# A characterization of $(\mu,\nu)$-dichotomies via admissibility

Lucas Backes  0000-0003-3275-1311 Departamento de Matemática, Universidade
Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto
Alegre, RS, Brazil<EMAIL_ADDRESS>and  Davor Dragičević
0000-0002-1979-4344 Department of Mathematics, University of Rijeka, Croatia
<EMAIL_ADDRESS>
###### Abstract.

We present a characterization of $(\mu,\nu)$-dichotomies in terms of the
admissibility of certain pairs of weighted spaces for nonautonomous discrete
time dynamics acting on Banach spaces. Our general framework enables us to
treat various settings in which no similar result has been previously obtained
as well as to recover and refine several known results. We emphasize that our
results hold without any bounded growth assumption and the statements make no
use of Lyapunov norms. Moreover, as a consequence of our characterization, we
study the robustness of $(\mu,\nu)$-dichotomies, i.e. we show that this notion
persists under small but very general linear perturbations.

###### Key words and phrases:

discrete nonautonomous systems, $(\mu,\nu)$-dichotomy, admissibility,
persistence

###### 2020 Mathematics Subject Classification:

Primary: 34D09, 39A06; Secondary: 37D25

##  1\. Introduction

Hyperbolicity is one of the cornerstone notions in the modern theory of
dynamical systems. Roughly speaking, a system is said to be hyperbolic if the
phase space splits into two complementary directions such that along one of
these directions we have exponential expansion with time, while in the other
one we have exponential contraction. On the other hand, sometimes it may be
very complicated to verify if a given system is hyperbolic. Consequently, an
important problem consists in presenting different characterizations of this
property.

In the present paper we are interested in characterizing the nonautonomous
version of hyperbolicity. More precisely, we will present a characterization
of the notion of _$(\mu,\nu)$ -dichotomy_. The notion of $(\mu,\nu)$-dichotomy
requires that the phase space splits (at each moment of time) into two
directions: the stable and the unstable direction. Along stable/unstable
direction dynamics contracts/expands with the rate of contraction/expansion
given by a function $\mu$, while the speed of contraction/expansion is
measured using a function $\nu$. We stress that the notion of
$(\mu,\nu)$-dichotomy includes the notion of (nonuniform) exponential
dichotomy as a very particular case. The type of characterization we are
looking for has a long history, which goes back to the work of Perron [33] for
ODEs, and is given in terms of what is nowadays called the _admissibility
property_.

Given a nonautonomous dynamical system

$x_{n+1}=A_{n}x_{n},\quad n\in I,$ (1)

where $A_{n}\colon X\to X$, $n\in I$, are linear maps acting on a Banach space
$(X,\|\cdot\|)$ and $I$ is an interval of ${\mathbb{Z}}$, we say that the pair
$(Y,Z)$ is _(properly) admissible_ for Eq. (1), where $Y$ and $Z$ are
subspaces of $X^{I}$, if for every sequence $(y_{n})_{n\in I}$ in $Y$ there
exists a (unique) sequence $(x_{n})_{n\in I}$ in $Z$ such that

$x_{n+1}=A_{n}x_{n}+y_{n+1},\quad\text{ for all }n\in I.$

Thus, a prototype result says that for $I={\mathbb{Z}}$, if
$Y=Z=\ell^{\infty}$ (the space of all bounded two-sided sequences in $X$) then
the proper admissibility of the pair $(Y,Z)$ is equivalent to the fact that
Eq. (1) admits an exponential dichotomy (see [19]) and the characterizations
of $(\mu,\nu)$-dichotomy that we are seeking have this flavor. Nevertheless,
our results involve two or three (depending on $I$) admissible pairs of some
special _weighted spaces_ instead of just $\ell^{\infty}$ (see Sections 3 and
4).

The importance of our results stems from our general framework. More
precisely, we are able to treat in a unified manner various settings in which
no similar result has been previously obtained as well as to recover and
refine several known results. We emphasize that our results hold without any
bounded growth assumption for (1) and the statements make no use of Lyapunov
norms. We explain a bit more the importance of these facts bellow. Moreover,
as a consequence of our characterization, we obtain that the notion of
$(\mu,\nu)$-dichotomy persists under small but very general linear
perturbations.

###  1.1. Relations with previous results

As already emphasized, characterizations of dichotomies in terms of
admissibility properties have a long history. For instance, a fundamental
contribution to this line of research is due to Massera and Schäffer [25, 26]
(see also Coppel [10]) who presented a complete characterization (in terms of
admissibility) of the notion of an exponential dichotomy, extending the
original work of Perron [33] (see also [21] for related results in the case of
discrete time) which dealt with exponentially stable systems. The case of
infinite-dimensional dynamics were first considered by Dalec′kiĭ and Kreĭn
[11] in the case of continuous time and by Coffman and Schäffer [9] as well as
Henry [19] in the case of noninvertible discrete time dynamics. More recent
contributions devoted to the characterization of uniform exponential
dichotomies include (but are not limited to) [1, 20, 22, 28, 29, 36, 39, 40].
We also refer to [5] for a detailed overview of this line of research and
additional references.

In the case of _nonuniform_ exponential dichotomies, there are essentially
three type of results available. Firstly, there are results which do not give
a complete characterization of nonuniform exponential dichotomies via
admissibility but rather only sufficient conditions for the existence of
dichotomy (see [27, 35, 37, 38] and references therein). Secondly, there are
various results in which a complete characterization of nonuniform dichotomies
via admissibility is obtained in which the corresponding input-output spaces
are constructed in terms of Lyapunov norms which are used to transform
nonuniform behavior into the uniform one (see [2, 3, 4, 23, 24, 45]). Finally,
since Lyapunov norms are difficult to construct without firstly establishing
the existence of nonuniform behavior, it was of interest to explore whether
nonuniform exponential dichotomies can be characterized in terms of
admissibility without the use of Lyapunov norms. The number of very recent
results show that this is indeed possible (see [17, 18, 44, 46]).

As already mentioned, besides exponential dichotomies, it of interest to study
dichotomic behavior when the rates of expansion/contraction along the
unstable/stable direction are not of exponential type. To the best of our
knowledge such dichotomies were first studied by Muldowney [31] and Naulin and
Pinto [32]. More recently, a systematic study of such behavior was initiated
by Barreira and Valls [6], as well as Bento and Silva [7, 8]. Dragičević [12]
characterized nonuniform polynomial dichotomies in terms of admissibility for
discrete dynamics (see [13] for related result for continuous time). An
alternative approach which relies on the relationship between exponential and
polynomial dichotomies was proposed in [15]. In the case of continuous time, a
very general class of dichotomies associated to differentiable growth rates
was characterized via admissibility in [14]. Finally, the most general results
in the case of discrete time were obtained by Silva [41]. We stress that all
these works rely on the usage of Lyapunov norms.

In the present paper we aim to characterize $(\mu,\nu)$-dichotomies in terms
of admissibility without the use of Lyapunov norms. Therefore, our results
complement those obtained by Silva [41]. We emphasize that even in the case of
uniform dichotomies associated to a growth rate $\mu$, our results do not
coincide with those in [41] as the input-output spaces are different.
Moreover, in contrast to [41] we:

  1. (1)

do not impose any bounded growth conditions;

  2. (2)

are able to treat the case of arbitrary growth rates, while in [41] it is
required that $\mu$ is “slowly growing”;

  3. (3)

discuss the case of two-sided dynamics which was not treated in [41].

Our proof was inspired by the work of Dragičević, Zhang and Zhou [17, 18] in
which the authors completely characterize, in terms of admissibility, the
notion of _nonuniform exponential dichotomy_ , which, as already mentioned, is
a particular example of $(\mu,\nu)$-dichotomy.

The paper is organized as follows. In Section 2, we introduce the notion of
$(\mu,\nu)$-dichotomy and the weighted spaces we are going to work with along
the text. Section 3 is devoted to present the characterization as well as the
robustness of $(\mu,\nu)$-dichotomies in the case of one-sided dynamics while
Section 4 is devoted to study the two-sided case.

##  2\. Preliminaries

Let $X=(X,\|\cdot\|)$ be an arbitrary Banach space and $I$ be either equal to
${\mathbb{Z}}$ or ${\mathbb{N}}$. By $\mathcal{B}(X)$ we will denote the space
of all bounded linear operators on $X$. The operator norm on $\mathcal{B}(X)$
will be denoted also by $\|\cdot\|$. Given a sequence $(A_{n})_{n\in I}$ of
bounded linear operators in $\mathcal{B}(X)$, let us consider the associated
linear difference equation

$x_{n+1}=A_{n}x_{n},\quad n\in I.$ (2)

For $m,n\in I$, the evolution operator associated to (2) is given by

${\mathcal{A}}(m,n)=\begin{cases}A_{m-1}\cdots A_{n}&\text{for $m>n$;}\\\
\text{Id}&\text{for $m=n$,}\\\ \end{cases}$

where Id denotes the identity operator on $X$.

###  2.1. Growth rates and $(\mu,\nu)$-dichotomy

Let $\mu=(\mu_{n})_{n\in I}$ be a strictly increasing sequence of positive
numbers such that

$\lim_{n\to+\infty}\mu_{n}=+\infty.$ (3)

Moreover, in the case when $I={\mathbb{Z}}$ we assume also that
$\lim\limits_{n\to-\infty}\mu_{n}=0$. We call such sequence $\mu$ a _growth
rate_. Furthermore, let $\nu=(\nu_{n})_{n\in I}$ be an arbitrary sequence with
$\nu_{n}\geq 1$ for every $n\in I$.

######  Definition 2.1.

We say that (2) admits a $(\mu,\nu)$-dichotomy if the following conditions are
satisfied:

  1. (1)

there exists a family of projections $P_{n}$, $n\in I$, such that

$A_{n}P_{n}=P_{n+1}A_{n};$ (4)

  2. (2)

$A_{n}\rvert_{\operatorname{Ker}P_{n}}\colon\operatorname{Ker}P_{n}\to\operatorname{Ker}P_{n+1}$
is an invertible operator for each $n\in I$;

  3. (3)

there exist $D,\lambda>0$ such that

$\|{\mathcal{A}}(m,n)P_{n}\|\leq
D\nu_{n}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\lambda}\quad\text{ for }m\geq
n$ (5)

and

$\|{\mathcal{A}}(m,n)(\text{Id}-P_{n})\|\leq
D\nu_{n}\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\lambda}\quad\text{ for }m\leq
n$ (6)

where

${\mathcal{A}}(m,n):=\big{(}{\mathcal{A}}(n,m)\rvert_{\operatorname{Ker}P_{m}}\big{)}^{-1}\colon\operatorname{Ker}P_{n}\to\operatorname{Ker}P_{m},$

for $m\leq n$.

######  Remark 2.2.

We observe that the notion of $(\mu,\nu)$-dichotomy, which appeared earlier,
for instance, in [7, 8, 34], generalizes several well-known notions of
dichotomy. To illustrate this claim, let us restrict ourselves to the case
when $I={\mathbb{N}}$ and suppose initially that $\nu_{n}=C$ for some $C\geq
1$ and $n\in{\mathbb{N}}$. Then, by taking $\mu_{n}=e^{n}$,
$n\in{\mathbb{N}}$, we recover the notion of _exponential dichotomy_ (see [10,
25, 26]); by taking $\mu_{n}=1+n$, $n\in{\mathbb{N}}$, we recover the notion
of _polynomial dichotomy_ (see [12]); by taking $\mu_{n}=\log(2+n)$,
$n\in{\mathbb{N}}$, we recover the notion of _logarithmic dichotomy_ (see
[41]). In all these cases, if we take $\nu_{n}=\mu_{n}^{\varepsilon}$ instead
of $\nu_{n}=C$ for some small $\varepsilon>0$ and $n\in{\mathbb{N}}$, we get
_nonuniform_ versions of those dichotomies.

