1. Introduction
Let 
$\mathbb Z$ and 
$\mathbb C$ denote the set of integers and the set of complex numbers, respectively. For 
$k\in\mathbb Z$, define 
$\mathbb Z^+_k=\{\,n\in\mathbb Z:n\geq k\,\}$ and 
$\mathbb Z^-_k=\{\,n\in\mathbb Z:n\leq k\,\}$. Throughout the article, we shall assume as a standing assumption that 
$(X,|\cdot|)$ is a Banach space. The symbol 
$\mathcal L(X)$ will denote the space of all bounded linear operators 
$A\colon X\to X$ equipped with the operator norm, 
$
|A|=\sup_{|x|=1}|Ax|
$ for 
$A\in\mathcal L(X)$.
Consider the linear autonomous difference equation
\begin{equation}
x(n+1)=Ax(n),
\end{equation} where 
$A\in\mathcal L(X)$. Given 
$\delta\geq0$, by a δ-pseudosolution of Eq. (1.1) on 
$\mathbb Z^+_0$, we mean a function 
$y\colon \mathbb Z^+_0\to X$ such that
\begin{equation*}
\sup_{n\geq 0}
|y(n+1)-Ay(n)|\leq\delta.
\end{equation*} Note that for δ = 0, the pseudosolution becomes a true solution of Eq. (1.1) on 
$\mathbb Z^+_0$. We say that Eq. (1.1) is shadowable on 
$\mathbb Z^+_0$ or that it has the positive shadowing property if, for every ϵ > 0, there exists δ > 0 such that for every δ-pseudosolution y of (1.1) on 
$\mathbb Z^+_0$, there exists a true solution x of (1.1) on 
$\mathbb Z^+_0$ such that
\begin{equation*}
\sup_{n\geq 0}|x(n)-y(n)|\leq\epsilon.
\end{equation*} The shadowing of Eq. (1.1) is closely related to its hyperbolicity. Recall that Eq. (1.1) is hyperbolic if 
$\sigma(A)$ does not intersect the unit circle 
$|\lambda|=1$ in 
$\mathbb C$, where 
$\sigma(A)$ denotes the spectrum of A.
In a recent article [Reference Bernardes, Cirilo, Darji, Messaoudi and Pujals5], Bernardes et al. have studied various shadowing properties of Eq. (1.1). Among others, they have shown that if Eq. (1.1) is hyperbolic, then it is shadowable on 
$\mathbb Z^+_0$ (see [Reference Bernardes, Cirilo, Darji, Messaoudi and Pujals5, Theorem 13]). Moreover, if the coefficient 
$A\in\mathcal L(X)$ in Eq. (1.1) is a compact operator, then the converse is also true. As usual, an operator 
$A\colon X\to X$ is compact if, for every bounded set 
$S\subset X$, the image A(S) has compact closure in X. Thus, we have the following theorem:
Theorem 1.1 [Reference Bernardes, Cirilo, Darji, Messaoudi and Pujals5, Theorem 15] Let 
$A\in\mathcal L(X)$ be a compact operator. Then, the following statements are equivalent.
Remark 1.2. As noted previously, the implication (ii) 
$\Rightarrow$ (i) in Theorem 1.1 is true if we assume merely that 
$A\in\mathcal L(X)$. The compactness of 
$A\in\mathcal L(X)$ is important only for the validity of the converse implication (i) 
$\Rightarrow$ (ii) (see [Reference Bernardes, Cirilo, Darji, Messaoudi and Pujals5, Remark 14]).
Now let us consider the finite dimensional case 
$X=\mathbb C^d$, where d is a positive integer and 
$\mathbb C^d$ denotes the d-dimensional space complex column vectors. Then, the space 
$\mathcal L(\mathbb C^d)$ can be identified with 
$\mathbb C^{d\times d}$, the space of d × d matrices with complex entries. Since linear operators between finite dimensional spaces are compact and the spectrum of a square matrix 
$A\in\mathbb C^{d\times d}$ consists of the roots of its characteristic equation
\begin{equation}
\det(\lambda E-A)=0,
\end{equation} where 
$E\in\mathbb C^{d\times d}$ is the unit matrix, in this case, Theorem 1.1 can be reformulated as follows:
Theorem 1.3 Let 
$A\in\mathbb C^{d\times d}$. Then, the following statements are equivalent:
Our aim in this article is to extend Theorems 1.1 and 1.3 to more general classes of linear difference equations with delay.
In §2, we will generalize Theorem 1.1 to the non-autonomous linear difference equation with finite delays
\begin{equation}
x(n+1)=\sum_{j=0}^r A_j(n)x(n-j),
\end{equation} where 
$r\in\mathbb Z^+_0$ is the maximum delay and the coefficients 
$A_j(n)\in\mathcal L(X)$, 
$0\leq j\leq r$, 
$n\in\mathbb Z^+_0$, are compact linear operators that are uniformly bounded, i.e., there exists 
$K\geq1$ such that
\begin{equation}
|A_j(n)|\leq K,\qquad n\in\mathbb Z^+_0,\quad 0\leq j\leq r.
\end{equation} The main result of §2 is formulated in Theorem 2.3, which may be viewed as a discrete analogue of our recent shadowing theorem for delay differential equations in 
$\mathbb R^d$ [Reference Backes, Dragičević and Pituk2, Theorem 2.2]. It says that, under the above hypotheses, Eq. (1.3) is shadowable on 
$\mathbb Z^+_0$ if and only if it has an exponential dichotomy, which is a non-autonomous variant of hyperbolicity. To the best of our knowledge, this result is new even for ordinary difference equations (r = 0). We note that in the particular case when r = 0 and X is finite-dimensional, a version of this result was established in [Reference Backes and Dragičević1] (see [Reference Backes and Dragičević1, Corollary 2] and [Reference Backes and Dragičević1, Proposition 4]).
In §3, we will extend Theorem 1.3 to the linear Volterra difference equation with infinite delay
\begin{equation}
x(n+1)=\sum_{j=-\infty}^n A(n-j)x(j),
\end{equation} where 
$A\colon\mathbb Z^+_0\to\mathbb C^{d\times d}$ satisfies
\begin{equation}
\sum_{j=0}^\infty|A(j)|e^{\gamma j} \lt \infty\qquad\text{for~some}~\gamma \gt 0.
\end{equation}The characteristic equation of Eq. (1.5) has the form
\begin{equation}
\det\varDelta(\lambda)=0 ,\qquad|\lambda| \gt e^{-\gamma},
\end{equation}where
\begin{equation}
\varDelta(\lambda)=\lambda E-\sum_{j=0}^\infty\lambda^{-j}A(j),
\qquad|\lambda| \gt e^{-\gamma}.
\end{equation} The main result of §3, Theorem 3.2, says that in the natural (infinite dimensional) phase space 
$\mathcal B_\gamma$ defined below Eq. (1.5) is shadowable on 
$\mathbb Z^+_0$ if and only if its characteristic Eq. (1.7) has no root on the unit circle 
$|\lambda|=1$.
