_{\theta,\beta}^{2}$
$\displaystyle=$
$\displaystyle\varepsilon\theta^{\varepsilon}\sum_{n=0}^{\infty}(n+\theta)^{\beta}\left|\sum_{k=1}^{\infty}\frac{1}{(k+n)^{\lambda}+1}\cdot(k+\theta)^{-\frac{\alpha+1+\varepsilon}{2}}\right|^{2}$
$\displaystyle\succeq$
$\displaystyle\varepsilon\theta^{\varepsilon}\sum_{n=0}^{\infty}(n+\theta)^{\beta}\left|\sum_{k=1}^{\infty}\frac{1}{(k+n+2\theta)^{\lambda}}\cdot(k+\theta)^{-\frac{\alpha+1+\varepsilon}{2}}\right|^{2}$

On the other hand, we notice that

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{(k+n+2\theta)^{\lambda}}\cdot(k+\theta)^{-\frac{\alpha+1+\varepsilon}{2}}$
$\displaystyle\geq$
$\displaystyle\int_{1}^{\infty}\frac{1}{(x+\theta+n+\theta)^{\lambda}}\cdot(x+\theta)^{-\frac{\alpha+1+\varepsilon}{2}}\,dx$
$\displaystyle=$
$\displaystyle\int_{1+\theta}^{\infty}\frac{1}{(s+n+\theta)^{\lambda}}\cdot
s^{-\frac{\alpha+1+\varepsilon}{2}}\,ds$ $\displaystyle=$
$\displaystyle(n+\alpha)^{1-\lambda-\frac{\alpha+1+\varepsilon}{2}}\int_{\frac{1+\theta}{n+\theta}}^{\infty}\frac{1}{(1+t)^{\lambda}}\cdot
t^{-\frac{\alpha+1+\varepsilon}{2}}\,dt$

Combine (3) and (3), we get that

$\displaystyle\|\mathbf{H}_{\lambda}f_{\varepsilon}\|_{\theta,\beta}^{2}$
$\displaystyle\succeq$
$\displaystyle\varepsilon\theta^{\varepsilon}\sum_{n=0}^{\infty}(n+\theta)^{(\beta-\alpha)+2(1-\lambda)-1-\varepsilon}\int_{\frac{1+\theta}{n+\theta}}^{\infty}\frac{1}{(1+t)^{\lambda}}\cdot
t^{-\frac{\alpha+1+\varepsilon}{2}}\,dt$ $\displaystyle\geq$
$\displaystyle\varepsilon\theta^{\varepsilon}\sum_{n=0}^{\infty}(n+\theta)^{(\beta-\alpha)+2(1-\lambda)-1-\varepsilon}\int_{1+\frac{1}{\theta}}^{\infty}\frac{1}{(1+t)^{\lambda}}\cdot
t^{-\frac{\alpha+1+\varepsilon}{2}}\,dt$

If $\lambda<1+\frac{1}{2}(\beta-\alpha)$, we see that
$(\beta-\alpha)+2(1-\lambda)>0$. We suppose that
$\mathbf{H}_{\lambda}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is bounded, then there exists a constant $C_{4}>0$ such that

$\displaystyle C_{4}$ $\displaystyle\geq$
$\displaystyle\frac{\|\mathbf{H}_{\lambda}f_{\varepsilon}\|_{\theta,\beta}^{2}}{\|f_{\varepsilon}\|_{\theta,\alpha}^{2}}$
$\displaystyle\geq$
$\displaystyle\varepsilon\theta^{\varepsilon}\sum_{n=0}^{\infty}(n+\theta)^{(\beta-\alpha)+2(1-\lambda)-1-\varepsilon}\int_{1+\frac{1}{\theta}}^{\infty}\frac{1}{(1+t)^{\lambda}}\cdot
t^{-\frac{\alpha+1+\varepsilon}{2}}\,dt.$

But when $\varepsilon<(\beta-\alpha)+2(1-\lambda)$, we have

$\sum_{n=0}^{\infty}(n+\theta)^{(\beta-\alpha)+2(1-\lambda)-1-\varepsilon}=\infty.$

Hence we get that (3) is a contradiction. This implies that
$\mathbf{H}_{\lambda}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
can not be bounded when $\lambda<1+\frac{1}{2}(\beta-\alpha)$. This proves the
”only if” part of Theorem 1.5.