###  2.2. Weighted spaces

In order to describe our main results, we need to introduce a number of
special _weighted spaces_. Given a growth rate $\mu=(\mu_{n})_{n\in I}$ and
$\beta\in{\mathbb{R}}$, let us consider the weighted space
$\ell_{\beta}^{\infty}$ which consist of all sequences
$\mathbf{x}=(x_{n})_{n\in I}\subset X$ such that

$\|\mathbf{x}\|_{\infty,\beta}:=\sup_{n\in
I}(\mu_{n}^{\beta}\|x_{n}\|)<+\infty.$

Then, it is not difficult to check that $\|\cdot\|_{\infty,\beta}$ is a norm
in $\ell_{\beta}^{\infty}$ and
$(\ell_{\beta}^{\infty},\|\cdot\|_{\infty,\beta})$ is a Banach space.
Similarly, if $\nu=(\nu_{n})_{n\in I}$ is a sequence such that $\nu_{n}\geq 1$
for each $n\in I$, we consider the weighted space $\ell^{1}_{\beta}$ which
consist of all sequences $\mathbf{x}=(x_{n})_{n\in I}\subset X$ such that

$\|\mathbf{x}\|_{1,\beta}:=\sum_{n\in
I}\mu_{n}^{\beta}\nu_{n}\|x_{n}\|<+\infty.$

Again it is not difficult to verify that
$(\ell_{\beta}^{1},\|\cdot\|_{1,\beta})$ is a Banach space. Moreover, given a
closed subspace $Z\subset X$, for $j\in\\{1,\infty\\}$, let
$\ell_{\beta,Z}^{j}$ be the space which consists of all sequences
$\mathbf{x}=(x_{n})_{n\in I}\in\ell_{\beta}^{j}$ such that $x_{0}\in Z$. Then,
$\ell_{\beta,Z}^{j}$ is a closed subspace of $\ell_{\beta}^{j}$ and, in
particular, $(\ell_{\beta,Z}^{j},\|\cdot\|_{j,\beta})$ is a Banach space. In
the particular case when $Z=\\{0\\}$, $\ell^{1}_{\beta,Z}$ will be denoted by
$\ell^{1}_{\beta,0}$.

In the case when $I={\mathbb{Z}}$, we will also need to consider some extra
spaces described as follows. Observe initially that, since
$(\mu_{n})_{n\in{\mathbb{Z}}}$ is strictly increasing and
$\lim_{n\to-\infty}\mu_{n}=0$ and $\lim_{n\to+\infty}\mu_{n}=+\infty$, there
exists $n_{0}\in{\mathbb{Z}}$ such that $\mu_{n}<1$ for every $n<n_{0}$ and
$\mu_{n}\geq 1$ for every $n\geq n_{0}$. Then, we consider the space
$\ell_{\beta,|\cdot|}^{\infty}$ which consist of all sequences
$\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\subset X$ such that

$\|\mathbf{x}\|_{\infty,\beta,|\cdot|}:=\max\left\\{\sup_{n<n_{0}}(\mu_{n}^{|\beta|}\|x_{n}\|),\sup_{n\geq
n_{0}}(\mu_{n}^{-|\beta|}\|x_{n}\|)\right\\}<+\infty.$

Similarly, let $\ell^{1}_{\beta,|\cdot|}$ be the space consisting of all
sequences $\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\subset X$ such that

$\|\mathbf{x}\|_{1,\beta,|\cdot|}:=\sum_{n<n_{0}}\mu_{n}^{|\beta|}\nu_{n}\|x_{n}\|+\sum_{n\geq
n_{0}}\mu_{n}^{-|\beta|}\nu_{n}\|x_{n}\|<+\infty.$

Once again one can easily check that
$(\ell_{\beta,|\cdot|}^{\infty},\|\cdot\|_{\infty,\beta,|\cdot|})$ and
$(\ell_{\beta,|\cdot|}^{1},\|\cdot\|_{1,\beta,|\cdot|})$ are Banach spaces.
Observe that, even though all the spaces defined above depend on $\mu$, $\nu$
and $I$, we do not write these dependence explicitly in order to simplify
notation. The dependence will be clear from the context.

##  3\. The case of one-sided dynamics

In this section we will restrict our attention to the case of one-sided
dynamics, that is, to the case when $I={\mathbb{N}}$.

###  3.1. Characterization of $(\mu,\nu)$-dichotomy

We will now present a characterization of $(\mu,\nu)$-dichotomy in terms of
the admissibility of the spaces $\ell^{1}_{\beta,0}$ and
$\ell^{\infty}_{\beta,Z}$ for some appropriate subspace $Z\subset X$ and
$\beta\in{\mathbb{R}}$.

######  Theorem 3.1.

Suppose that $(A_{n})_{n\in\mathbb{N}}$ admits a $(\mu,\nu)$-dichotomy with
respect to projections $P_{n}$ and let $\lambda>0$ be such that (5) and (6)
hold. Moreover, suppose that there exists $\varepsilon\in[0,\lambda)$ such
that

$\sup_{n\in\mathbb{N}}(\mu_{n}^{-\varepsilon}\nu_{n})<+\infty.$ (7)

Set $Z:=\operatorname{Ker}P_{0}$. Then, for each
$\beta\in(-(\lambda-\varepsilon),\lambda)$ and
$\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{\beta,0}$, there exists a
unique $\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{\beta,Z}^{\infty}$ such
that

$x_{n+1}-A_{n}x_{n}=y_{n+1},\quad n\in\mathbb{N}.$ (8)

###### Proof.

Take $\beta\in(-(\lambda-\varepsilon),\lambda)$,
$\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{\beta,0}$ and set

$x_{n}:=\sum_{k=0}^{n}\mathcal{A}(n,k)P_{k}y_{k}-\sum_{k=n+1}^{\infty}\mathcal{A}(n,k)(\text{Id}-P_{k})y_{k},\quad
n\in\mathbb{N}.$ (9)

Then,

$\begin{split}\mu_{n}^{\beta}\|x_{n}\|&\leq
D\mu_{n}^{\beta}\sum_{k=0}^{n}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-\lambda}\nu_{k}\|y_{k}\|+D\mu_{n}^{\beta}\sum_{k=n+1}^{\infty}\left(\frac{\mu_{k}}{\mu_{n}}\right)^{-\lambda}\nu_{k}\|y_{k}\|\\\
&\leq
D\sum_{k=0}^{n}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-(\lambda-\beta)}\mu_{k}^{\beta}\nu_{k}\|y_{k}\|+D\sum_{k=n+1}^{\infty}\left(\frac{\mu_{k}}{\mu_{n}}\right)^{-(\lambda+\beta)}\mu_{k}^{\beta}\nu_{k}\|y_{k}\|,\\\
\end{split}$

which implies (since $\lambda-\beta>0$ and $\lambda+\beta>0$) that

$\mu_{n}^{\beta}\|x_{n}\|\leq
D\sum_{k=0}^{\infty}\mu_{k}^{\beta}\nu_{k}\|y_{k}\|,\quad n\in\mathbb{N}.$

Note also that $x_{0}\in Z$ and, consequently,
$\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell^{\infty}_{\beta,Z}$. Moreover, it
is easy to verify that $\mathbf{x}$ satisfies (8). Indeed, given
$n\in{\mathbb{N}}$,

$\begin{split}A_{n}x_{n}&=A_{n}\sum_{k=0}^{n}{\mathcal{A}}(n,k)P_{k}y_{k}-A_{n}\sum_{k=n+1}^{\infty}{\mathcal{A}}(n,k)(\text{Id}-P_{k})y_{k}\\\
&=\sum_{k=0}^{n}{\mathcal{A}}(n+1,k)P_{k}y_{k}-\sum_{k=n+1}^{\infty}{\mathcal{A}}(n+1,k)(\text{Id}-P_{k})y_{k}\\\
&=\sum_{k=0}^{n+1}{\mathcal{A}}(n+1,k)P_{k}y_{k}-P_{n+1}y_{n+1}\\\
&\phantom{=}-\sum_{k=n+2}^{\infty}{\mathcal{A}}(n+1,k)(\text{Id}-P_{k})y_{k}-(\text{Id}-P_{n+1})y_{n+1}\\\
&=x_{n+1}-y_{n+1},\end{split}$

as claimed. It remains to establish the uniqueness of such sequence
$\mathbf{x}$. Suppose that
$\tilde{\mathbf{x}}=(\tilde{x}_{n})_{n\in\mathbb{N}}$ is another sequence in
$\ell^{\infty}_{\beta,Z}$ that satisfies (8). Then,

$x_{n}-\tilde{x}_{n}=\mathcal{A}(n,0)(x_{0}-\tilde{x}_{0}),\quad
n\in\mathbb{N}.$

Consequently, since $x_{0}-\tilde{x}_{0}\in Z=\operatorname{Ker}P_{0}$,
condition (6) gives us that

$\begin{split}\mu_{0}^{-\lambda}\|x_{0}-\tilde{x}_{0}\|&\leq
D\mu_{n}^{-\lambda}\nu_{n}\|x_{n}-\tilde{x}_{n}\|\\\ &\leq
D\mu_{n}^{-(\lambda+\beta)}\nu_{n}\|\mathbf{x}-\tilde{\mathbf{x}}\|_{\infty,\beta}\\\
&\leq
D\mu_{n}^{-(\lambda+\beta-\varepsilon)}\|\mathbf{x}-\tilde{\mathbf{x}}\|_{\infty,\beta}\cdot\sup_{n\in\mathbb{N}}(\mu_{n}^{-\varepsilon}\nu_{n}),\end{split}$

for $n\in\mathbb{N}$. Letting $n\to\infty$, using (7) and
$\beta>-(\lambda-\varepsilon)$ yields that $x_{0}=\tilde{x}_{0}$ and thus
$\mathbf{x}=\tilde{\mathbf{x}}$. The proof of the theorem is completed. ∎

We now establish the converse result.

######  Theorem 3.2.

Suppose that there exist a closed subspace $Z\subset X$ and $\beta>0$ such
that:

  1. (1)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{\beta,0}$, there
exists a unique
$\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{\beta,Z}^{\infty}$ such that (8)
holds;

  2. (2)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{-\beta,0}$, there
exists a unique
$\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{-\beta,Z}^{\infty}$ such that (8)
holds.

Then, $(A_{n})_{n\in\mathbb{N}}$ admits a $(\mu,\nu)$-dichotomy.

###### Proof.

We will split the proof into a series of auxiliary results starting with the
following lemma which follows directly from our assumptions and the simple
observation that $\ell^{\infty}_{\beta,Z}\subset\ell^{\infty}_{-\beta,Z}$.

######  Lemma 3.3.

Let
$\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{\beta,0}\cap\ell^{1}_{-\beta,0}$,
$\mathbf{x}^{1}=(x_{n}^{1})_{n\in\mathbb{N}}\in\ell^{\infty}_{\beta,Z}$ and
$\mathbf{x}^{2}=(x_{n}^{2})_{n\in\mathbb{N}}\in\ell^{\infty}_{-\beta,Z}$ be
such that both pairs $(\mathbf{x}^{i},\mathbf{y})$, $i\in\\{1,2\\}$ satisfy
(8). Then, $x_{n}^{1}=x_{n}^{2}$ for every $n\in\mathbb{N}$.

Let us consider

$S(n):=\left\\{v\in X:\sup_{m\geq n}\|\mathcal{A}(m,n)v\|<+\infty\right\\}$
(10)

and

$U(n)=\mathcal{A}(n,0)Z.$

It is easy to check that

$A_{n}S(n)\subset S(n+1)\text{ and }A_{n}U(n)\subset U(n+1)$ (11)

for every $n\in{\mathbb{N}}$.