The fact that non-autonomous linear delay difference equations, including (1.3) and (1.5), are shadowable whenever they are hyperbolic follows from [Reference Dragičević and Pituk10, Theorem 1]. Therefore, in both cases (1.3) and (1.5), we need to prove only the converse result. In the case of the non-autonomous equation with finite delays (1.3), the proof follows similar lines as the proof our continuous time result [Reference Backes, Dragičević and Pituk2, Theorem 2.2] with non-trivial modifications because, instead of 
$\mathbb R^d$, we consider Eq. (1.3) in a general infinite dimensional Banach space. It is based on the eventual compactness of the solution operator, combined with an input–output technique [Reference Barreira, Dragičević and Valls3] and Schäffer’s result about the existence of regular covariant subspaces of linear difference equation in a Banach space [Reference Schäffer27]. A similar argument for the Volterra equation with infinite delay (1.5) does not apply since its solution operator is not eventually compact. In this case, the proof will be based on the duality between Eq. (1.5) and its formal adjoint equation, which has been established by Matsunaga et. al. [Reference Matsunaga, Murakami, Nagabuchi and Nakano18].
2. Shadowing of non-autonomous linear difference equation with finite delays
In this section, we consider the shadowing of the non-autonomous linear difference equation with finite delays (1.3), where 
$A_j\colon\mathbb Z^+_0\to\mathcal L(X)$, 
$0\leq j\leq r$, satisfy condition (1.4). The phase space for Eq. (1.3) is 
$(\mathcal B_r,\|\cdot\|)$, where 
$\mathcal B_r$ is the set of all functions 
$\phi\colon[-r,0]\cap\mathbb Z\to X$ and
\begin{equation*}
\|\phi\|=\max_{-r\leq\theta\leq0}|\phi(\theta)|,\qquad\phi\in\mathcal B_r.
\end{equation*}Eq. (1.3) can be written equivalently in a form of a functional difference equation
\begin{equation}
x(n+1)=L_n (x_n) ,
\end{equation} where the solution segment 
$x_n\in\mathcal B_r$ is defined by
\begin{equation*}
x_n(\theta)=x(n+\theta),\qquad \theta \in[-r,0]\cap\mathbb Z,
\end{equation*} and 
$L_n\colon\mathcal B_r\to X$ is a bounded linear functional defined by
\begin{equation*}
L_n(\phi)=\sum_{j=0}^r A_j(n)\phi(-j),\qquad \phi\in\mathcal B_r,\quad n\in\mathbb Z^+_0,\quad 0\leq j\leq r.
\end{equation*}In view of (1.4), we have that
\begin{equation}
\|L_n\|\leq M:=(r+1)K,\qquad n\in\mathbb Z^+_0.
\end{equation} Given 
$m\in\mathbb Z^+_0$ and 
$\phi\in\mathcal B_r$, there exists a unique function 
$x\colon\mathbb Z^+_{m-r}\to X$ satisfying Eq. (1.3) and the initial condition 
$x(m+\theta)=\phi(\theta)$ for 
$\theta\in[-r,0]\cap\mathbb Z$. We shall call x the solution of Eq. (1.3) with initial value 
$x_m=\phi$. By a solution of Eq. (1.3) on 
$\mathbb Z_m^+$, we mean a solution x with initial value 
$x_m=\phi$ for some 
$\phi\in\mathcal B_r$.
For each 
$n, m\in\mathbb Z^+_0$ with 
$n\geq m$, the solution operator 
$T(n,m)\colon\mathcal B_r\to\mathcal B_r$ is defined by 
$T(n,m)\phi=x_n$ for 
$\phi\in\mathcal B_r$, where x is the unique solution of Eq. (1.3) with initial value 
$x_m=\phi$. It is easily seen that for all 
$n,k,m\in\mathbb Z^+_0$ with 
$n\geq k\geq m$,
\begin{align}
T(m,m)=I,
\end{align}
\begin{align}
T(n,m)=T(n,k)T(k,m),
\end{align}
\begin{align}
\|T(n,m)\|\leq e^{\omega(n-m)},
\end{align} where I denotes the identity operator on 
$\mathcal B_r$ and 
$\omega=\log (M(1+r))$.
Now we can introduce the definitions of shadowing and exponential dichotomy for Eq. (1.3) (equivalently, (2.1)).
Definition 2.1. We say that Eq. (1.3) is shadowable on 
$\mathbb Z^+_0$ if, for each ϵ > 0, there exists δ > 0 such that for every function 
$y\colon \mathbb Z^+_{-r}\to X$ satisfying
\begin{equation*}
\sup_{n\geq 0}|y(n+1)-L_n(y_n)| \le \delta,
\end{equation*} there exists a solution x of (1.3) on 
$\mathbb Z^+_0$ such that
\begin{equation*}
\sup_{n\geq 0} \|x_n-y_n\| \le \epsilon.
\end{equation*}Definition 2.2. We say that Eq. (1.3) admits an exponential dichotomy (on 
$\mathbb Z^+_0$) if there exist a sequence of projections 
$(P_n)_{n\in\mathbb Z^+_0}$ on 
$\mathcal B_r$ and constants 
$D, \lambda \gt 0$ with the following properties:
• for
$n,m\in \mathbb Z^+_0$ with $n\geq m$,
(2.6)\begin{equation}
P_n T(n,m)=T(n,m)P_m,
\end{equation}and
$T(n,m)\rvert_{\operatorname{ker} P_m} \colon \operatorname{ker} P_m\to
\operatorname{ker} P_n$ is onto and invertible;• for
$n,m\in \mathbb Z^+_0$ with $n\geq m$,
(2.7)\begin{equation}
\|T(n,m)P_m\| \le De^{-\lambda (n-m)};
\end{equation}• for
$n,m\in \mathbb Z^+_0$ with $n\leq m$,
(2.8)\begin{equation}
\|T(n,m)Q_m\| \le De^{-\lambda (m-n)},
\end{equation}where
$Q_m=I-P_m$ and $T(n,m):=\left (T(m,n)\rvert_{\operatorname{ker} P_n} \right )^{-1}$.
The main result of this section is the following shadowing theorem for Eq. (1.3).
Theorem 2.3 Suppose that the coefficients 
$A_j(n)\in\mathcal L(X)$, 
$n\in\mathbb Z^+_0$, 
$0\leq j\leq r$ , of Eq. (1.3) are compact operators satisfying condition (1.4). Then, the following statements are equivalent.
Remark 2.4. It is well known that the autonomous linear Eq. (1.1) admits an exponential dichotomy if and only if the spectrum 
$\sigma(A)$ does not intersect the unit circle 
$|\lambda|=1$. Therefore, Theorem 2.3 is a generalization of Theorem 1.1 to the non-autonomous delay difference Eq. (1.3). Its conclusion is new even for ordinary difference equations (r = 0).
Remark 2.5. The implication (ii) 
$\Rightarrow$ (i) in Theorem 2.3 is a consequence of [Reference Dragičević and Pituk10, Theorem 1] with f = 0, c = 0, and µ = 1, which does not require the compactness of the coefficients. Thus, this implication is true even without the compactness assumption. However, for the validity of the converse implication (i) 
$\Rightarrow$ (ii), the compactness of the coefficient operators of Eq. (1.3) is essential (see Remark 1.2).