The Theorem 1.5 is now proved.

##  4\. Proof of Theorem 1.6

###### Proof of ”if” part of Theorem 1.6.

By combining Lemma 2.2 and Lemma 2.3, we easily see that, for any $\theta>0$,
$\mathbf{H}_{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is bounded if $\mu$ is a $[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on
$[0,1)$. The ”if” part of Theorem 1.6 is proved. ∎

###### Proof of ”only if” part of Theorem 1.6.

In our proof, we need the following well-known estimate, see [18, Page 54].
Let $0<w<1$. For any $c>0$, we have

(4.1) $\sum_{n=1}^{\infty}n^{c-1}w^{2n}\asymp\frac{1}{(1-w^{2})^{c}}.$

For any $0<w<1$, we set
${f}_{w}=\sum_{k=0}^{\infty}{a}_{k}z^{k}\in\mathcal{H}(\mathbb{C})$ with

${a}_{k}=(1-w^{2})^{\frac{1}{2}}(k+\theta)^{-\frac{\alpha}{2}}w^{k}\frac{1}{\sqrt{k!}},\;k\in\mathbb{N}_{0}.$

Then we see from (4.1) that $\|{f}_{w}\|_{\theta,\alpha}^{2}\asymp 1.$

In view of the boundedness of
$\mathbf{H}_{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$,
we obtain that

$\displaystyle 1$ $\displaystyle\succeq$
$\displaystyle\|\mathbf{H}_{\mu}{f}_{w}\|_{\theta,\beta}^{2}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}(n+\theta)^{\beta}\left[\sum_{k=0}^{\infty}{a}_{k}\sqrt{k!}\int_{0}^{1}t^{k+n}d\mu(t)\right]^{2}$
$\displaystyle=$
$\displaystyle(1-w^{2})\sum_{n=0}^{\infty}(n+\theta)^{\beta}\left[\sum_{k=0}^{\infty}(k+\theta)^{-\frac{\alpha}{2}}w^{k}\int_{0}^{1}t^{k+n}d\mu(t)\right]^{{2}}$
$\displaystyle\succeq$
$\displaystyle(1-w^{2})\sum_{n=1}^{\infty}n^{\beta}\left[\sum_{k=1}^{\infty}k^{-\frac{\alpha}{2}}w^{k}\int_{w}^{1}t^{k+n}d\mu(t)\right]^{{2}}.$

On the other hand, by again (4.1), we have

$\displaystyle\sum_{n=1}^{\infty}n^{\beta}\left[\sum_{k=1}^{\infty}k^{-\frac{\alpha}{2}}w^{k}\int_{w}^{1}t^{k+n}d\mu(t)\right]^{{2}}$
$\displaystyle\geq$
$\displaystyle[\mu([w,1))]^{2}\sum_{n=1}^{\infty}n^{\beta}\left[\sum_{k=1}^{\infty}k^{-\frac{\alpha}{2}}w^{k}\cdot
w^{k+n}\right]^{{2}}$ $\displaystyle=$
$\displaystyle[\mu([w,1))]^{2}\left[\sum_{n=1}^{\infty}n^{\beta}w^{2n}\right]\left[\sum_{k=1}^{\infty}k^{-\frac{\alpha}{2}}w^{2k}\right]^{{2}}$
$\displaystyle\asymp$
$\displaystyle[\mu([w,1))]^{2}\cdot\frac{1}{(1-w^{2})^{1+\beta}}\cdot\frac{1}{(1-w^{2})^{2-\alpha}}.$

We see from (4) and (4) that

$\mu([w,1))\preceq(1-w^{2})^{1+\frac{1}{2}(\beta-\alpha)}$

holds for all $0<w<1$. It follows that $\mu$ is a
$[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on $[0,1)$ and the ”only if”
part is proved. ∎

Now, the proof of Theorem 1.6 is finished.