######  Lemma 3.4.

We have that

$X=S(n)\oplus U(n),\quad n\in\mathbb{N}.$ (12)

###### Proof of Lemma 3.4.

Take initially $n\geq 1$. Given $v\in X$, we define
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}$ by $y_{n}=v$ and $y_{m}=0$ for $m\neq
n$. Then, $\mathbf{y}\in\ell^{1}_{\beta,0}$. By our assumption, there exists
$\mathbf{x}=(x_{m})_{m\in\mathbb{N}}\in\ell^{\infty}_{\beta,Z}$ such that

$x_{m+1}-A_{m}x_{m}=y_{m+1},\quad m\in\mathbb{N}.$

In particular,

$x_{n}-A_{n-1}x_{n-1}=v.$

Then, since $x_{0}\in Z$, we have that
$A_{n-1}x_{n-1}=\mathcal{A}(n,0)x_{0}\in U(n)$. Moreover, since
$x_{m}=\mathcal{A}(m,n)x_{n}$ for $m\geq n$ and
$\mathbf{x}\in\ell^{\infty}_{\beta,Z}$, we conclude that $x_{n}\in S(n)$.
Thus, $v\in S(n)+U(n)$.

Suppose now that $v\in S(n)\cap U(n)$. Then, there exists $w\in Z$ such that
$v=\mathcal{A}(n,0)w$. We define

$x_{m}:=\mathcal{A}(m,0)w,\quad m\in\mathbb{N}.$

Then, $\mathbf{x}=(x_{m})_{m\in\mathbb{N}}\in\ell^{\infty}_{-\beta,Z}$ and (8)
holds with $\mathbf{y}=0$. Therefore, using the uniqueness in our assumption,
we conclude that $\mathbf{x}=0$ and $v=0$ proving that (12) holds for every
$n\geq 1$.

Let us now consider the case when $n=0$. Given $v\in X$, we define sequences
$\mathbf{x}=(x_{m})_{m\in\mathbb{N}}$ and
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}$ given by $x_{0}=v$ and $x_{m}=0$ for
$m\neq n$ and $y_{1}=-A_{0}v$ and $y_{m}=0$ for $m\neq 1$. Then,

$x_{m+1}-A_{m}x_{m}=y_{m+1},\quad m\in{\mathbb{N}}.$

Moreover, since $\mathbf{y}\in\ell^{1}_{\beta,0}$, there exists
$\tilde{\mathbf{x}}=(\tilde{x}_{m})_{m\in{\mathbb{N}}}\in\ell^{\infty}_{\beta,Z}$
such that (8) holds. Then,

$x_{m}-\tilde{x}_{m}=\mathcal{A}(m,0)(v-\tilde{x}_{0}),\quad\text{ for all
}m\in{\mathbb{N}}.$

Consequently, since $\mathbf{x}-\tilde{\mathbf{x}}\in\ell^{\infty}_{\beta}$,
it follows that $v-\tilde{x}_{0}\in S(0)$. Therefore, since $\tilde{x}_{0}\in
Z$, we conclude that $v\in S(0)+U(0)$. Finally, consider $v\in S(0)\cap U(0)$
and let $x_{m}=\mathcal{A}(m,0)v$ for $m\in{\mathbb{N}}$. Then, as in the case
when $n\geq 1$,
$\mathbf{x}=(x_{m})_{m\in\mathbb{N}}\in\ell^{\infty}_{-\beta,Z}$ and (8) holds
with $\mathbf{y}=0$. Therefore, using the uniqueness in our assumption, we
conclude that $\mathbf{x}=0$ and $v=0$. This completes the proof of the lemma.
∎

######  Lemma 3.5.

For every $n\in{\mathbb{N}}$, the operator $A_{n}|_{U(n)}\colon U(n)\to
U(n+1)$ is an isomorphism.

###### Proof of Lemma 3.5.

Fix $n\in{\mathbb{N}}$. Let us start proving that $A_{n}|_{U(n)}$ is
injective. Suppose there exists $v\in U(n)$ such that $A_{n}v=0$. By the
definition of $U(n)$ there exists $z_{0}\in Z$ such that
$v=\mathcal{A}(n,0)z_{0}$. Then, the sequence $(x_{m})_{m\in{\mathbb{N}}}$
given by

$x_{m}=\mathcal{A}(m,0)z_{0},\quad m\in{\mathbb{N}},$

belongs to $\ell^{\infty}_{\beta,Z}$ and satisfies (8) with $\mathbf{y}=0$.
Consequently, by the uniqueness in our hypothesis, it follows that $x_{m}=0$
for every $m\in{\mathbb{N}}$. In particular, $v=0$ and $A_{n}|_{U(n)}$ is
injective.

Now, given $x\in U(n+1)$, let $z_{0}\in Z$ be such that
$x=\mathcal{A}(n+1,0)z_{0}$. Set $x^{\prime}:=\mathcal{A}(n,0)z_{0}$. Then, it
follows directly from the definition that $x^{\prime}\in U(n)$ and
$A_{n}x^{\prime}=x$ which proves that $A_{n}|_{U(n)}\colon U(n)\to U(n+1)$ is
surjective. Consequently, $A_{n}|_{U(n)}\colon U(n)\to U(n+1)$ is an
isomorphism as claimed. ∎

######  Lemma 3.6.

For each $n\in{\mathbb{N}}$, let $P_{n}\colon X\to S(n)$ be the projection
associated with (12). There exists $D>0$ such that

$\sup_{n\in\mathbb{N}}\|P_{n}\|\leq D\nu_{n},\quad n\in\mathbb{N}.$ (13)

###### Proof of Lemma 3.6.

Let us consider the map
$T_{\beta}\colon\ell^{1}_{\beta,0}\to\ell^{\infty}_{\beta,Z}$ given by
$T_{\beta}(\mathbf{y})=\mathbf{x}$ where $\mathbf{x}$ is the unique element in
$l^{\infty}_{\beta,Z}$ such that (8) holds. Then, one can easily check that
$T_{\beta}$ is a linear operator. Moreover, we observe that $T_{\beta}$ is a
closed operator. In fact, let $\mathbf{y}^{k}=(y^{k}_{n})_{n\in{\mathbb{N}}}$
be a sequence of elements in $\ell^{1}_{\beta,Z_{0}}$ such that
$\mathbf{y}^{k}\to\mathbf{y}$ for some
$\mathbf{y}=(y_{n})_{n\in{\mathbb{N}}}\in\ell^{1}_{\beta,Z_{0}}$ and
$T_{\beta}(\mathbf{y}^{k})\to\mathbf{x}$ for some
$\mathbf{x}=(x_{n})_{n\in{\mathbb{N}}}\in\ell^{\infty}_{\beta,Z}$. Then,
writing
$T_{\beta}(\mathbf{y}^{k})=\mathbf{x}^{k}=(x^{k}_{n})_{n\in{\mathbb{N}}}$ we
get that

$x_{n+1}^{k}-A_{n}x^{k}_{n}=y^{k}_{n+1},\text{ for every }n,k\in{\mathbb{N}}.$

Thus, letting $k\to+\infty$ we conclude that

$x_{n+1}-A_{n}x_{n}=y_{n+1},\text{ for every }n\in{\mathbb{N}}$

which implies that $T_{\beta}(\mathbf{y})=\mathbf{x}$. In particular,
$T_{\beta}$ is a closed operator and thus, according to the Closed Graph
Theorem (see, e.g., [43, Theorem 4.2-I, p. 181]), $T_{\beta}$ is bounded.

Now, given $n\geq 1$ and $v\in X$, consider
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}$ given by $y_{n}=v$ and $y_{m}=0$ for
$m\neq n$. Then, $\mathbf{y}\in\ell^{1}_{\beta,0}$. Considering
$T_{\beta}(\mathbf{y})=\mathbf{x}=(x_{n})_{n\in{\mathbb{N}}}$, it follows by
the proof of Lemma 3.4 that $P_{n}v=x_{n}$. Thus,

$\mu_{n}^{\beta}\|P_{n}v\|=\mu_{n}^{\beta}\|x_{n}\|\leq\|\mathbf{x}\|_{\infty,\beta}=\|T_{\beta}(\mathbf{y})\|_{\infty,\beta}\leq\|T_{\beta}\|\cdot\|\mathbf{y}\|_{1,\beta}=\|T_{\beta}\|\mu_{n}^{\beta}\nu_{n}\|v\|.$

This yields that (13) holds with $D=\|T_{\beta}\|$ for $n\geq 1$. On the other
hand, (13) holds with $D=\|P_{0}\|$ for $n=0$ (recall that $\nu_{0}\geq 1$).
We conclude that (13) holds with

$D=\max\left\\{\|T_{\beta}\|,\|P_{0}\|\right\\}>0.$

∎

######  Lemma 3.7.

There exists $C>0$ such that

$\|\mathcal{A}(m,n)v\|\leq
C\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}\|v\|,$

for every $m\geq n$ and $v\in S(n)$.

###### Proof of Lemma 3.7.

Let $n\geq 1$ and $v\in S(n)$. We define sequences $(y_{m})_{m\in\mathbb{N}}$
and $(x_{m})_{m\in\mathbb{N}}$ by

$y_{m}=\begin{cases}v&m=n;\\\ 0&m\neq n,\end{cases}\quad\text{and}\quad
x_{m}=\begin{cases}\mathcal{A}(m,n)v&m\geq n;\\\ 0&m<n.\end{cases}$

Then,
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}\in\ell^{1}_{\beta}\cap\ell^{1}_{-\beta}$
and $\mathbf{x}=(x_{m})_{m\in\mathbb{Z}}\in\ell^{\infty}_{-\beta,Z}$.
Moreover, (8) holds. Then, from Lemma 3.3 it follows that
$\mathbf{x}\in\ell^{\infty}_{\beta,Z}$. Thus, letting
$T_{\beta}\colon\ell^{1}_{\beta,0}\to\ell^{\infty}_{\beta,Z}$ be the linear
operator defined in the proof of Lemma 3.6 and considering
$K=\|T_{\beta}\|+1$, we get that

$\mu_{m}^{\beta}\|x_{m}\|\leq\|\mathbf{x}\|_{\infty,\beta}=\|T_{\beta}(\mathbf{y})\|_{\infty,\beta}\leq
K\|y\|_{1,\beta}=K\mu_{n}^{\beta}\nu_{n}\|v\|$

for $m\geq n$. Thus,

$\|\mathcal{A}(m,n)v\|\leq
K\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}\|v\|$

for every $m\geq n$ and $n\geq 1$.

Now, for the case when $n=0$, if $m=n=0$ then the same estimate as before
holds since $K\geq 1$. Take $m\geq 1$. Then, given $v\in S(0)$, by (11) we
have that $A_{0}v\in S(1)$. Thus, using the estimate obtained above we get
that

$\begin{split}\|\mathcal{A}(m,0)v\|&=\|\mathcal{A}(m,1)A_{0}v\|\leq
K\left(\frac{\mu_{m}}{\mu_{1}}\right)^{-\beta}\nu_{1}\|A_{0}v\|\\\ &\leq
K\|A_{0}\|\frac{\nu_{1}}{\nu_{0}}\left(\frac{\mu_{0}}{\mu_{1}}\right)^{-\beta}\left(\frac{\mu_{m}}{\mu_{0}}\right)^{-\beta}\nu_{0}\|v\|.\end{split}$

Therefore, taking
$C=\max\\{K,K\nu_{1}\nu_{0}^{-1}\|A_{0}\|\mu_{0}^{-\beta}\mu_{1}^{\beta}\\}$
we get the desired result. ∎

######  Lemma 3.8.

There exists $\tilde{C}>0$ such that

$\|\mathcal{A}(m,n)v\|\leq\tilde{C}\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}\|v\|,$

for every $m\leq n$ and $v\in U(n)$.