Remark 2.6. It follows from (2.7) and (2.8) that if Eq. (1.3) admits an exponential dichotomy, then the solution operator 
$T(m, n)$ of Eq. (1.3) exhibits the (one-sided) domination property in the sense of [Reference Quas, Thieullen and Zarrabi22, p. 2]. In [Reference Quas, Thieullen and Zarrabi22, Theorem 1.2], the authors have formulated sufficient conditions under which the solution operator associated with a non-autonomous difference equation (without delay)
\begin{equation*}
x_{n+1}=A_n x_n, \qquad n\in \mathbb Z,
\end{equation*} on an arbitrary Banach space X exhibits the domination property. We stress that no compactness assumptions on the coefficients An, 
$n\in \mathbb Z$, are assumed. These sufficient conditions are expressed in terms of the so-called uniform singular valued gap property (see [Reference Quas, Thieullen and Zarrabi22, (SVG)]). For related results in the case of linear cocycles over topological dynamical systems, we refer to the works of Bochi and Gourmelon [Reference Bochi and Gourmelon7] and Blumenthal and Morris [Reference Blumenthal and Morris6], where the connection between this type of results and the Oseledets multiplicative ergodic theorem is discussed. Our Theorem 2.3 provides a characterization of the more restrictive notion of uniform exponential dichotomy, which is expressed in terms of the shadowing property instead of the singular values.
Proof of Theorem 2.3
As noted in Remark 2.5, we need to prove only the implication (i) 
$\Rightarrow$ (ii). Suppose that Eq. (1.3) is shadowable on 
$\mathbb Z^+_0$. We will show that it admits an exponential dichotomy. We split the proof into several auxiliary results, which we now briefly describe.
In Claim 1, we show that the shadowing property implies the so-called Perron property, which guarantees that for each bounded function 
$z\colon \mathbb Z_0^+\to X$, the non-homogeneous Eq. (2.9) has at least one bounded solution 
$x\colon \mathbb Z_0^+\to X$.
The next four claims are preparatory results for the proof of the crucial Claim 6, which shows that the subspace 
$\mathcal S(0)$ of those initial functions in 
$\mathcal B_r$, which generate bounded solutions, is closed and complemented in 
$\mathcal B_r$. Claims 2 and 4 are rather simple observations, while Claim 3 is a straightforward consequence of Claim 1. A more involved argument is needed for the proof of Claim 5, which asserts that the solution operator 
$T(n, m)$ of Eq. (1.3) is a compact operator on 
$\mathcal B_r$ whenever 
$n\ge m+r+1$. The proof of Claim 6 follows directly from Claims 2, 3, 4, and 5 by applying an abstract result from [Reference Schäffer27] formulated in Lemma 2.13.
As a consequence of Claim 6, we are able to construct the unstable subspace 
$\mathcal U$ at time n = 0 as a topological complement of 
$\mathcal S(0)$ (see (2.14)), and we can revisit the Perron property established in Claim 1. More precisely, in Claim 7, we show that for each bounded 
$z\colon \mathbb{Z}_0^+\to X$, there exists a unique bounded solution 
$x\colon \mathbb{Z}_0^+\to X$ of Eq. (2.9) with 
$x_0\in \mathcal U$. Moreover, the supremum norm of x can be controlled by the supremum norm of z (see (2.15)).
In the next step, we construct the unstable subspace 
$\mathcal U(n)$ at each time 
$n\in \mathbb Z^+$. In Claims 8 and 9, we prove that 
$T(n, m)\rvert_{\mathcal U(m)}\colon \mathcal U(m)\to \mathcal U(n)$ is an isomorphism whenever 
$n\ge m$ and the phase space 
$\mathcal B_r$ splits into stable and unstable subspaces at each moment 
$n\in \mathbb Z^+$. Note that both claims are consequences of Claim 7 we show that for each bound.
The desired exponential estimates along the stable and unstable directions are obtained in Claims 11 and 13, respectively. As a preparation for the proofs of these results, in Claims 10 and 12, we show that the dynamics along the stable and unstable directions is uniformly bounded forward and backward in time, respectively. These results also rely on Claim 7.
Finally, in Claim 14, we prove that there is a uniform bound for the norms of the projections onto the stable subspaces along the unstable ones.
We now proceed with the details.
Claim 1. Eq. (1.3) has the following Perron-type property: for each bounded function 
$z\colon \mathbb Z^+_0\to X$, there exists a bounded function 
$x\colon \mathbb Z^+_{-r}\to X$, which satisfies
\begin{equation}
x(n+1)=L_n(x_n)+z(n), \qquad n\in \mathbb Z^+_0.
\end{equation}Proof of claim 1
Let 
$z\colon \mathbb Z^+_0\to X$ be an arbitrary bounded function. If 
$z(n)=0$ for every 
$n\in \mathbb Z^+_0$, then (2.9) is trivially satisfied with 
$x(n)=0$ for every 
$n\in \mathbb Z^+_{-r}$. Now suppose that 
$z(n)\neq0$ for some 
$n\in \mathbb Z^+_0$ so that 
$\|z\|_\infty:=\sup_{n\in \mathbb Z^+_0}|z(n)| \gt 0$. Choose a constant δ > 0 corresponding to the choice of ϵ = 1 in Definition 2.1. Take an arbitrary solution 
$y\colon \mathbb Z^+_{-r}\to X$ of the non-homogeneous equation
\begin{equation*}
y(n+1)=L_n(y_n)+\frac{\delta}{\|z\|_\infty}z(n), \qquad n\in \mathbb Z^+_0.
\end{equation*} (The unique solution y with initial value 
$y_0=0$ is sufficient for our purposes.) Since
\begin{equation*}
\sup_{n\geq 0}|y(n+1)-L_n(y_n)| \le \delta,
\end{equation*} according to Definition 2.1, there exists a solution 
$\tilde x$ of Eq. (2.1) on 
$\mathbb Z^+_0$ such that
\begin{equation*}
\sup_{n\geq{-r}}|\tilde x(n)-y(n)|=\sup_{n\geq 0}\|\tilde x_n-y_n\| \le 1.
\end{equation*} Define a function 
$x\colon \mathbb Z^+_{-r}\to X$ by
\begin{equation*}
x(n):=\frac{\|z\|_\infty}{\delta}(\,y(n)-\tilde x(n)\,), \qquad n\in \mathbb Z^+_{-r}.
\end{equation*}It can be easily verified that x satisfies (2.9) and
\begin{equation*}
\sup_{n\in \mathbb Z^+_{-r}}|x(n)|\le \frac{\|z\|_\infty}{\delta} \lt \infty.
\end{equation*}The proof of the claim is complete.
For each 
$m\in \mathbb Z^+_0$, define
\begin{equation*}
\mathcal S(m)=\bigl \{\phi\in \mathcal B_r: \sup_{n\ge m}\|T(n,m)\phi \| \lt \infty \bigr \}.
\end{equation*} Clearly, 
$\mathcal S(m)$ is a subspace of 
$\mathcal B_r$, which will be called the stable subspace of Eq. (2.1) at 
$m\in \mathbb Z^+_0$.