##  5\. Proof of Theorem 1.7

We first show the ”if” part. For any
$f=\sum_{k=0}^{\infty}a_{k}z^{k}\in\mathcal{F}_{\theta,\alpha}^{2}$. Let
$\mathfrak{N}\in\mathbb{N}$, we consider

$\mathbf{H}_{\mu}^{[\mathfrak{N}]}(f)(z):=\sum_{n=0}^{\mathfrak{N}}\left[\frac{1}{\sqrt{n!}}\sum_{k=0}^{\infty}\mu[k+n]a_{k}\sqrt{k!}\right]z^{n},z\in\mathbb{C}.$

Then we see that $\mathbf{H}_{\mu}^{[\mathfrak{N}]}$ is a finite rank operator
and hence it is compact from $\mathcal{F}_{\theta,\alpha}^{2}$ to
$\mathcal{F}_{\theta,\beta}^{2}$.

By Lemma 2.2, we see that, for any $\epsilon>0$, there is an
$N_{0}\in\mathbb{N}$ such that

$\mu[n]\preceq\frac{\epsilon}{(n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}$

holds for all $n>N_{0}$.

Note that

$\displaystyle\|(\mathbf{H}_{\mu}-\mathbf{H}_{\mu}^{[\mathfrak{N}]})(f)\|_{\theta,\beta}^{2}=\sum_{n=\mathfrak{N}+1}^{\infty}(n+\theta)^{\beta}\left|\frac{1}{\sqrt{n!}}\sum_{k=0}^{\infty}\mu[k+n]a_{k}\sqrt{k!}\right|^{2}.$

When $\mathfrak{N}>N_{0}$, we get that

$\displaystyle\|(\mathbf{H}_{\mu}-\mathbf{H}_{\mu}^{[\mathfrak{N}]})(f)\|_{\theta,\beta}^{2}\preceq{\epsilon}^{2}\sum_{n=\mathfrak{N}+1}^{\infty}(n+\theta)^{\beta}\left|\sum_{k=0}^{\infty}\frac{a_{k}}{(k+n+2\theta)^{1+\frac{1}{2}(\beta-\alpha)}}\right|^{2}.$

Consequently, by Lemma 2.3, we see that, for any $\epsilon>0$, there is an
$N_{0}\in\mathbb{N}$ such that

$\displaystyle\|(\mathbf{H}_{\mu}-\mathbf{H}_{\mu}^{[\mathfrak{N}]})(f)\|_{\theta,\beta}^{2}\preceq{\epsilon}^{2}\|f\|^{2}_{\theta,\alpha}$

holds for all $\mathfrak{N}>N_{0}$. We conclude that $\mathbf{H}_{\mu}$ is
compact from $\mathcal{F}_{\theta,\alpha}^{2}$ to
$\mathcal{F}_{\theta,\beta}^{2}$. This proves the ”if” part.

Next, we show the ”only if” part. For $0<w<1$. We set
$\widetilde{f}_{w}=\sum_{k=0}^{\infty}\widetilde{a}_{k}z^{k}$ with

$\widetilde{a}_{k}=(1-w^{2})^{\frac{1+\alpha}{2}}w^{k}\frac{1}{\sqrt{k!}},\>k\in\mathbb{N}_{0}.$

We see from (4.1) that $\|\widetilde{f}_{w}\|_{\theta,\alpha}\asymp 1$, and we
conclude that $\widetilde{f}_{w}$ is convergent weakly to $0$ in
$\mathcal{F}_{\theta,\alpha}^{2}$ as $w\rightarrow 1^{-}$. Since
$\mathbf{H}_{\mu}$ is compact from $\mathcal{F}_{\theta,\alpha}^{2}$ to
$\mathcal{F}_{\theta,\beta}^{2}$, we get

(5.1)
$\lim_{w\rightarrow{1^{-}}}\|\mathbf{H}_{\mu}(\widetilde{f}_{w})\|_{\theta,\beta}=0.$