###### Proof of Lemma 3.8.

Take $n\geq 1$ and $v\in U(n)$. Then by Lemma 3.5 we may consider

$x_{m}=\begin{cases}\mathcal{A}(m,n)v&m<n;\\\ 0&m\geq
n,\end{cases}\quad\text{and}\quad y_{m}=\begin{cases}-v&m=n;\\\ 0&m\neq
n.\end{cases}$

Note that $\mathbf{x}=(x_{m})_{m\in\mathbb{N}}\in\ell^{\infty}_{-\beta,Z}$ and
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}\in\ell^{1}_{-\beta,0}$. Furthermore, (8)
holds. Let us now consider the map
$T_{-\beta}\colon\ell^{1}_{-\beta,Z_{0}}\to\ell^{\infty}_{-\beta,Z}$ given by
$T_{-\beta}(\mathbf{y})=\mathbf{x}$ where $\mathbf{x}$ is the unique element
in $\ell^{\infty}_{-\beta,Z}$ associated to $\mathbf{y}$ by our assumption (so
that (8) holds). Then, proceeding as in the proof of Lemma 3.6 we can prove
that $T_{-\beta}$ is a bounded linear operator. Thus, taking
$\tilde{C}=\|T_{-\beta}\|+1$ we get that

$\mu_{m}^{-\beta}\|x_{m}\|\leq\|\mathbf{x}\|_{\infty,-\beta}=\|T_{-\beta}(\mathbf{y})\|_{\infty,-\beta}\leq\tilde{C}\|y\|_{1,-\beta}=\tilde{C}\mu_{n}^{-\beta}\nu_{n}\|v\|,$

for every $m<n$. This easily implies that

$\|\mathcal{A}(m,n)v\|\leq\tilde{C}\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}\|v\|,$

for every $m<n$ as claimed. Finally, the cases when $m=n$ or $n=0$ trivially
holds since $\tilde{C}\geq 1$. ∎

For $m,n\in\mathbb{N}$, set

$\mathcal{G}(m,n):=\begin{cases}\mathcal{A}(m,n)P_{n}&m\geq n;\\\
-\mathcal{A}(m,n)(\text{Id}-P_{n})&m<n.\end{cases}$

Then, by Lemmas 3.6, 3.7 and 3.8 there exists $C^{\prime}>0$ such that

$\|\mathcal{G}(m,n)\|\leq
C^{\prime}\begin{cases}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}^{2}&m\geq
n;\\\
\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}^{2}&m<n.\end{cases}$ (14)

The purpose of the following lemma is to replace $\nu_{n}^{2}$ by $\nu_{n}$ in
(14).

######  Lemma 3.9.

There exists $C^{\prime\prime}>0$ such that

$\|\mathcal{G}(m,n)\|\leq
C^{\prime\prime}\begin{cases}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}&m\geq
n;\\\ \left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}&m<n.\end{cases}$

###### Proof of the Lemma 3.9.

Take $n\geq 1$ and $v\in X$. We define a sequence
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}$ by $y_{n}=v$ and $y_{m}=0$ for $m\neq
n$. Then, $\mathbf{y}\in\ell_{\beta,0}^{1}\cap\ell_{-\beta,0}^{1}$. Set

$x_{m}:=\mathcal{G}(m,n)v,\quad m\in\mathbb{N}.$

It is easy to verify that (8) holds. We claim that
$\mathbf{x}=(x_{m})_{m\in\mathbb{N}}\in\ell_{\beta,Z}^{\infty}$. Indeed, for
$m\geq n$ we have that

$\|x_{m}\|=\|\mathcal{G}(m,n)v\|\leq
C^{\prime}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}^{2}\|v\|,$

which implies that

$\sup_{m\in\mathbb{N}}(\mu_{m}^{\beta}\|x_{m}\|)<+\infty.$

Consequently, $\mathbf{x}=(x_{m})_{m\in\mathbb{N}}\in\ell_{\beta,Z}^{\infty}$
and $T_{\beta}(\mathbf{y})=\mathbf{x}$, where $T_{\beta}$ is as in the proof
of Lemma 3.6. Therefore, for $m\geq n$ we have that

$\mu_{m}^{\beta}\|\mathcal{G}(m,n)v\|=\mu_{m}^{\beta}\|x_{m}\|\leq\|\mathbf{x}\|_{\infty,\beta}\leq\|T_{\beta}\|\cdot\|\mathbf{y}\|_{1,\beta}=\|T_{\beta}\|\mu_{n}^{\beta}\nu_{n}\|v\|,$

and thus

$\|\mathcal{G}(m,n)\|\leq\|T_{\beta}\|\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n},\quad
m\geq n\geq 1.$

We now consider the case when $n=0$. For $m>0$ we have

$\begin{split}\|\mathcal{G}(m,0)\|=\|\mathcal{G}(m,1)A_{0}\|&\leq\|T_{\beta}\|\cdot\|A_{0}\|\left(\frac{\mu_{m}}{\mu_{1}}\right)^{-\beta}\nu_{1}\\\
&=\|T_{\beta}\|\cdot\|A_{0}\|\frac{\nu_{1}}{\nu_{0}}\left(\frac{\mu_{0}}{\mu_{1}}\right)^{-\beta}\left(\frac{\mu_{m}}{\mu_{0}}\right)^{-\beta}\nu_{0}.\end{split}$

Hence, taking

$C^{\prime\prime}\geq\max\left\\{\|T_{\beta}\|,\|T_{\beta}\|\cdot\|A_{0}\|\frac{\nu_{1}}{\nu_{0}}\left(\frac{\mu_{0}}{\mu_{1}}\right)^{-\beta},\|P_{0}\|\right\\}$

we get that

$\|\mathcal{G}(m,n)\|\leq
C^{\prime\prime}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n},$

for $m\geq n$. Similarly, one can treat the case when $m<n$. ∎

Now the proof of Theorem 3.2 can be easily completed by combining the previous
auxiliary results. ∎

###  3.2. An alternative characterization of $(\mu,\nu)$-dichotomy for compact
operators

We now present an alternative to Theorem 3.2 in the case when at least one of
the operators $A_{n}$ is compact. For this purpose, given
$\beta\in\mathbb{R}$, let

$S_{\beta}(0)=\left\\{v\in X:\
\sup_{n\in\mathbb{N}}(\mu_{n}^{\beta}\|{\mathcal{A}}(n,0)v\|)<+\infty\right\\}.$

Note that $S_{\beta}(0)$ is a subspace of $X$.

######  Theorem 3.10.

Suppose that $A_{n}$ is compact _for some_ $n\in\mathbb{N}$ and that there
exists $\beta>0$ such that:

  1. (1)

$S_{0}(0)=S_{\beta}(0)$;

  2. (2)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{\beta,0}$, there
exists $\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{\beta}^{\infty}$ such that
(8) holds;

  3. (3)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{N}}\in\ell^{1}_{-\beta,0}$, there
exists $\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{-\beta}^{\infty}$ such
that (8) holds.

Then, $(A_{n})_{n\in\mathbb{N}}$ admits a $(\mu,\nu)$-dichotomy.

###### Proof.

For $n\in\mathbb{N}$, let $S(n)$ be given by (10). We first claim that

$A_{n}^{-1}(S(n+1))=S(n),\quad n\in\mathbb{N}.$

To this end, take $v\in A_{n}^{-1}(S(n+1))$. Then, $A_{n}v\in S(n+1)$ and thus

$\sup_{m\geq n+1}\|{\mathcal{A}}(m,n+1)A_{n}v\|=\sup_{m\geq
n+1}\|{\mathcal{A}}(m,n)v\|<+\infty,$

which implies that $\sup_{m\geq n}\|{\mathcal{A}}(m,n)v\|<+\infty$, i.e. $v\in
S(n)$. The converse inclusion can be established in a similar manner.

Next, we claim that for $m\geq n$,

$X={\mathcal{A}}(m,n)X+S(m).$

One can easily see that it is sufficient to consider the case when $n=0$.
Then, if $m=0$, there is nothing to show since ${\mathcal{A}}(m,n)X=X$. Let us
now consider the case when $m>0$. Take $v\in X$ and define
$\mathbf{y}=(y_{k})_{k\in\mathbb{N}}$ by $y_{m}=v$ and $y_{k}=0$ for $k\neq
m$. Then, $\mathbf{y}\in\ell_{\beta,0}^{1}$. Consequently, there exists
$\mathbf{x}=(x_{k})_{k\in\mathbb{N}}\in\ell_{\beta}^{\infty}$ such that (8)
holds. In particular,

$x_{m}-A_{m-1}x_{m-1}=v$

and

$x_{k}=A_{k-1}x_{k-1},\quad\text{for }k\neq m.$

Hence, since $\mathbf{x}\in\ell_{\beta}^{\infty}$, we get that $x_{m}\in
S(m)$. On the other hand, $A_{m-1}x_{m-1}={\mathcal{A}}(m,0)x_{0}$, yielding
that

$v=x_{m}-A_{m-1}x_{m-1}\in S(m)+{\mathcal{A}}(m,0)X.$

Thus, the desired claim holds.

We proceed by noting that each $S(m)$ is an image of a Banach space under the
action of a bounded linear operator. Indeed, let $\mathcal{C}(m)$ denote the
space of all sequences $\mathbf{x}=(x_{n})_{n\geq m}\subset X$ such that

$\|\mathbf{x}\|_{\mathcal{C}(m)}:=\sup_{n\geq m}\|x_{n}\|<+\infty.$

Then, $(\mathcal{C}(m),\|\cdot\|_{\mathcal{C}(m)})$ is a Banach space. Set
$\mathcal{C}^{\prime}(m)$ to be the set of all sequences
$\mathbf{x}=(x_{n})_{n\geq m}\in\mathcal{C}(m)$ satisfying

$x_{n+1}=A_{n}x_{n},\quad n\geq m.$

It is straightforward to verify that $\mathcal{C}^{\prime}(m)$ is a closed
subspace of $\mathcal{C}(m)$ and therefore also a Banach space. Then, we
observe that $S(m)=\Phi(\mathcal{C}^{\prime}(m))$, where
$\Phi\colon\mathcal{C}(m)\to X$ is a bounded linear operator given by

$\Phi(\mathbf{x})=x_{m},\quad\mathbf{x}=(x_{n})_{n\geq m}\in\mathcal{C}(m).$

It follows now from [42, Lemma 3.4.] that $S(0)=S_{0}(0)$ is closed and
complemented in $X$. Therefore, there exists a closed subspace $Z\subset X$
such that

$X=S(0)\oplus Z.$

Take an arbitrary
$\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell_{\beta,0}^{\infty}$. By our
assumption, there exists
$\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{\beta}^{\infty}$ such that (8)
holds. Write $x_{0}=v_{1}+v_{2}$ with $v_{1}\in S(0)$ and $v_{2}\in Z$. Set

$\tilde{x}_{n}:=x_{n}-{\mathcal{A}}(n,0)v_{1}.$

Since $v_{1}\in S(0)=S_{0}(0)=S_{\beta}(0)$, we easily get that
$\tilde{\mathbf{x}}=(\tilde{x}_{n})_{n\in\mathbb{N}}\in\ell_{\beta}^{\infty}$.
Moreover, $\tilde{x}_{0}=v_{2}\in Z$. Therefore,
$\tilde{\mathbf{x}}\in\ell_{\beta,Z}^{\infty}$ and it is straightforward to
verify that the pair $(\tilde{\mathbf{x}},\mathbf{y})$ satisfies (8). In
addition, let us suppose that the pair $(\overline{\mathbf{x}},\mathbf{y})$
satisfies (8) with
$\overline{\mathbf{x}}=(\overline{x}_{n})_{n\in\mathbb{N}}\in\ell_{\beta,Z}^{\infty}$.
Then, we have that $\overline{x}_{0}-\tilde{x}_{0}\in S(0)\cap Z$, which gives
that $\overline{x}_{0}-\tilde{x}_{0}=0$. Therefore,
$\overline{\mathbf{x}}=\tilde{\mathbf{x}}$. We have thus proved that the first
assumption in the statement of Theorem 3.2 holds. Similarly, one can establish
the second assumption in Theorem 3.2 (for this it is sufficient to observe
that for $v\in S(0)$, $n\mapsto{\mathcal{A}}(n,0)v\in\ell_{-\beta}^{\infty}$).
The desired conclusion now follows from Theorem 3.2. ∎

######  Remark 3.11.

  * •

It is easy to verify that under the assumptions of Theorem 3.1 we have that
$S_{0}(0)=S_{\beta}(0)$, making this a reasonable assumption in the statement
of Theorem 3.10;

  * •

in contrast to Theorem 3.2, the admissibility assumptions in Theorem 3.10 do
not require uniqueness.