Claim 2.
For each 
$n,m\in \mathbb Z^+_0$ with 
$n\ge m $, we have that
\begin{equation*}
[\,T(n,m)\,]^{-1}(\mathcal S(n))=\mathcal S(m).
\end{equation*}Proof of claim 2
Let n and m be as in the statement. If 
$\phi \in \mathcal S(m)$, then (see (2.4))
\begin{equation*}
\sup_{k\geq n}\|T(k,n)T(n, m)\phi \|=\sup_{k\geq n}\|T(k,m)\phi \|\le \sup_{k\geq m}\|T(k,m)\phi \| \lt \infty.
\end{equation*} This shows that 
$T(n,m)\phi \in \mathcal S(n)$, and hence, 
$\phi \in [\,T(n,m)\,]^{-1}(\mathcal S(n))$.
Now suppose that 
$\phi \in [\,T(n,m)\,]^{-1}(\mathcal S(n))$. Then, 
$T(n,m)\phi \in \mathcal S(n)$, which implies that
\begin{equation*}
\sup_{k\geq n}\|T(k,m)\phi\|=\sup_{k\geq n}\|T(k,n)T(n,m)\phi\| \lt \infty.
\end{equation*}Hence,
\begin{equation*}
\sup_{k\geq m}\|T(k,m)\phi\|\leq \max_{m\leq k\leq n-1}\|T(k,m)\phi\|+\sup_{k\geq n}\|T(k,m)\phi\| \lt \infty.
\end{equation*} Thus, 
$\phi \in \mathcal S(m)$.
Claim 3. For 
$n,m\in \mathbb Z^+_0$ with 
$n\ge m$, we have the algebraic sum decomposition
\begin{equation}
\mathcal B_r=T(n,m)\mathcal B_r+\mathcal S(n).
\end{equation}Proof of claim 3
It is sufficient to prove the claim for m = 0. Indeed, assuming that the desired conclusion holds for m= 0, we now fix an arbitrary 
$m\geq 1$. Then, for every 
$n\geq m$ and 
$\phi \in \mathcal B_r$, there exist 
$\phi_1\in \mathcal B_r$ and 
$\phi_2\in \mathcal S(n)$ such that 
$\phi=T(n,0)\phi_1+\phi_2$. Hence,
\begin{equation*}
\phi=T(n,0)\phi_1+\phi_2=T(n,m)T(m,0)\phi_1+\phi_2\in T(n,m)\mathcal B_r+\mathcal S(n).
\end{equation*} Thus, (2.10) holds. Therefore, from now on, we suppose that m = 0. Evidently, for every 
$\phi\in \mathcal B_r$,
\begin{equation*}
\phi=T(0,0)\phi+0\in T(0,0)\mathcal B_r+\mathcal S(0).
\end{equation*} Thus, (2.10) holds for 
$n=m=0$. Now suppose that n > 0 and let 
$\phi \in \mathcal B_r$ be arbitrary. Define 
$v\colon \mathbb Z^+_{n-r}\to X$ by
\begin{equation*}
v(k)=\begin{cases}
\phi(k-n) &\quad\text{for}~n-r\leq k\leq n,\\
0 & \quad \text{for}~k\geq n+1
\end{cases}
\end{equation*} so that 
$v_n=\phi$ and 
$z\colon\mathbb Z^+_0\to X$ by
\begin{equation*}
z(k)=\begin{cases}
0&\quad\text{for}~0\leq k\leq n-1,\\
v(k+1)-L_k(v_k) &\quad\text{for}~k\geq n.
\end{cases}
\end{equation*} Clearly, v is bounded on 
$\mathbb Z^+_{n-r}$. From this and (2.2), we find that 
$\sup_{k\geq 0} |z(k)| \lt \infty$. By Claim 1, there exists a function 
$x\colon \mathbb Z^+_{-r} \to X$ such that 
$\sup_{k\geq-r}|x(k)| \lt \infty$ and (2.9) holds. Moreover, it follows from the definition of z that
\begin{equation*}
v(k+1)=L_k(v_k)+z(k), \qquad k\geq n.
\end{equation*} From this and (2.9), we conclude that x − v is a solution of Eq. (2.1) on 
$\mathbb Z^+_n$ and thus
\begin{equation}
x_k-v_k=T(k,n)(x_n-v_n)=T(k,n)(x_n-\phi), \qquad k\geq n.
\end{equation} Since both x and v are bounded on 
$\mathbb Z^+_{n-r}$, this implies that 
$x_n-\phi \in \mathcal S(n)$. On the other hand, since 
$z(k)=0$ for 
$0\leq k \lt n$, we have that 
$x_n=T(n, 0)x_0$. This implies that
\begin{equation*}
\phi=x_n+(\phi-x_n)=T(n,0)x_0+(\phi-x_n)\in T(n,0)\mathcal B_r+ \mathcal S(n).
\end{equation*} Since 
$\phi \in \mathcal B_r$ was arbitrary, we conclude that (2.10) holds for m = 0.
Claim 4. For each 
$m\in \mathbb Z^+_0$, 
$\mathcal S(m)$ is the image of a Banach space under the action of a bounded linear operator.
Proof of claim 4
Fix 
$m\in \mathbb Z^+_0$ and let 
$\mathcal X$ denote the Banach space of all bounded functions 
$x\colon \mathbb Z^+_{m-r} \to X$ equipped with the supremum norm,
\begin{equation*}
\|x\|_{\mathcal X}:=\sup_{k\geq m-r}|x(k)| \lt \infty,\qquad x\in\mathcal X.
\end{equation*} Let 
$\mathcal X'$ denote the set of all 
$x\in \mathcal X$, which are solutions of (2.1) on 
$\mathbb Z^+_0$. We will show that 
$\mathcal X'$ is a closed subspace of 
$\mathcal X$. To this end, let 
$(x^j)_{j\in \mathbb Z^+_0}$ be a sequence in 
$\mathcal X'$ such that 
$x^j \to y$ in 
$\mathcal X$ as 
$j\to\infty$ for some 
$y\in\mathcal X$. Then, 
$x^j(k)\to y(k)$ as 
$j\to\infty$ for every 
$k\geq m-r$. Moreover, for each 
$n\geq m$, we have that 
$x^j_n\to y_n$ in 
$\mathcal B_r$ as 
$j\to\infty$. From this, by letting 
$j\to\infty$ in the equation
\begin{equation*}
x^j(n+1)=L_n(x^j_n),\qquad n\geq m,
\end{equation*}and using the continuity of the coefficients Ln, we conclude that
\begin{equation*}
y(n+1)=L_n(y_n),\qquad n\geq m.
\end{equation*} Thus, y is a solution of Eq. (2.1) on 
$\mathbb Z^+_m$ and hence 
$y\in\mathcal X'$. This shows that 
$\mathcal X'$ is a closed subspace of 
$\mathcal X$, and hence, it is a Banach space. Now define 
$\Phi \colon \mathcal X' \to \mathcal B_r$ by 
$\Phi(x)=x_m$ for 
$x\in\mathcal X'$. Clearly, 
$\Phi$ is a bounded linear operator with 
$\| \Phi \| \le 1$ and 
$\Phi(\mathcal X')=\mathcal S(m)$.