On the other hand, we have

$\displaystyle\|\mathbf{H}_{\mu}(\widetilde{f}_{w})\|_{\theta,\beta}^{2}$
$\displaystyle=$
$\displaystyle(1-w^{2})^{1+\alpha}\sum_{n=0}^{\infty}(n+\theta)^{\beta}\left[\sum_{k=0}^{\infty}w^{k}\int_{0}^{1}t^{k+n}d\mu(t)\right]^{{2}}$
$\displaystyle\succeq$
$\displaystyle(1-w^{2})^{1+\alpha}\sum_{n=1}^{\infty}n^{\beta}\left[\sum_{k=1}^{\infty}w^{k}\int_{w}^{1}t^{k+n}d\mu(t)\right]^{{2}}$
$\displaystyle\geq$
$\displaystyle(1-w^{2})^{1+\alpha}[\mu([w,1))]^{2}\left[\sum_{n=1}^{\infty}n^{\beta}w^{2n}\right]\left[\sum_{k=1}^{\infty}w^{2k}\right]^{{2}}$
$\displaystyle\succeq$
$\displaystyle(1-w^{2})^{1+\alpha}[\mu([w,1))]^{2}\cdot\frac{1}{(1-w^{2})^{\beta+1}}\cdot\frac{1}{(1-w^{2})^{2}}.$

Then we get that

$\displaystyle\mu([w,1))\preceq\|\mathbf{H}_{\mu}(\widetilde{f}_{w})\|_{\theta,\beta}(1-w^{2})^{1+\frac{1}{2}(\beta-\alpha)}.$

It follows from (5.1) that $\mu$ is a vanishing
$[1+\frac{1}{2}(\beta-\alpha)$-Carleson measure on $[0,1)$. This proves the
”only if” part of Theorem 1.7 and the proof of Theorem 1.7 is completed.

##  6\. Final remarks

######  Remark 6.1.

For $\lambda>0$, we define the Hilbert-type operator
$\mathbf{\widehat{H}}_{\lambda}(f)$ as

$\mathbf{\widehat{H}}_{\lambda}(f)(z):=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_{k}}{(k+n+1)^{\lambda}}\right)z^{n}.$

When $\lambda=1$, we get the classical Hilbert operator. We next will study
the boundedness of $\mathbf{\widehat{H}}_{\lambda}(f)$ acting on certain
spaces of holomorphic functions on $\mathbb{D}$.

Let $p$ be a positive number and $X_{p}$ be a Banach space of holomorphic
functions on $\mathbb{D}$. For any $f\in X_{p}$, we assume that the norm
$\|f\|_{X_{p}}$ of $f$ is determined by $f,p$ and other finite parameters
$\beta_{1},\beta_{2},\cdots,\beta_{m}$; $m\in\mathbb{N}_{0}$($m=0$ means that
there is no parameter).

For any $f=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{H}(\mathbb{D})$ with
$\\{a_{n}\\}_{n=0}^{\infty}$ is a decreasing sequence of non-negative real
numbers, we say $X_{p}$ have the property $(\star)$, if there exist a function
$\mathbf{G}_{X}=\mathbf{G}_{X}(p,\beta_{1},\beta_{2},\cdots,\beta_{m})$ with
$\mathbf{G}_{X}>-1$ such that $f\in X_{p}$ if and only if

$\sum_{n=1}^{\infty}n^{\mathbf{G}_{X}}a_{n}^{p}<\infty.$

We list several classical spaces of holomorphic functions on $\mathbb{D}$,
which have the property $(\star)$. Let
$f=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{H}(\mathbb{D})$ with
$\\{a_{n}\\}_{n=0}^{\infty}$ is a decreasing sequence of non-negative real
numbers. For example,

(1) the Hardy space $H^{p}(\mathbb{D}),1<p<\infty$, we know that $f\in
H^{p}(\mathbb{D})$ if and only if

$\sum_{n=1}^{\infty}n^{p-2}a_{n}^{p}<\infty.$

See [16].

(2) For $1<p<\infty$, let $p-2<\alpha\leq p-1$. It holds that $f\in
D^{p}_{\alpha}(\mathbb{D})$ if and only if

$\sum_{n=1}^{\infty}n^{2p-3-\alpha}a_{n}^{p}<\infty.$

Here $D^{p}_{\alpha}(\mathbb{D})$ is the Dirichlet-type space, defined as

$D^{p}_{\alpha}(\mathbb{D})=\left\\{f\in\mathcal{H}({\mathbb{D}}):\|f\|_{D^{p}_{\alpha}}=|f(0)|+\left[\int_{\mathbb{D}}|f^{\prime}(z)|^{p}(1-|z|^{2})^{\alpha}dA(z)\right]^{\frac{1}{p}}<\infty\right\\}.$

See [12].