###  3.3. Persistence of $(\mu,\nu)$-dichotomy

In this section, as a consequence of the characterization of
$(\mu,\nu)$-dichotomy given in Section 3.1, we are going to show that the
notion of $(\mu,\nu)$-dichotomy persists under small linear perturbations.

######  Theorem 3.12.

Let $(\gamma_{n})_{n\in{\mathbb{N}}}$ be a sequence of numbers with
$\gamma_{n}>0$ for every $n\in{\mathbb{N}}$ such that

$\sum_{n=0}^{\infty}\gamma_{n}<+\infty.$ (15)

Suppose that (2) admits a $(\mu,\nu)$-dichotomy with $\lambda>0$ and condition
(7) is satisfied and take $\beta\in(0,\lambda-\varepsilon)$. Moreover, let
$(B_{n})_{n\in{\mathbb{N}}}$ be a sequence of operators in $\mathcal{B}(X)$
with the property that there exists $c>0$ such that

$\|B_{n}\|\leq\frac{c\gamma_{n}\mu_{n}^{\beta}}{\nu_{n+1}\mu_{n+1}^{\beta}},\quad\text{for
$n\in\mathbb{N}$.}$ (16)

Then, if $c$ is sufficiently small we have that the nonautonomous difference
equation

$x_{n+1}=(A_{n}+B_{n})x_{n},\quad n\in{\mathbb{N}}$ (17)

also admits a $(\mu,\nu)$-dichotomy.

######  Remark 3.13.

Observe that in the case when $\mu_{n+1}\leq K\mu_{n}$ for every
$n\in{\mathbb{N}}$ and some $K>0$, that is, the sequence
$(\mu_{n})_{n\in{\mathbb{N}}}$ does not grow faster than exponential,
condition (16) can be reduced to

$\|B_{n}\|\leq\frac{c\gamma_{n}}{\nu_{n}},$ (18)

for $n\in{\mathbb{N}}$ and $c>0$ small enough. In particular, this comment
applies to the classical settings of exponential, polynomial and logarithmic
dichotomy. Observe moreover that whenever we are in the case of an _uniform_
dichotomy, meaning that the sequence $(\nu_{n})_{n\in{\mathbb{N}}}$ is
constant, condition (18) is basically saying that the classical
$\ell^{1}$-norm of $(\|B_{n}\|)_{n\in{\mathbb{N}}}$ is small. This indicates
that our robustness is, in general, not optimal as, it is for example well-
known, that the notion of uniform exponentially dichotomy persists under the
requirement that $\sup_{n}\|B_{n}\|$ is small. On the other hand, in the case
of polynomial dichotomy it does generalize existing results [12, 15, 16, 41].

In order to prove Theorem 3.12 let us introduce some terminology. Given a
closed subspace $Z\subset X$ and $\beta>0$, let us consider consider the
linear operators
$\mathbf{A}_{\beta}\colon\mathcal{D}(\mathbf{A}_{\beta})\subset\ell^{\infty}_{\beta,Z}\to\ell^{1}_{\beta,0}$
and
$\mathbf{A}_{-\beta}\colon\mathcal{D}(\mathbf{A}_{-\beta})\subset\ell^{\infty}_{-\beta,Z}\to\ell^{1}_{-\beta,0}$
given by

$(\mathbf{A}_{j}\mathbf{x})_{n}=x_{n}-A_{n-1}x_{n-1}\text{ for }n\geq 1$

and $(\mathbf{A}_{j}\mathbf{x})_{0}=0$ for $j=\beta,-\beta$ where
$\mathcal{D}(\mathbf{A}_{\beta})=\\{\mathbf{x}\in\ell^{\infty}_{\beta,Z}:\mathbf{A}_{\beta}\mathbf{x}\in\ell^{1}_{\beta,0}\\}$
and
$\mathcal{D}(\mathbf{A}_{-\beta})=\\{\mathbf{x}\in\ell^{\infty}_{-\beta,Z}:\mathbf{A}_{-\beta}\mathbf{x}\in\ell^{1}_{-\beta,0}\\}$.
Then, using this terminology we can reformulate Theorems 3.1 and 3.2 as
follows.

######  Theorem 3.14.

Let us consider the following conditions:

  * i)

Eq. (2) admits a $(\mu,\nu)$-dichotomy.

  * ii)

There exist a closed subspace $Z\subset X$ and $\beta>0$ such that for every
$\mathbf{y}^{1}\in\ell^{1}_{\beta,0}$ there exists a unique
$\mathbf{x}^{1}\in\ell^{\infty}_{\beta,Z}$ satisfying
$\mathbf{A}_{\beta}\mathbf{x}^{1}=\mathbf{y}^{1}$ and for every
$\mathbf{y}^{2}\in\ell^{1}_{-\beta,0}$ there exists a unique
$\mathbf{x}^{2}\in\ell^{\infty}_{-\beta,Z}$ satisfying
$\mathbf{A}_{-\beta}\mathbf{x}^{2}=\mathbf{y}^{2}$. In other words, operators
$\mathbf{A}_{\beta}$ and $\mathbf{A}_{-\beta}$ are invertible.

Thus, if $i)$ holds and (7) is satisfied then $ii)$ also holds. Reciprocally,
if $ii)$ holds then $i)$ also holds.

###### Proof of Theorem 3.12.

Let $Z\subset X$ and $\beta>0$ be given by Theorem 3.14. From Theorem 3.1 it
follows that $\beta$ may be any value in $(0,\lambda-\varepsilon)$. In
particular, we may assume without loss of generality that $\beta$ is such that
(16) is satisfied. Let us also consider the operators $\mathbf{A}_{\beta}$ and
$\mathbf{A}_{-\beta}$ given above. We endow $\mathcal{D}(\mathbf{A}_{\beta})$
and $\mathcal{D}(\mathbf{A}_{-\beta})$ with the graph norms

$\|\mathbf{x}\|_{\mathbf{A}_{\beta}}=\|\mathbf{x}\|_{\infty,\beta}+\|\mathbf{A}_{\beta}(\mathbf{x})\|_{1,\beta}$

and

$\|\mathbf{x}\|_{\mathbf{A}_{-\beta}}=\|\mathbf{x}\|_{\infty,-\beta}+\|\mathbf{A}_{-\beta}(\mathbf{x})\|_{1,-\beta},$

respectively. By proceeding as in the proof of Lemma 3.6 we can conclude that
$\mathbf{A}_{\beta}$ and $\mathbf{A}_{-\beta}$ are closed operators. In
particular, $(\mathcal{D}(\mathbf{A}_{j}),\|\cdot\|_{\mathbf{A}_{j}})$,
$j=\beta,-\beta$, are Banach spaces. Consider also the operators
$\mathbf{B}_{\beta}\colon\ell^{\infty}_{\beta,Z}\to\ell^{\infty}_{\beta,0}$
and
$\mathbf{B}_{-\beta}\colon\ell^{\infty}_{-\beta,Z}\to\ell^{\infty}_{-\beta,0}$
given by

$(\mathbf{B}_{j}\mathbf{x})_{n}=B_{n-1}x_{n-1}\text{ for }n\geq 1$

and $(\mathbf{B}_{j}\mathbf{x})_{0}=0$ for $j=\beta,-\beta$. Observe that
condition (16) guarantee that both $\mathbf{B}_{\beta}$ and
$\mathbf{B}_{-\beta}$ are well-defined.

Now, using (16) we get that for every
$\mathbf{x}\in\mathcal{D}(\mathbf{A}_{\beta})$,

$\begin{split}\|((\mathbf{A}_{\beta}-\mathbf{B}_{\beta})\mathbf{x})_{n}\|&\leq\|(\mathbf{B}_{\beta}\mathbf{x})_{n}\|+\|(\mathbf{A}_{\beta}\mathbf{x})_{n}\|\\\
&\leq\frac{c\gamma_{n-1}\mu_{n-1}^{\beta}}{\nu_{n}\mu_{n}^{\beta}}\|x_{n-1}\|+\|(\mathbf{A}_{\beta}\mathbf{x})_{n}\|\end{split}$
(19)

for every $n\geq 1$. Consequently, using (15) and that
$\mathbf{x}\in\ell^{\infty}_{\beta}$ and
$\mathbf{A}_{\beta}\mathbf{x}\in\ell^{1}_{\beta,0}$ we get that

$\begin{split}\|(\mathbf{A}_{\beta}-\mathbf{B}_{\beta})\mathbf{x}\|_{1,\beta}&\leq\sum_{n=0}^{\infty}c\gamma_{n}\mu_{n}^{\beta}\|x_{n}\|+\|\mathbf{A}_{\beta}\mathbf{x}\|_{1,\beta}\\\
&\leq
c\|\mathbf{x}\|_{\infty,\beta}\sum_{n=0}^{\infty}\gamma_{n}+\|\mathbf{A}_{\beta}\mathbf{x}\|_{1,\beta}<+\infty\end{split}$
(20)

and $(\mathbf{A}_{\beta}-\mathbf{B}_{\beta})\mathbf{x}\in\ell^{1}_{\beta,0}$.
In particular, the operator
$\mathbf{A}_{\beta}-\mathbf{B}_{\beta}\colon(\mathcal{D}(\mathbf{A}_{\beta}),\|\cdot\|_{\mathbf{A}_{\beta}})\to\ell^{1}_{\beta,0}$
given by $(\mathbf{A}_{\beta}-\mathbf{B}_{\beta})\mathbf{x}$ is well-defined.
Moreover, by (20) we have that

$\|(\mathbf{A}_{\beta}-\mathbf{B}_{\beta})\mathbf{x}\|_{1,\beta}\leq
K\|\mathbf{x}\|_{\mathbf{A}_{\beta}}$

for some $K>0$ and, consequently, $\mathbf{A}_{\beta}-\mathbf{B}_{\beta}$ is
bounded. Furthermore, using part of the estimate obtained in (19) we get that

$\begin{split}\|\mathbf{A}_{\beta}\mathbf{x}-(\mathbf{A}_{\beta}-\mathbf{B}_{\beta})\mathbf{x}\|_{1,\beta}\leq
c\sum_{n=0}^{\infty}\gamma_{n}\|\mathbf{x}\|_{\infty,\beta}\leq
c\sum_{n=0}^{\infty}\gamma_{n}\|\mathbf{x}\|_{\mathbf{A}_{\beta}}.\end{split}$

Therefore, since $\mathbf{A}_{\beta}$ is invertible (recall Theorem 3.14), we
conclude that for $c>0$ small enough $\mathbf{A}_{\beta}-\mathbf{B}_{\beta}$
is also invertible. Proceeding in a similar manner we conclude that
$\mathbf{A}_{-\beta}-\mathbf{B}_{-\beta}\colon(\mathcal{D}(\mathbf{A}_{-\beta}),\|\cdot\|_{\mathbf{A}_{-\beta}})\to\ell^{1}_{-\beta,0}$
is also a well-defined bounded and invertible linear operator. Then, combining
these two observations with Theorem 3.14 we get that (17) admits a
$(\mu,\nu)$-dichotomy. ∎

##  4\. The case of two-sided dynamics

In this section we are going to consider the case of two-sided dynamics, that
is, the case when $I={\mathbb{Z}}$.