Claim 5. For 
$n, m\in \mathbb Z^+_0$ with 
$n\geq m+r+1$, the solution operator 
$T(n,m)\colon \mathcal B_r\to \mathcal B_r$ is compact.
Proof of claim 5
Suppose that 
$m\in\mathbb Z^+_0$ and 
$\phi\in \mathcal B_r$. Let x denote the unique solution of (2.1) with initial value 
$x_m=\phi$. From (2.1) and (2.2), we obtain for 
$n\geq m$,
\begin{equation*}
|x(n+1)|\leq\|L_n\|\|x_n\|\leq M\|x_n\|,
\end{equation*}and hence,
\begin{equation*}
\|x_{n+1}\|\leq|x(n+1)|+\|x_n\|\leq(1+M)\|x_n\|.
\end{equation*} Since 
$\|x_m\|=\|\phi\|$, this implies by induction on n that
\begin{equation}
\|x_n\|\leq(1+M)^{n-m}\|\phi\|\qquad\text{for}~n\geq m.
\end{equation} Clearly, 
$\iota(\phi)=(\phi(-r),\phi(-r+1),\dots,\phi(0))$ is an isometric isomorphism between the phase space 
$\mathcal B_r$ and the 
$(r+1)$-fold product space 
$X^{r+1}$ endowed with the maximum norm, 
$\|x\|=\max_{1\leq j\leq r+1}|x_j|$ for 
$x=(x_1,x_2,\dots,x_{r+1})\in X^{r+1}$. With this identification, we have that
\begin{equation*}
T(m+r+1,m)\phi=x_{m+r+1}=(x(m+1),x(m+2),\dots,x(m+r+1)),
\end{equation*}which, together with Eq. (2.1), implies that
\begin{equation}
T(m+r+1,m)\phi=\bigl(L_m(x_m),L_{m+1}(x_{m+1}),\dots,L_{m+r}(x_{m+r})\bigr).
\end{equation} Let S be an arbitrary bounded subset of 
$\mathcal B_r$. Then, there exists ρ > 0 such that 
$\|\phi\|\leq\rho$ for all 
$\phi\in S$. From this and (2.11), we obtain that
\begin{equation*}\Arrowvert x_{m+j}\Arrowvert\leq(1+M)^r\rho\qquad\text{whenever}\;\phi\in S\;\text{and}\;\;0\leq j\leq r.\end{equation*} Therefore, if 
$\phi\in S$, then the segments 
$x_m, x_{m+1},\dots,x_{m+r}$ of the corresponding solution x of (2.1) with initial value 
$x_m=\phi$ belong to the closed ball of radius 
$(1+M)^{r}\rho$ around zero in 
$\mathcal B_r$, which will be denoted by D. From this and (2.12), we conclude that
\begin{equation}
T(m+r+1,m)(S)\subset C:=\overline{L_m(D)}\times\overline{L_{m+1}(D)}\times\dots\times\overline{L_{m+r}(D)}.
\end{equation} Since 
$L_m,L_{m+1},\dots,L_{m+r}$ are compact operators and D is a bounded set, the closures of the image sets 
$L_m(D), L_{m+1}(D),\dots,L_{m+r}(D)$ are compact subsets of X. Therefore, C is a product of r + 1 compact subsets of X, which is a compact subset of the product space 
$X^{r+1}$. In view of (2.13), 
$T(m+r+1,m)(S)$ is a subset of the compact set 
$C\subset X^{r+1}$, and hence, it is relatively compact in 
$X^{r+1}$. Since S was an arbitrary bounded subset of 
$\mathcal B_r$, this proves that 
$T(m+r+1,m)\colon \mathcal B_r\to \mathcal B_r$ is a compact operator. Finally, if 
$n \gt m+r+1$, then
\begin{equation*}
T(n,m)=T(n,m+r+1)T(m+r+1,m)
\end{equation*} is a product of the bounded linear operator 
$T(n,m+r+1)$ and the compact operator 
$T(m+r+1,m)$, and hence, it is compact (see, e.g., [Reference Taylor28]).
Claim 6. The stable subspace 
$\mathcal S(0)$ of Eq. (2.1) is closed and has finite codimension in 
$\mathcal B_r$.
Before giving a proof of Claim 6, let us recall the following notions [Reference Schäffer27]. Let B be a Banach space. A subspace S of B is called subcomplete in B if there exist a Banach space Z and a bounded linear operator 
$\Phi\colon Z\to B$ such that 
$\Phi(Z)=S$.
Let 
$\mathcal A\colon\mathbb Z^+_0\to\mathcal L(B)$ be an operator-valued map. For 
$n,m\in \mathbb Z^+_0$ with 
$n\geq m$, define the corresponding transition operator 
$U(n,m)\colon B\to B$ by
\begin{equation*}
U(n,m)=\mathcal A(n-1)\mathcal A(n-2)\cdots \mathcal A(m)\qquad\text{for }\textit{n}\,\ge\,\textit{m}
\end{equation*} and 
$U(n,n)=I$ for 
$n\in\mathbb Z^+_0$, where I denotes the identity operator on B. A sequence 
$Y=(Y(n))_{n\in\mathbb Z^+_0}$ of subspaces in B is called a covariant sequence for 
$\mathcal A$ if
\begin{equation*}
[\,\mathcal A(n)\,]^{-1}(Y(n+1))=Y(n) \qquad\text{for all $n\in \mathbb Z^+_0$}.
\end{equation*} A covariant sequence 
$Y=(Y(n))_{n\in\mathbb Z^+_0}$ for 
$\mathcal A$ is called algebraically regular if
\begin{equation*}
U(n,0)B+Y(n)=B\qquad\text{for each $n\in \mathbb Z^+_0$}.
\end{equation*} Finally, a covariant sequence 
$Y=(Y(n))_{n\in\mathbb Z^+_0}$ for 
$\mathcal A$ is called subcomplete if the subspace Y(n) is subcomplete in B for all 
$n\in\mathbb Z^+_0$.
The proof of Claim 6 will be based on the following result due to Schäffer [Reference Schäffer27].
Lemma 2.13. ([Reference Schäffer27, Lemma 3.4]) Let B be a Banach space and 
$\mathcal A\colon\mathbb Z^+_0\to\mathcal L(B)$. Suppose that 
$Y=(Y(n))_{n\in\mathbb Z^+}$ is a subcomplete algebraically regular covariant sequence for 
$\mathcal A$. If the transition operator 
$U(n,m)\colon B\to B$ is compact for some 
$n,m\in\mathbb Z^+_0$ with n > m, then the subspaces Y(n),
$n\in\mathbb Z^+_0$, are closed and have constant finite codimension in B.
Now we can give a proof of Claim 6.
Proof of claim 6
Claims 2, 3, and 4 guarantee that the stable subspaces 
$Y(n):=\mathcal S(n)\subset \mathcal B_r$ of Eq. (2.1) form a subcomplete algebraically regular covariant sequence for 
$\mathcal A\colon\mathbb Z^+\to\mathcal L(\mathcal B_r)$ defined by
\begin{equation*}
\mathcal A(n):=T(n+r+1, n), \qquad n\in \mathbb Z^+_0.