(3) For $1<p<\infty$, when $-1<\alpha<p-2$, we have $f\in
A^{p}_{\alpha}(\mathbb{D})$ if and only if

$\sum_{n=1}^{\infty}n^{2p-3-\alpha}a_{n}^{p}<\infty.$

Here $A^{p}_{\alpha}(\mathbb{D})$ is the Bergman space, defined as

$A^{p}_{\alpha}(\mathbb{D})=\left\\{f\in\mathcal{H}({\mathbb{D}}):\|f\|_{A^{p}_{\alpha}}=\left[(\alpha+1)\int_{\mathbb{D}}|f(z)|^{p}(1-|z|^{2})^{\alpha}dA(z)\right]^{\frac{1}{p}}<\infty\right\\}.$

See [12].

We obtain that

######  Proposition 6.2.

Let $\lambda$, $p$ be two positive numbers, $X_{p}$ be a Banach space of
holomorphic functions on $\mathbb{D}$ having the property $(\star)$ and
$\mathbf{\widehat{H}}_{\lambda}$ be as above. Then the necessary condition of
$\mathbf{\widehat{H}}_{\lambda}:X_{p}\rightarrow X_{p}$ is bounded is
$\lambda\geq 1$.

###### Proof.

It is enough to prove that, $\mathbf{\widehat{H}}_{\lambda}:X_{p}\rightarrow
X_{p}$ can not be bounded, if $0<\lambda<1$.

Let $\varepsilon>0$ and set
$\widehat{f}_{\varepsilon}=\sum_{k=0}^{\infty}\widehat{a}_{k}z^{k}$ with
$\widehat{a}_{0}=(\frac{\varepsilon}{1+\varepsilon})^{\frac{1}{p}},\,\widehat{a}_{k}=(\frac{\varepsilon}{1+\varepsilon})^{\frac{1}{p}}k^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}},\,k\geq
1.$ It is easy to see that $\\{\widehat{a}_{k}\\}_{k=0}^{\infty}$ is a
decreasing sequence and
$\sum_{k=1}^{\infty}k^{\mathbf{G}_{X}}\widehat{a}_{k}^{p}<\infty.$ Hence
$\widehat{f}_{\varepsilon}\in X_{p}$.

Set

$b_{n}=\sum_{k=0}^{\infty}\frac{\widehat{a}_{k}}{(k+n+1)^{\lambda}},\,n\in\mathbb{N}_{0}.$

We suppose that $\mathbf{\widehat{H}}_{\lambda}:X_{p}\rightarrow X_{p}$ is
bounded. Then, by the fact that $\\{b_{n}\\}_{n=0}^{\infty}$ is a decreasing
sequence, we see that $\widehat{f}=\sum_{n=0}^{\infty}b_{n}z^{n}\in X_{p}$ and
hence

$\sum_{n=1}^{\infty}n^{\mathbf{G}_{X}}b_{n}^{p}<\infty.$

Then

$\displaystyle\infty$ $\displaystyle>$
$\displaystyle\sum_{n=1}^{\infty}n^{\mathbf{G}_{X}}\left[\sum_{k=0}^{\infty}\frac{\widehat{a}_{k}}{(k+n+1)^{\lambda}}\right]^{p}$
$\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{1+\varepsilon}\sum_{n=1}^{\infty}n^{\mathbf{G}_{X}}\left[\sum_{k=1}^{\infty}\frac{1}{(k+n+1)^{\lambda}}\cdot
k^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\right]^{p}$

On the other hand, from the fact that $k+n+1\leq 2(k+n)$ for all
$k,n\in\mathbb{N}$, we see that

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{(k+n+1)^{\lambda}}\cdot
k^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2^{\lambda}}\sum_{k=1}^{\infty}\frac{1}{(k+n)^{\lambda}}\cdot
k^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2^{\lambda}}\int_{1}^{\infty}\frac{1}{(x+n)^{\lambda}}\cdot
x^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\,dx$ $\displaystyle=$
$\displaystyle
n^{(1-\lambda)-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\cdot\frac{1}{2^{\lambda}}\int_{\frac{1}{n}}^{\infty}\frac{1}{(1+s)^{\lambda}}\cdot
s^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\,ds$ $\displaystyle\geq$
$\displaystyle
n^{(1-\lambda)-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\cdot\frac{1}{2^{\lambda}}\int_{1}^{\infty}\frac{1}{(1+s)^{\lambda}}\cdot
s^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\,ds$