###  4.1. Characterization of $(\mu,\nu)$-dichotomy

Following the ideas of Section 3.1, we will now present a characterization of
$(\mu,\nu)$-dichotomy in terms of the admissibility of certain weighted
spaces. In the present context, in addition to the admissibility of spaces of
the form $\ell^{1}_{\beta}$ and $\ell^{\infty}_{\beta}$ for some appropriate
values of $\beta\in{\mathbb{R}}$, we will also have to consider the
admissibility of spaces $\ell^{1}_{\beta,|\cdot|}$ and
$\ell^{\infty}_{\beta,|\cdot|}$.

######  Theorem 4.1.

Suppose that $(A_{n})_{n\in\mathbb{Z}}$ admits a $(\mu,\nu)$-dichotomy with
respect to projections $P_{n}$ and let $\lambda>0$ be such that (5) and (6)
hold. Moreover, suppose that there exists $\varepsilon\in[0,\lambda)$ such
that

$\sup_{n\in\mathbb{N}}(\mu_{-n}^{\varepsilon}\nu_{-n})<+\infty\;\text{ and
}\;\sup_{n\in\mathbb{N}}(\mu_{n}^{-\varepsilon}\nu_{n})<+\infty.$ (21)

Then, for each $\beta\in(-(\lambda-\varepsilon),\lambda-\varepsilon)$ the
following holds:

  1. (1)

for every $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta}$, there
exists a unique $\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{\beta}^{\infty}$
such that

$x_{n+1}-A_{n}x_{n}=y_{n+1}\quad n\in\mathbb{Z}.$ (22)

  2. (2)

for every $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta,|\cdot|}$,
there exists a unique
$\mathbf{x}=(x_{n})_{n\in\mathbb{N}}\in\ell_{\beta,|\cdot|}^{\infty}$ such
that (22) holds.

###### Proof.

Take $\beta\in(-(\lambda-\varepsilon),\lambda-\varepsilon)$,
$\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta}$ and set

$x_{n}:=\sum_{k=-\infty}^{n}\mathcal{A}(n,k)P_{k}y_{k}-\sum_{k=n+1}^{\infty}\mathcal{A}(n,k)(\text{Id}-P_{k})y_{k},\quad
n\in\mathbb{Z}.$ (23)

Then, proceeding as in the proof of Theorem 3.1 we obtain that
$\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in\ell^{\infty}_{\beta}$ and that (22) is
satisfied. We now establish the uniqueness of such $\mathbf{x}$. For this
purpose, by linearity, all we have to do is to prove that the unique sequence
$\mathbf{z}=(z_{n})_{n\in\mathbb{Z}}$ in $\ell^{\infty}_{\beta}$ that
satisfies

$z_{n+1}=A_{n}z_{n},\quad n\in\mathbb{Z}$

is the null sequence. That is, $z_{n}=0$ for all $n\in{\mathbb{Z}}$. Let us
consider

$z^{s}_{n}=P_{n}z_{n}\;\text{ and }\;z^{u}_{n}=(\text{Id}-P_{n})z_{n}\;\text{
for }n\in{\mathbb{Z}}.$

Then, $z_{n}=z^{s}_{n}+z^{u}_{n}$ and, by (4),

$z^{s}_{n+1}=A_{n}z^{s}_{n}\;\text{ and }\;z^{u}_{n+1}=A_{n}z^{u}_{n}\;\text{
for all }n\in{\mathbb{Z}}.$

Thus, using (5), we get that for every $m\geq n$,

$\begin{split}\mu_{m}^{\lambda}\|z^{s}_{m}\|&=\mu_{m}^{\lambda}\|\mathcal{A}(m,n)z^{s}_{n}\|\\\
&=\mu_{m}^{\lambda}\|\mathcal{A}(m,n)P_{n}z_{n}\|\\\ &\leq
D\nu_{n}\mu_{n}^{\lambda}\|z_{n}\|\\\ &\leq
D\nu_{n}\mu_{n}^{\lambda-\beta}\|\mathbf{z}\|_{\infty,\beta}\\\ &\leq
D\mu_{n}^{\lambda-\beta-\varepsilon}\|\mathbf{z}\|_{\infty,\beta}\cdot\sup_{k\leq
m}(\mu_{k}^{\varepsilon}\nu_{k}).\end{split}$

Consequently, since $\lim_{n\to-\infty}\mu_{n}=0$, letting $n\to-\infty$ and
using (21) and $\beta<\lambda-\varepsilon$ it follows that $z^{s}_{m}=0$ for
every $m\in{\mathbb{Z}}$. Similarly, using (6), we get that for every $m<n$,

$\begin{split}\mu_{m}^{-\lambda}\|z^{u}_{m}\|&=\mu_{m}^{-\lambda}\|\mathcal{A}(m,n)z^{u}_{n}\|\\\
&=\mu_{m}^{-\lambda}\|\mathcal{A}(m,n)(\text{Id}-P_{n})z_{n}\|\\\ &\leq
D\nu_{n}\mu_{n}^{-\lambda}\|z_{n}\|\\\ &\leq
D\nu_{n}\mu_{n}^{-(\lambda+\beta)}\|\mathbf{z}\|_{\infty,\beta}\\\ &\leq
D\mu_{n}^{-(\lambda+\beta-\varepsilon)}\|\mathbf{z}\|_{\infty,\beta}\cdot\sup_{k\geq
m}(\mu_{k}^{-\varepsilon}\nu_{k}).\end{split}$

Therefore, since $\lim_{n\to+\infty}\mu_{n}=+\infty$, letting $n\to+\infty$
and using (21) and $\beta>-(\lambda-\varepsilon)$ it follows that
$z^{u}_{m}=0$ for every $m\in{\mathbb{Z}}$. Combining these observations we
conclude that $z_{m}=z^{s}_{m}+z^{u}_{m}=0$ for every $m\in{\mathbb{Z}}$
completing the proof of the first claim in the theorem. Let us now prove the
second one.

Given $\beta\in(-(\lambda-\varepsilon),\lambda-\varepsilon)$ and
$\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta,|\cdot|}$, let us
consider $x_{n}$, $n\in{\mathbb{Z}}$, defined by (23). Then, using (5) and
(6), if $n\geq n_{0}$ we get that

$\begin{split}\|x_{n}\|&\leq
D\sum_{k=-\infty}^{n}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-\lambda}\nu_{k}\|y_{k}\|+D\sum_{k=n+1}^{\infty}\left(\frac{\mu_{k}}{\mu_{n}}\right)^{-\lambda}\nu_{k}\|y_{k}\|\\\
&=D\sum_{k=-\infty}^{n_{0}-1}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-\lambda}\nu_{k}\|y_{k}\|+D\sum_{k=n_{0}}^{n}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-\lambda}\nu_{k}\|y_{k}\|\\\
&\phantom{=}+D\sum_{k=n+1}^{\infty}\left(\frac{\mu_{k}}{\mu_{n}}\right)^{-\lambda}\nu_{k}\|y_{k}\|\\\
&=D\sum_{k=-\infty}^{n_{0}-1}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-(\lambda-|\beta|)}\mu_{n}^{-|\beta|}\mu_{k}^{|\beta|}\nu_{k}\|y_{k}\|\\\
&\phantom{=}+D\sum_{k=n_{0}}^{n}\left(\frac{\mu_{n}}{\mu_{k}}\right)^{-(\lambda+|\beta|)}\mu_{n}^{|\beta|}\mu_{k}^{-|\beta|}\nu_{k}\|y_{k}\|\\\
&\phantom{=}+D\sum_{k=n+1}^{\infty}\left(\frac{\mu_{k}}{\mu_{n}}\right)^{-(\lambda-|\beta|)}\mu_{n}^{|\beta|}\mu_{k}^{-|\beta|}\nu_{k}\|y_{k}\|.\\\
\end{split}$

Thus, since $(\mu_{k})_{k\in{\mathbb{Z}}}$ is strictly increasing,
$|\beta|<\lambda$ and $\mu_{n}\geq 1$ for every $n\geq n_{0}$, it follows that

$\begin{split}\|x_{n}\|&\leq
D\mu_{n}^{|\beta|}\left(\sum_{k=-\infty}^{n_{0}-1}\mu_{k}^{|\beta|}\nu_{k}\|y_{k}\|+\sum_{k=n_{0}}^{\infty}\mu_{k}^{-|\beta|}\nu_{k}\|y_{k}\|\right)\\\
\end{split}$

which implies that

$\mu_{n}^{-|\beta|}\|x_{n}\|\leq D\|\textbf{y}\|_{1,\beta,|\cdot|}.$

Similarly, in the case when $n<n_{0}$ we can prove that

$\mu_{n}^{|\beta|}\|x_{n}\|\leq D\|\textbf{y}\|_{1,\beta,|\cdot|}.$

Consequently,
$\textbf{x}=(x_{n})_{n\in{\mathbb{Z}}}\in\ell^{\infty}_{\beta,|\cdot|}$.
Moreover, proceeding again as in the proof of Theorem 3.1 we conclude that
(22) is satisfied. Finally, the uniqueness of this
$\mathbf{x}\in\ell^{\infty}_{\beta,|\cdot|}$ can be obtained via an argument
similar to the one we did in the proof of the first claim of the theorem. The
proof of the theorem is completed. ∎

Our next theorem gives us the converse result of Theorem 4.1.

######  Theorem 4.2.

Suppose that there exists $\beta>0$ such that:

  1. (1)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta}$, there
exists a unique $\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in\ell_{\beta}^{\infty}$
such that (22) holds;

  2. (2)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{-\beta}$, there
exists a unique $\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in\ell_{-\beta}^{\infty}$
such that (22) holds;

  3. (3)

for each $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta,|\cdot|}$,
there exists a unique
$\mathbf{x}=(x_{n})_{n\in\mathbb{Z}}\in\ell_{\beta,|\cdot|}^{\infty}$ such
that (22) holds.

Then, $(A_{n})_{n\in\mathbb{Z}}$ admits a $(\mu,\nu)$-dichotomy.

###### Proof.

We will proceed as in the proof of Theorem 3.2. We start with two auxiliary
observations.

######  Lemma 4.3.

Let
$\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}\in\ell^{1}_{\beta}\cap\ell^{1}_{-\beta}$,
$\mathbf{x}^{1}=(x_{n}^{1})_{n\in\mathbb{Z}}\in\ell^{\infty}_{\beta}$ and
$\mathbf{x}^{2}=(x_{n}^{2})_{n\in\mathbb{Z}}\in\ell^{\infty}_{-\beta}$ be such
that both pairs $(\mathbf{x}^{i},\mathbf{y})$, $i\in\\{1,2\\}$ satisfy (22).
Then, $x_{n}^{1}=x_{n}^{2}$ for every $n\in\mathbb{Z}$.