\end{equation*} According to Claim 5, the associated transition operator 
$U(n+1,n)=\mathcal A(n)$ is compact. By the application of Lemma 2.13, we conclude that 
$Y(0)=\mathcal S(0)$ is closed and has finite codimension in 
$\mathcal B_r$.
By Claim 6, the stable subspace 
$\mathcal S(0)$ is closed and has finite codimension in 
$\mathcal B_r$. This implies that 
$\mathcal S(0)$ is complemented in 
$\mathcal B_r$ (see, e.g., [Reference Rudin23, Lemma 4.21, p. 106]). More precisely, there exists a subspace 
$\mathcal U$ of 
$\mathcal B_r$ such that 
$\dim\mathcal U=\operatorname{codim}\mathcal S(0) \lt \infty$ and
\begin{equation}
\mathcal B_r=\mathcal S(0)\oplus \mathcal U.
\end{equation}Claim 7.
For each bounded function 
$z\colon \mathbb Z^+_0\to X$, there exists a unique bounded function 
$x\colon \mathbb Z^+_{-r} \to X$ with 
$x_0\in \mathcal U$ which satisfies (2.9). Moreover, there exists a constant C > 0, independent of z, such that
\begin{equation}
\sup_{n\geq-r}|x(n)| \le C\sup_{n\geq0}|z(n)|.
\end{equation}Proof of claim 7
By Claim 1, there exists a bounded function 
$\tilde x\colon\mathbb Z^+_{-r}\to X$ that satisfies
\begin{equation*}
\tilde x(n+1)=L_n(\tilde x_n)+z(n), \qquad n\in \mathbb Z^+_0.
\end{equation*} On the other hand, (2.14) implies the existence of 
$\phi_1\in \mathcal S(0)$ and 
$\phi_2\in \mathcal U$ such that
\begin{equation*}
\tilde x_0=\phi_1+\phi_2.
\end{equation*} Define 
$x\colon\mathbb Z^+_{-r} \to X$ by
\begin{equation*}
x(n)=\tilde x(n)-y(n), \qquad n\geq-r,
\end{equation*} where y is a solution of Eq. (2.1) with initial value 
$y_0=\phi_1$. Since 
$y_0=\phi_1\in\mathcal S(0)$, we have that 
$\sup_{n\geq-r} |y(n)| \lt \infty$. Then, x satisfies (2.9), 
$x_0=\tilde x_0-\phi_1=\phi_2\in \mathcal U$ and 
$\sup_{n\geq-r}|x(n)| \lt \infty$. We claim that x with the desired properties is unique. Indeed, if 
$\bar x$ is an arbitrary function with the desired properties, then 
$x_0-\bar x_0\in \mathcal U\cap \mathcal S(0)=\{0\}$. Thus, 
$x_0=\bar x_0$ and hence 
$x=\bar x$ identically on 
$\mathbb Z^+_{-r}$.
Finally, we show the existence of a constant C > 0 such that (2.15) holds. Let 
$\mathcal X_0$ and 
$\mathcal X_{-r}$ denote the Banach space of all bounded X-valued functions defined on 
$\mathbb Z^+_0$ and 
$\mathbb Z^+_{-r}$, respectively, equipped with the supremum norm. For 
$z\in\mathcal X_0$, define 
$\mathcal F(z)=x$, where x is the unique bounded solution of the non-homogeneous Eq. (2.9) with 
$x_0\in\mathcal U$. (The existence and uniqueness of x is guaranteed by the first part of the proof.) Evidently, 
$\mathcal F(z)=x\in\mathcal X_{-r}$ for 
$z\in\mathcal X_0$ and 
$\mathcal F \colon \mathcal X_0\to \mathcal X_{-r}$ is a linear operator. We will now observe that 
$\mathcal F$ is a closed operator. Indeed, let 
$(z^k)_{k\in \mathbb Z^+}$ be a sequence in 
$\mathcal X_0$ such that 
$z^k\to z$ for some 
$z\in\mathcal X_0$ and 
$x^k:=\mathcal F (z^k) \to x$ for some 
$x\in\mathcal X_{-r}$. Then, letting 
$k\to +\infty$ in
\begin{equation*}x^k(n+1)=L_n(x_n^k)+z^k(n) \end{equation*}for each fixed n, we get that
\begin{equation*}x(n+1)=L_n(x_n)+z(n), \qquad n\in \mathbb Z^+_0.\end{equation*} That is, x satisfies (2.9). Now, since 
$x^k_0\in\mathcal U$ for 
$k\in\mathbb Z^+_0$ and 
$\mathcal U$ is a finite-dimensional and hence a closed subset of 
$\mathcal B_r$, we have that 
$x_0=\lim_{k\to\infty}x^k_0\in\mathcal U$. Therefore, 
$x\in\mathcal X_{-r}$ is a bounded function satisfying (2.9) with 
$x_0\in\mathcal U$. Hence 
$\mathcal F(z)=x$, which shows that 
$\mathcal F \colon \mathcal X_0\to \mathcal X_{-r}$ is a closed operator. According to the Closed Graph Theorem (see, e.g., [Reference Taylor28, Theorem 4.2-I, p. 181]), 
$\mathcal F$ is bounded, which implies that (2.15) holds with 
$C=\|\mathcal F\|$, the operator norm of 
$\mathcal F$.
For 
$n\in \mathbb Z^+$, define
\begin{equation*} \mathcal U(n)=T(n,0)\mathcal U
\end{equation*} so that 
$\mathcal U(0)=\mathcal U$. It is easily seen that
\begin{equation}
T(n,m)\mathcal S(m)\subset \mathcal S(n) \quad \text{and} \quad T(n,m)\mathcal U(m)=\mathcal U(n)
\end{equation} whenever 
$n,m\in\mathbb Z^+_0$ with 
$n\geq m$.
Claim 8. For 
$n,m\in \mathbb Z^+_0$ with 
$n\geq m$, 
$T(n,m)\rvert_{\mathcal U(m)} \colon \mathcal U(m)\to \mathcal U(n)$ is invertible.
Proof of claim 8
In view of (2.16), we only need to show that the operator above is injective. Let 
$n,m\in \mathbb Z^+_0$ with 
$n\geq m$ and 
$\phi \in \mathcal U(m)$ be such that 
$T(n,m)\phi=0$. Since 
$\phi \in \mathcal U(m)$, there exists 
$\bar{\phi}\in \mathcal U(0)=\mathcal U$ such that 
$\phi=T(m,0)\bar{\phi}$. Let 
$x\colon \mathbb Z^+_{-r}\to X$ be the solution of (2.1) with initial value 
$x_0=\bar{\phi}$. Since 
$x(k)=0$ for all sufficiently large 
$k\in \mathbb Z^+$, we have that 
$\sup_{k\geq-r}|x(k)| \lt \infty$. It follows from the uniqueness in Claim 7, applied for 
$z\equiv 0$, that 
$x\equiv 0$. This implies that 
$\bar{\phi}=\phi= 0$.