Combine (6.1) and (6.1), we get that

(6.3) $\displaystyle\infty$ $\displaystyle>$
$\displaystyle\frac{\varepsilon}{1+\varepsilon}\left[\sum_{n=1}^{\infty}n^{p(1-\lambda)-1-\varepsilon}\right]\cdot\left[\frac{1}{2^{\lambda}}\int_{1}^{\infty}\frac{1}{(1+s)^{\lambda}}\cdot
s^{-\frac{\mathbf{G}_{X}+1+\varepsilon}{p}}\,ds\right]^{p}.$

If $\lambda<1$, when $\varepsilon<p(1-\lambda)$, we have
$\sum_{n=1}^{\infty}n^{p(1-\lambda)-1-\varepsilon}=\infty.$ Thus (6.3) is a
contradiction. This means that
$\mathbf{\widehat{H}}_{\lambda}:X_{p}\rightarrow X_{p}$ can not be bounded if
$\lambda<1$. The Theorem 6.2 is proved. ∎

######  Remark 6.3.

From Theorem 1.5, we know that $\mathbf{H}_{\lambda}$ is not bounded from
$\mathcal{F}_{\theta,\alpha}^{2}$ to $\mathcal{F}_{\theta,\beta}^{2}$ if
$0<\lambda<1+\frac{1}{2}(\beta-\alpha)$. On the other hand, we notice that,
for $\lambda>0$, it holds that

$\frac{1}{(k+n)^{\lambda}+1}\asymp\frac{1}{(k+n+1)^{\lambda}}\asymp\int_{0}^{1}t^{k+n}(1-t)^{\lambda-1}dt$

for all $k,n\in\mathbb{N}_{0}$. To make $\mathbf{H}_{\lambda}$ bounded from
$\mathcal{F}_{\theta,\alpha}^{2}$ to $\mathcal{F}_{\theta,\beta}^{2}$ when
$0<\lambda<1+\frac{1}{2}(\beta-\alpha)$. For
$\lambda>0,f=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{H}(\mathbb{C})$, let
$\mu$ be a positive Bore measure on $[0,1)$, we define a new Hilbert-type
operator $\mathbf{H}_{\lambda}^{\mu}$ as

$\mathbf{H}_{\lambda}^{\mu}(f)(z):=\sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{n!}}\sum_{k=0}^{\infty}\mu_{\lambda}[k+n]a_{k}\sqrt{k!}\right)z^{n}.$

Here

$\mu_{\lambda}{[k+n]}=\int_{0}^{1}t^{k+n}(1-t)^{\lambda-1}d\mu(t),\>k,n\in\mathbb{N}_{0}.$

In view of the proofs of Theorem 1.6 and 1.7, we can get the following result,
which is a generalization of Theorem 1.6 and 1.7.

######  Proposition 6.4.

Let $\lambda>0,-1<\alpha,\beta<1$. Let $\mu$ be a positive Bore measure on
$[0,1)$ such that $d\nu(t)=(1-t)^{\lambda-1}d\mu(t)$ is a finite measure on
$[0,1)$, and $\mathbf{H}_{\lambda}^{\mu}$ be as above. Then, for any
$\theta>0$, it holds that

(1)
$\mathbf{H}_{\lambda}^{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is bounded if and only if $\nu$ is a $[1+\frac{1}{2}(\beta-\alpha)]$-Carleson
measure on $[0,1)$;

(2)
$\mathbf{H}_{\lambda}^{\mu}:\mathcal{F}_{\theta,\alpha}^{2}\rightarrow\mathcal{F}_{\theta,\beta}^{2}$
is compact if and only if $\nu$ is a vanishing
$[1+\frac{1}{2}(\beta-\alpha)]$-Carleson measure on $[0,1)$.

## Acknowledgement

We thank the referee for invaluable comments and useful suggestions on this
paper.

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