###### Proof of Lemma 22.

Since $\ell^{1}_{\beta}\subset\ell^{1}_{\beta,|\cdot|}$ and
$\ell^{1}_{-\beta}\subset\ell^{1}_{\beta,|\cdot|}$, it follows that
$\mathbf{y}\in\ell^{1}_{\beta,|\cdot|}$. Similarly, since
$\ell^{\infty}_{\beta}\subset\ell^{\infty}_{\beta,|\cdot|}$ and
$\ell^{\infty}_{-\beta}\subset\ell^{\infty}_{\beta,|\cdot|}$, it follows that
$\mathbf{x}^{1},\mathbf{x}^{2}\in\ell^{\infty}_{\beta,|\cdot|}$. Thus, by the
uniqueness given in the third assumption of the statement of Theorem 4.2 it
follows that $x_{n}^{1}=x_{n}^{2}$ for every $n\in\mathbb{Z}$ as claimed. ∎

Let us consider $T_{j}\colon\ell^{1}_{j}\to\ell^{\infty}_{j}$ given by
$T_{j}(\mathbf{y})=\mathbf{x}$ where $\mathbf{x}$ is the unique element in
$l^{\infty}_{j}$ such that (22) holds for $j=\beta,-\beta$.

######  Lemma 4.4.

Then $T_{\beta}$ and $T_{-\beta}$ are bounded linear operators.

###### Proof of Lemma 4.4.

The proof can be obtained by proceeding as in the proof of Lemma 3.6. ∎

Given $n\in{\mathbb{Z}}$, let us consider

$S(n):=\left\\{v\in X:\;\sup_{m\geq
n}\|{\mathcal{A}}(m,n)v\|<+\infty\right\\}.$

Similarly, let $U(n)$ be the space of all $v\in X$ for which there exists a
sequence $(z_{m})_{m\leq n}$ such that $z_{n}=v$, $z_{m}=A_{m-1}z_{m-1}$ for
every $m\leq n$ and $\sup_{m\leq n}\|z_{m}\|<+\infty$.

It is easy to see that

$A_{n}S(n)\subset S(n+1)\quad\text{ and }\quad A_{n}U(n)\subset U(n+1)$

for every $n\in{\mathbb{Z}}$. Moreover, we have the following observations.

######  Lemma 4.5.

For every $n\in{\mathbb{Z}}$,

$X=S(n)\oplus U(n).$ (24)

###### Proof of Lemma 4.5.

Fix $n\in{\mathbb{Z}}$. Given $v\in X$, we define
$\mathbf{y}=(y_{m})_{m\in\mathbb{Z}}$ by $y_{n}=v$ and $y_{m}=0$ for $m\neq
n$. Then, $\mathbf{y}\in\ell^{1}_{\beta}\cap\ell^{1}_{-\beta}$. By our
assumption and Lemma 4.3, there exists
$\mathbf{x}=(x_{m})_{m\in\mathbb{Z}}\in\ell^{\infty}_{\beta}\cap\ell^{\infty}_{-\beta}$
such that

$x_{m+1}-A_{m}x_{m}=y_{m+1},\quad m\in\mathbb{Z}.$

In particular,

$x_{n}-v=A_{n-1}x_{n-1}$ (25)

and

$x_{m+1}=A_{m}x_{m}\quad\text{ for all }m\neq n-1.$ (26)

Thus, (26) combined with the fact that
$(x_{m})_{m\in{\mathbb{Z}}}\in\ell_{\beta}^{\infty}$ implies that $x_{n}\in
S(n)$. Similarly, (25) and (26) combined with the fact that
$(x_{m})_{m\in{\mathbb{Z}}}\in\ell_{-\beta}^{\infty}$ implies that $x_{n}-v\in
U(n)$. Therefore,

$v=x_{n}+(v-x_{n})\in S(n)+U(n)$

and $X=S(n)+U(n)$.

Suppose now that $v\in S(n)\cap U(n)$. Then there exists $(z_{m})_{m\leq n}$
such that $z_{n}=v$, $z_{m}=A_{m-1}z_{m-1}$ for every $m\leq n$ and
$\sup_{m\leq n}\|z_{m}\|<+\infty$. Define

$x_{m}=\left\\{\begin{array}[]{ll}z_{m},&\text{ for }m\leq n;\\\
{\mathcal{A}}(m,n)v,&\text{ for }m>n.\end{array}\right.$

Thus, since $v\in S(n)$ and $\sup_{m\leq n}\|z_{m}\|<+\infty$, it follows that
$\sup_{m\in{\mathbb{Z}}}\|x_{m}\|<+\infty$. In particular,
$(x_{m})_{m\in{\mathbb{Z}}}\subset\ell^{\infty}_{\beta,|\cdot|}$ and

$x_{m+1}-A_{m}x_{m}=0\;\text{ for every }m\in{\mathbb{Z}}.$

Therefore, from the uniqueness given in the third hypothesis of Theorem 4.2 we
get that $x_{m}=0$ for every $m\in{\mathbb{Z}}$ which implies that $v=0$ and
$S(n)\cap U(n)=\\{0\\}$. This concludes the proof. ∎

######  Lemma 4.6.

For every $n\in{\mathbb{Z}}$, the operator $A_{n}|_{U(n)}\colon U(n)\to
U(n+1)$ is an isomorphism.

###### Proof of Lemma 4.6.

Fix $n\in{\mathbb{Z}}$. Let us start by observing that $A_{n}|_{U(n)}$ is
injective. Suppose there exists $v\in U(n)$ such that $A_{n}v=0$. By the
definition of $U(n)$ there exists $(z_{m})_{m\leq n}$ such that $z_{n}=v$,
$z_{m}=A_{m-1}z_{m-1}$ for every $m\leq n$ and $\sup_{m\leq
n}\|z_{m}\|<+\infty$. Then the sequence $(x_{m})_{m\in{\mathbb{Z}}}$ given by

$x_{m}=\left\\{\begin{array}[]{ll}z_{m},&\text{ for }m\leq n;\\\ 0,&\text{ for
}m>n,\end{array}\right.$

is in $l^{\infty}_{\beta}$ and satisfies (22) with
$\mathbf{y}=(y_{m})_{m\in{\mathbb{Z}}}$ and $y_{m}=0$ for every
$m\in{\mathbb{Z}}$. Consequently, by the uniqueness in the hypothesis of
Theorem 4.2, it follows that $x_{m}=0$ for every $m\in{\mathbb{Z}}$. In
particular, $v=0$ and $A_{n}|_{U(n)}$ is injective.

Now, given $v\in U(n+1)$, let us consider a sequence $(z_{m})_{m\leq n+1}$
such that $z_{n+1}=v$, $z_{m}=A_{m-1}z_{m-1}$ for every $m\leq n+1$ and
$\sup_{m\leq n+1}\|z_{m}\|<+\infty$. Then, $z_{n}\in U(n)$ and $A_{n}z_{n}=v$
which proves that $A_{n}|_{U(n)}\colon U_{n}\to U(n+1)$ is surjective. ∎

######  Lemma 4.7.

For each $n\in{\mathbb{Z}}$, let $P_{n}\colon X\to S(n)$ be the projection
associated with (24). Then, there exists $D>0$ such that

$\|P_{n}\|\leq D\nu_{n},\quad n\in\mathbb{Z}.$ (27)

###### Proof of Lemma 4.7.

The proof of this result is the same, mutatis mutandis, as the proof of Lemma
3.6. ∎

######  Lemma 4.8.

There exists $C>0$ such that

$\|\mathcal{A}(m,n)v\|\leq
C\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}\|v\|,$

for every $m\geq n$ and $v\in S(n)$.

###### Proof of Lemma 4.8.

Let $n\in{\mathbb{Z}}$ and $v\in S(n)$. We define sequences
$(y_{m})_{m\in\mathbb{Z}}$ and $(x_{m})_{m\in\mathbb{Z}}$ by

$y_{m}=\begin{cases}v&m=n;\\\ 0&m\neq n,\end{cases}\quad\text{and}\quad
x_{m}=\begin{cases}\mathcal{A}(m,n)v&m\geq n;\\\ 0&m<n.\end{cases}$

Then,
$\mathbf{y}=(y_{m})_{m\in\mathbb{N}}\in\ell^{1}_{\beta}\cap\ell^{1}_{-\beta}$
and $\mathbf{x}=(x_{m})_{m\in\mathbb{Z}}\in\ell^{\infty}_{-\beta}$. Moreover,
(22) holds. Then, from Lemma 4.3 it follows that
$\mathbf{x}\in\ell^{\infty}_{\beta}$. Thus, considering
$T_{\beta}\colon\ell^{1}_{\beta}\to\ell^{\infty}_{\beta}$ as in Lemma 4.4 and
$C=\|T_{\beta}\|+1$ we get that

$\mu_{m}^{\beta}\|x_{m}\|\leq\|\mathbf{x}\|_{\infty,\beta}=\|T_{\beta}(\mathbf{y})\|_{\infty,\beta}\leq
C\|y\|_{1,\beta}=C\mu_{n}^{\beta}\nu_{n}\|v\|$

for $m\geq n$. Thus,

$\|\mathcal{A}(m,n)v\|\leq
C\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}\|v\|$

for every $m\geq n$ as claimed. ∎

######  Lemma 4.9.

There exists $\tilde{C}>0$ such that

$\|\mathcal{A}(m,n)v\|\leq\tilde{C}\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}\|v\|,$

for every $m\leq n$ and $v\in U(n)$.

###### Proof of Lemma 4.9.

Take $n\in{\mathbb{Z}}$ and $v\in U(n)$. Then by Lemma 4.6 we may consider

$y_{m}=\begin{cases}-v&m=n;\\\ 0&m\neq n,\end{cases}\quad\text{and}\quad
x_{m}=\begin{cases}\mathcal{A}(m,n)v&m<n;\\\ 0&m\geq n.\end{cases}$

Note that
$\mathbf{y}=(y_{m})_{m\in\mathbb{Z}}\in\ell^{1}_{\beta}\cap\ell^{1}_{-\beta}$
and $\mathbf{x}=(x_{m})_{m\in\mathbb{Z}}\in\ell^{\infty}_{\beta}$.
Furthermore, (22) holds. Thus, by Lemma 4.3 it follows that
$\mathbf{x}\in\ell^{\infty}_{-\beta}$. Then, considering
$T_{-\beta}\colon\ell^{1}_{-\beta}\to\ell^{\infty}_{-\beta}$ given in Lemma
4.4 and taking $\tilde{C}=\|T_{-\beta}\|+1$ we get that

$\mu_{m}^{-\beta}\|x_{m}\|\leq\|\mathbf{x}\|_{\infty,-\beta}=\|T_{-\beta}(\mathbf{y})\|_{\infty,-\beta}\leq\tilde{C}\|y\|_{1,-\beta}=\tilde{C}\mu_{n}^{-\beta}\nu_{n}\|v\|,$

for every $m<n$. This easily implies that

$\|\mathcal{A}(m,n)v\|\leq\tilde{C}\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}\|v\|,$

for every $m<n$ as claimed. ∎

For $m,n\in\mathbb{Z}$, set

$\mathcal{G}(m,n):=\begin{cases}\mathcal{A}(m,n)P_{n}&m\geq n;\\\
-\mathcal{A}(m,n)(\text{Id}-P_{n})&m<n.\end{cases}$

Then, by Lemmas 4.7, 4.8 and 4.9 there exists $C^{\prime}>0$ such that

$\|\mathcal{G}(m,n)\|\leq
C^{\prime}\begin{cases}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}^{2}&m\geq
n;\\\
\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}^{2}&m<n.\end{cases}$ (28)

######  Lemma 4.10.

There exists $C^{\prime\prime}>0$ such that

$\|\mathcal{G}(m,n)\|\leq
C^{\prime\prime}\begin{cases}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}&m\geq
n;\\\ \left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}&m<n.\end{cases}$

###### Proof of the Lemma 4.10.

Take $n\in{\mathbb{Z}}$ and $v\in X$. We define a sequence
$\mathbf{y}=(y_{m})_{m\in\mathbb{Z}}$ by $y_{n}=v$ and $y_{m}=0$ for $m\neq
n$. Then, $\mathbf{y}\in\ell_{\beta}^{1}\cap\ell_{-\beta}^{1}$. Set