Claim 9. For each 
$n\in \mathbb Z^+_0$, 
$\mathcal B_r$ can be decomposed into the direct sum
\begin{equation}
\mathcal B_r=\mathcal S(n)\oplus \mathcal U(n).
\end{equation}Proof of claim 9
Since 
$\mathcal U(0)=\mathcal U$, for n = 0, the decomposition (2.17) follows immediately from (2.14). Now suppose that 
$n\geq 1$ and let 
$\phi \in \mathcal B_r$ be arbitrary. Let 
$v\colon \mathbb Z^+_{n-r}\to X$ and 
$z\colon \mathbb Z^+_0\to X$ be as in the proof of Claim 3. Since 
$\sup_{k\geq0} |z(k)| \lt \infty$, by Claim 7, there exists a unique function 
$x\colon\mathbb Z^+_{-r} \to X$ such that 
$x_0\in \mathcal U$, 
$\sup_{k\geq{-r}}|x(k)| \lt \infty$, and (2.9) holds. By the same reasoning as in the proof of Claim 3, we have that 
$x_n-\phi\in \mathcal S(n)$. Moreover, 
$x_n=T(n,0)x_0\in \mathcal U(n)$. Consequently,
\begin{equation*}
\phi=(\phi-x_n)+x_n\in \mathcal S(n)+\mathcal U(n).
\end{equation*} Suppose now that 
$\phi \in \mathcal S(n)\cap \mathcal U(n)$. Then, there exists 
$\bar{\phi}\in \mathcal U$ such that 
$\phi=T(n,0)\bar{\phi}$. Consider the unique solution 
$x\colon \mathbb Z^+_{-r}\to X$ of Eq. (2.1) with 
$x_0=\bar{\phi}$. Then, x satisfies (2.9) with 
$z\equiv 0$, 
$x_0=\bar\phi\in \mathcal U$ and 
$\sup_{k\geq{-r}}|x(k)| \lt \infty$. By the uniqueness in Claim 7, we conclude that 
$x\equiv 0$. Therefore, 
$\bar\phi=0$, which implies that ϕ = 0. The proof of the Claim is completed.
Claim 10. There exists Q > 0 such that
\begin{equation*}
\|T(n,m)\phi \| \le Q\|\phi \|,
\end{equation*} for every 
$n,m\in \mathbb Z^+_0$ with 
$n\ge m$ and 
$\phi \in \mathcal S(m)$.
Proof of claim 10
Fix 
$m\in \mathbb Z^+_0$ and 
$\phi \in \mathcal S(m)$. Let 
$u\colon \mathbb Z^+_{m-r} \to X$ be the solution of Eq. (2.1) with initial value 
$u_m=\phi$. Define 
$x\colon \mathbb Z^+_{-r} \to X$ and 
$z\colon\mathbb Z^+_0\to X$ by
\begin{equation*}
x(k)=\begin{cases}
u(k)&\text{for $k\geq{m+1}$;} \\
0 &\text{for $-r\leq k\leq m$}
\end{cases}
\end{equation*}and
\begin{equation*}
z(k)=x(k+1)-L_k(x_k),\qquad k\geq0,
\end{equation*} respectively. Evidently, (2.9) is satisfied, and since 
$\phi \in \mathcal S(m)$, we have that 
$\sup_{k\geq{-r}}|x(k)| \lt \infty$. Moreover, 
$x_0=0\in \mathcal U$. Furthermore, 
$z(k)=0$ for 
$0\leq k\leq m-1$ and 
$k\geq m+r+1$. Thus, using (2.2), (2.5), and the fact that 
$\|x_k\|\leq \|u_k\|$, we find that
\begin{equation*}
\begin{split}
\sup_{k\geq0}|z(k)|&=\sup_{m\leq k \leq m+r}|z(k)|\\
&\le \sup_{m\leq k \leq m+r }|x(k+1)|+\sup_{m\leq k \leq m+r }|L_kx_k|\\
&\leq \sup_{m\leq k \leq m+r}|u(k+1)|+M\sup_{m\leq k \leq m+r }\|x_k\|\\
&\leq \sup_{m\leq k \leq m+r}|L_k(u_{k})|+M\sup_{m\leq k \leq m+r}\|u_k\|\\
&\leq 2M\sup_{m\leq k \leq m+r }\|u_k\|\\
&= 2 M\sup_{m\leq k \leq m+r}\|T(k,m)\phi\|\\
&\leq 2Me^{\omega r}\| \phi\|.
\end{split}
\end{equation*}From the last inequality and conclusion (2.15) of Claim 7, we conclude that
\begin{equation*}
\sup_{k\geq{m+1}}|u(k)| \le \sup_{k\geq{-r}}|x(k)| \le C\sup_{k\geq 0}|z(k)| \le 2CMe^{\omega r}\| \phi\|.
\end{equation*}Hence,
\begin{equation*}
\|T(n,m)\phi \|=\|u_n\| \le 2CMe^{\omega r}\| \phi\|, \qquad n\ge m+r+1.
\end{equation*}Since (2.2) implies that
\begin{equation*}
\|T(n,m)\phi \| \le e^{\omega r}\|\phi\|, \qquad m\leq n\leq m+r,
\end{equation*}the conclusion of the claim holds with
\begin{equation*}
Q:=\max \{\,e^{\omega r}, 2CMe^{\omega r}\,\} \gt 0.
\end{equation*}Claim 11. There exist 
$D, \lambda \gt 0$ such that
\begin{equation*}
\|T(n,m)\phi \| \le De^{-\lambda (n-m)}\|\phi\|
\end{equation*} for every 
$n,m\in \mathbb Z^+_0$ with 
$n\geq m$ and 
$\phi \in \mathcal S(m)$.
Proof of claim 11
We claim that if
\begin{equation}
N \gt eCMQ^2(r+1)+r+1
\end{equation} with 
$Q$ as in Claim 10, then for every 
$m\in \mathbb Z^+_0$ and 
$\phi\in\mathcal S(m)$,
\begin{equation}
\|T(n,m)\phi \| \le \frac 1 e \|\phi \|\qquad\text{for}~n\geq m+N.
\end{equation} Suppose, for the sake of contradiction, that (2.18) holds and there exist 
$m\in \mathbb Z^+_0$ and 
$\phi\in\mathcal S(m)$ such that
\begin{equation}
\|T(n,m)\phi \| \gt \frac 1 e \|\phi \|\qquad\text{for~some}~n\geq m+N.
\end{equation} Fix 
$n\geq m+N$ such that the first inequality in (2.20) holds and let u denote the solution of the homogeneous equation Eq. (2.1) with initial value 
$u_m=\phi$ so that 
$u_{n}=T(n,m)\phi$. Therefore, the first inequality in (2.20) can be written as
\begin{equation}
\|u_n\| \gt \frac{1}{e}\|\phi\|.
\end{equation} In view of (2.16), 
$\phi\in\mathcal S(m)$ implies that 
$u_j=T(j,m)\phi\in\mathcal S(j)$ for 
$j\geq m$. Therefore, by Claim 10, we have that
\begin{equation*}
\|u_{n}\|=\|T(n,j)u_j\|\leq Q\|u_j\|\qquad \text{whenever}~m\leq j\leq n.