$x_{m}:=\mathcal{G}(m,n)v,\quad m\in\mathbb{Z}.$

It is easy to verify that (22) holds. We claim that
$\mathbf{x}=(x_{m})_{m\in\mathbb{Z}}\in\ell_{\beta}^{\infty}$. Indeed, by
(28), for $m\geq n$ we have that

$\|x_{m}\|=\|\mathcal{G}(m,n)v\|\leq
C^{\prime}\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n}^{2}\|v\|,$

which implies that

$\sup_{m\geq n}(\mu_{m}^{\beta}\|x_{m}\|)<+\infty.$

Similarly, for $m<n$ we have that

$\|x_{m}\|=\|\mathcal{G}(m,n)v\|\leq
C^{\prime}\left(\frac{\mu_{n}}{\mu_{m}}\right)^{-\beta}\nu_{n}^{2}\|v\|,$

which implies that

$\mu_{m}^{-\beta}\|x_{m}\|\leq C^{\prime}\mu_{n}^{-\beta}\nu_{n}^{2}\|v\|.$

Thus, since $\lim_{m\to-\infty}\mu_{m}=0$ and, in particular,
$\mu_{m}^{\beta}\leq\mu_{m}^{-\beta}$ for $m$ sufficiently small, the previous
inequality implies that

$\sup_{m<n}(\mu_{m}^{\beta}\|x_{m}\|)<+\infty.$

Combining these observations we conclude that
$\mathbf{x}=(x_{m})_{m\in\mathbb{Z}}\in\ell_{\beta}^{\infty}$ and
$T_{\beta}(\mathbf{y})=\mathbf{x}$, where $T_{\beta}$ is as in Lemma 4.4.
Therefore, for $m\geq n$ we have that

$\mu_{m}^{\beta}\|\mathcal{G}(m,n)v\|=\mu_{m}^{\beta}\|x_{m}\|\leq\|\mathbf{x}\|_{\infty,\beta}\leq\|T_{\beta}\|\cdot\|\mathbf{y}\|_{1,\beta}=\|T_{\beta}\|\mu_{n}^{\beta}\nu_{n}\|v\|,$

and thus

$\|\mathcal{G}(m,n)\|\leq\|T_{\beta}\|\left(\frac{\mu_{m}}{\mu_{n}}\right)^{-\beta}\nu_{n},\quad
m\geq n.$

Similarly, one can treat the case when $m<n$. ∎

Now the proof of Theorem 4.2 can be easily completed by combining the previous
auxiliary results. ∎

###  4.2. Persistence of $(\mu,\nu)$-dichotomy

We now present a version of Theorem 3.12 in the case of two-sided dynamics.

######  Theorem 4.11.

Let $(\gamma_{n})_{n\in{\mathbb{Z}}}$ be a sequence of numbers with
$\gamma_{n}>0$ for every $n\in{\mathbb{Z}}$ such that

$\sum_{n\in{\mathbb{Z}}}\gamma_{n}<+\infty.$

Suppose that (2) admits a $(\mu,\nu)$-dichotomy with $\lambda>0$ and condition
(21) is satisfied and take $\beta\in(0,\lambda-\varepsilon)$. Moreover, let
$(B_{n})_{n\in{\mathbb{Z}}}$ be a sequence of operators in $\mathcal{B}(X)$
with the property that there exists $c>0$ such that

$\|B_{n}\|\leq\frac{c\gamma_{n}\mu_{n}^{\beta}}{\nu_{n+1}\mu_{n+1}^{\beta}},\quad\text{for
$n\in\mathbb{Z}$.}$ (29)

Then, if $c$ is small enough we have that the nonautonomous difference
equation

$x_{n+1}=(A_{n}+B_{n})x_{n},\quad n\in{\mathbb{Z}}$

also admits a $(\mu,\nu)$-dichotomy.

###### Proof.

The proof of this result is similar to the proof of Theorem 3.12 and,
therefore, we refrain from writing it. ∎

######  Remark 4.12.

A comment similar to Remark 3.13 also applies to condition (29).

Acknowledgements. We would like to thank C. Silva for useful discussions
related to [41]. D.D is also grateful to W. Zhang and L. Zhou for ideas
related to [17, 18] which inspired results of the present paper. L. Backes was
partially supported by a CNPq-Brazil PQ fellowship under Grant No.
307633/2021-7. D. Dragičević was supported in part by University of Rijeka
under the project uniri-iskusni-prirod-23-98 3046.

Statements and Declarations

Competing Interests: no potential conflict of interest was reported by the
authors.

## References

  * [1] B. Aulbach and N. Van Minh, _The concept of spectral dichotomy for linear difference equations II_ , J. Differ. Equ. Appl. 2 (1996) 251–262.
  * [2] L. Barreira, D. Dragičević and C. Valls, _Nonuniform hyperbolicity and admissibility_ , Adv. Nonlinear Stud. 14 (2014), 791–811.
  * [3] L. Barreira, D. Dragičević and C. Valls, _Nonuniform hyperbolicity and one-sided admissibility_ , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016), 235–247.
  * [4] L. Barreira, D. Dragičević and C. Valls, _Admissibility on the half line for evolution families_ , J. Anal. Math. 132 (2017), 157–176.
  * [5] L. Barreira, D. Dragičević and C. Valls, Admissibility and Hyperbolicity, SpringerBriefs Math., Springer, Cham, 2018.
  * [6] L. Barreira and C. Valls, _Growth rates and nonuniform hyperbolicity_ , Discrete Contin. Dyn. Syst. 22 (2008), 509–528.
  * [7] A. J. Bento and C. M. Silva, _Nonuniform $(\mu,\nu)$-dichotomies and local dynamics of difference equations_, Nonlinear Anal. 75 (2012), 78–90.
  * [8] A. J. Bento and C. M. Silva, _Generalized Nonuniform Dichotomies and Local Stable Manifolds_ , J. Dyn. Diff. Equat. 25 (2013), 1139–1158.
  * [9] C.V. Coffman and J.J. Schäffer, _Dichotomies for linear difference equations_ , Math. Ann. 172 (1967) 139–166.
  * [10] W. Coppel, Dichotomies in Stability Theory, Lect. Notes in Math. 629, Springer, 1978.
  * [11] Ju. Dalec′kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974.
  * [12] D. Dragičević, _Admissibility and nonuniform polynomial dichotomies_ , Math. Nachr. 293 (2019), 226–243.
  * [13] D. Dragičević, _Admissibility and polynomial dichotomies for evolution families_ , Commun. Pure Appl. Anal. 19 (2020), 1321–1336.
  * [14] D. Dragičević, N. Jurčević Peček and N. Lupa, _Admissibility and general dichotomies for evolution families_ , Electron. J. Qual. Theory Differ. Equ. (2020), Paper No. 58, 19 pp.
  * [15] D. Dragičević, A. L. Sasu and B. Sasu, _Admissibility and polynomial dichotomy of discrete nonautonomous systems_ , Carpathian J. Math. 38 (2022), 737–762.
  * [16] D. Dragičević, A. L. Sasu and B. Sasu, _On the Robustness of Polynomial Dichotomy of Discrete Nonautonomous Systems_ , Carpathian J. Math. 40 (2024), 643–654.
  * [17] D. Dragičević, W. Zhang and L. Zhou, _Admissibility and nonuniform exponential dichotomies_ , J. Differ. Equ. 326 (2022), 201-226.
  * [18] D. Dragičević, W. Zhang and L. Zhou, _Measurable weighted shadowing for random dynamical systems on Banach spaces_ , J. Differential Equations 392 (2024), 364–386.
  * [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981.
  * [20] N. T. Huy, _Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line_ , J. Funct. Anal. 235 (2006), 330–354.
  * [21] Ta Li, _Die Stabilitätsfrage bei Differenzengleichungen_ , Acta Math. 63 (1934) 99–141.
  * [22] Y. Latushkin, T. Randolph and R. Schnaubelt, _Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces_ , J. Dynam. Differential Equations 10 (1998), 489–510.
  * [23] N. Lupa and L. H. Popescu, _A complete characterization of exponential stability for discrete dynamics_ , J. Difference Equ. Appl. 23 (2017), 2072–2092.
  * [24] N. Lupa and L. H. Popescu, _Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line_ , Semigroup Forum 98 (2019), 184–208.
  * [25] J. Massera and J. Schäffer, _Linear differential equations and functional analysis, I_ , Ann. of Math. 67 (1958), 517–573.
  * [26] J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, New York-London, 1966.
  * [27] M. Megan, B. Sasu and A. L. Sasu, _On nonuniform exponential dichotomy of evolution operators in Banach spaces_ , Integral Equations Operator Theory 44 (2002), 71–78.
  * [28] M. Megan, A. L. Sasu and B. Sasu, _Discrete admissibility and exponential dichotomy for evolution families_ , Discrete Contin. Dyn. Syst. 9 (2003), 383–397.
  * [29] N. Van Minh, F. Räbinger and R. Schnaubelt, _Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line_ , Integral Equations Operator Theory 32 (1998), 332–353.
  * [30] N. Van Minh and N. T. Huy, _Characterizations of exponential dichotomies of evolution equations on the half-line_ , J. Math. Anal. Appl. 261 (2001) 28–44.
  * [31] J. S. Muldowney, _Dichotomies and asymptotic behaviour for linear differential systems_ , Trans. Amer. Math. Soc. 283 (1984), 465–484.
  * [32] R. Naulin, M. Pinto, _Roughness of $(h,k)$-dichotomies_, J. Differential Equations 118 (1995), 20–35.
  * [33] O. Perron, _Die stabilitätsfrage bei diﬀerentialgleichungen_ , Math. Z. 32 (1930), 703–728.
  * [34] M. Pinto, _Discrete dichotomies_ , Comput. Math. Appl., 28 (1994), 259–270.
  * [35] P. Preda and M. Megan, _Nonuniform dichotomy of evolutionary processes in Banach spaces_ , Bull. Austral. Math. Soc. 27 (1983), 31–52.
  * [36] A. L. Sasu, _Exponential dichotomy and dichotomy radius for difference equations_ , J. Math. Anal. Appl. 344 (2008), 906–920.
  * [37] A. L. Sasu, M. G. Babuţia and B. Sasu, _Admissibility and nonuniform exponential dichotomy on the half-line_ , Bull. Sci. Math. 137 (2013), 466–484.
  * [38] A. L. Sasu and B. Sasu, _Admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line_ , Appl. Math. Comput. 378 (2020), 125167.
  * [39] A. L. Sasu and B. Sasu, _Strong exponential dichotomy of discrete nonautonomous systems: input-output criteria and strong dichotomy radius_ , J. Math. Anal. Appl. 504 (2021), Paper No. 125373, 29 pp.
  * [40] B. Sasu and A. L. Sasu, _On the dichotomic behavior of discrete dynamical systems on the half-line_ , Discrete Contin. Dyn. Syst. 33 (2013), 3057–3084.
  * [41] C. M. Silva, _Admissibility and generalized nonuniform dichotomies for discrete dynamics_ , Commun. Pure Appl. Anal., 20 (2021), 3419-3443.
  * [42] J. Schäffer, _Linear difference equations: Closedness of covariant sequences_ , Math. Ann. 187 (1970), 69–76.
  * [43] A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York, 1958.
  * [44] M. Wu and Y. Xia, _Admissibility and nonuniform exponential dichotomies for difference equations without bounded growth or Lyapunov norms_ , Proc. Amer. Math. Soc. 151 (2023), 4389-4403.
  * [45] L. Zhou, K. Lu and W. Zhang, _Equivalences between nonuniform exponential dichotomy and admissibility_ , J. Differ. Equ. 262 (2017), 682–747.
  * [46] L. Zhou and W. Zhang, _Admissibility and roughness of nonuniform exponential dichotomies for difference equations_ , J. Funct. Anal. 271 (2016), 1087–1129.