\end{equation*} From the last inequality and (2.21), we find that 
$\|u_j\| \gt 0$ whenever 
$ m\leq j\leq n$. This, together with the fact that 
$n\ge m+N \gt m+r+1$ and hence, 
$n-r-1 \gt m$, implies that we can define a function 
$x\colon\mathbb Z^+_{-r}\to X$ by
\begin{equation*}
x(k)=\begin{cases}
\chi(k)u(k)& \text{for $k\geq m$,}\\
0 & \text{for $-r\leq k\leq m-1$,}
\end{cases}
\end{equation*}where
\begin{equation*}
\chi(k)=\begin{cases}
0\qquad&\text{for $-r\leq k\leq m$},\\[2pt]
\displaystyle\sum_{j=m}^{k-1} \|u_j\|^{-1}
\qquad&\text{for $m+1\leq k\leq n-r-1$}, \\[10pt]
\displaystyle \sum_{j=m}^{n-r-1} \|u_j\|^{-1}
\qquad&\text{for $k \geq n-r$}.
\end{cases}
\end{equation*}Evidently, x satisfies the non-homogeneous equation
\begin{equation}
x(k+1)=L_k(x_k)+z(k),\qquad k\in \mathbb Z^+_0,
\end{equation} where 
$z\colon \mathbb Z^+_0\rightarrow X$ is defined by
\begin{equation*}
z(k)=x(k+1)-L_k(x_k), \qquad k\in \mathbb Z^+_0.
\end{equation*} Since 
$u_m=\phi\in\mathcal S(m)$ implies that u is bounded on 
$\mathbb Z^+_{m-r}$ and 
$0\leq\chi(k)\leq \chi(n-r)$ for 
$k\geq{-r}$, it follows that x is bounded on 
$\mathbb Z^+_{-r}$. From this and (2.2), we obtain that z is also bounded on 
$\mathbb Z^+_0$. Since 
$x_0=0\in \mathcal{U}$, by Claim 7, we have that
\begin{equation}
\sup_{k\geq{-r}}|x(k)|\leq C\sup_{k\geq0}|z(k)|.
\end{equation} Our objective now is to estimate the norm of z(k). Since 
$x(k)=0$ for 
$-r\leq k\leq m$, we have that 
$z(k)=0$ for 
$0\leq k\leq m-1$. The function χ is constant on 
$\mathbb Z^+_{n-r}$, therefore x is a constant multiple of the solution u of the homogeneous Eq. (2.1) on 
$\mathbb Z^+_{n-r}$. Thus, x also satisfies the homogeneous Eq. (2.1) for 
$k\geq n$, and hence, 
$z(k)=0$ for 
$k\geq n$. It remains to consider the case when 
$m\leq k\leq n-1$. Let such a k be fixed. By definition, we have
\begin{equation*}
\begin{split}
z(k)&=x(k+1)-L_k(x_k)\\
&=\chi(k+1)u(k+1)-L_k(\chi_k u_k)\\
&=\chi(k+1)L_k(u_k)-L_k(\chi_k u_k)\\
&=L_k(\chi(k+1)u_k-\chi_ku_k).
\end{split}
\end{equation*}Therefore, using (2.2), it follows that
\begin{equation*}
|z(k)|\leq M\|\chi(k+1)u_k-\chi_ku_k\|.
\end{equation*} Let 
$\theta\in[-r,0]\cap\mathbb Z$. Then,
\begin{equation*}
\begin{split}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|&=|\left(\chi(k+1)-\chi (k+\theta)\right) u(k+\theta)|,
\end{split}
\end{equation*}and hence,
\begin{equation*}
\begin{split}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|&\leq \sum_{j=m}^{k} \|u_j\|^{-1}|u(k+\theta)| \qquad \text{whenever } k+\theta\leq m
\end{split}
\end{equation*}and
\begin{equation*}
\begin{split}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|&\leq \sum_{j=k+\theta}^{k} \|u_j\|^{-1}|u(k+\theta)| \qquad \text{whenever } k+\theta \gt m.
\end{split}
\end{equation*} In view of (2.16), 
$u_m=\phi\in\mathcal S(m)$ implies that 
$u_j=T(j,m)\phi\in\mathcal S(j)$ for 
$j\geq m$. Therefore, by Claim 10, for 
$m\leq j\leq k$, we have
\begin{equation*}
|u(k+\theta)|\leq\|u_{k}\|=\|T(k,j)u_j\|\leq Q\|u_j\|
\end{equation*} so that 
$\|u_j\|^{-1}|u(k+\theta)|\leq Q$. From this, we conclude that if 
$k+\theta\leq m$ so that 
$k-m\leq-\theta\leq r$, then
\begin{equation*}
\sum_{j=m}^{k} \|u_j\|^{-1}|u(k+\theta)|
\leq Q(k-m+1)\leq Q(r+1),
\end{equation*} while in case 
$k+\theta \gt m$, we have
\begin{equation*}
\sum_{j=k+\theta}^{k} \|u_j\|^{-1}|u(k+\theta)|
\leq Q(-\theta+1)\leq Q(r+1).
\end{equation*} Therefore, in both cases 
$k+\theta\leq m$ and 
$k+\theta \gt m$, we have that
\begin{equation*}
\begin{split}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|&\leq Q(r+1).
\end{split}
\end{equation*} Since 
$\theta\in[-r,0]\cap\mathbb Z$ was arbitrary, this implies that
\begin{equation*}
\|\chi(k+1)u_k-\chi_k u_k\|\leq Q(r+1),
\end{equation*}which, combined with (2.2), yields
\begin{equation*}
|z(k)|\leq MQ(r+1).
\end{equation*} We have shown that the last inequality is valid whenever 
$m\leq k\leq n-1$ and 
$z(k)=0$ otherwise. Hence,
\begin{equation*}
\sup_{k\geq0}|z(k)|\leq MQ(r+1).
\end{equation*}This, together with (2.23), implies that
\begin{equation}
\sup_{k\geq-r}|x(k)|\leq CMQ(r+1).
\end{equation} Since 
$n-r\geq n-N\geq m$ and χ is non-decreasing on 
$\mathbb Z^+_{-r}$, we have for 
$\theta\in[-r,0]\cap\mathbb Z$,
\begin{equation*}
|x(n+\theta)|=\chi(n+\theta)|u(n+\theta)|\geq\chi(n-r)|u(n+\theta)|
=|u(n+\theta)|\sum_{j=m}^{n-r-1} \|u_j\|^{-1}.
\end{equation*}Hence,
\begin{equation*}
\|x_n\|\geq\|u_n\|\sum_{j=m}^{n-r-1} \|u_j\|^{-1}.
\end{equation*} According to Claim 10, 
$u_m=\phi\in\mathcal S(m)$ implies that 
$\|u_j\|=\|T(j,m)\phi\|\leq Q\|\phi\|$ for 
$j\geq m$. This, together with the previous inequality, yields
\begin{equation*}
\|x_n\|\geq\|u_n\|\sum_{j=m}^{n-r-1} \|u_j\|^{-1}
\geq\|u_n\|\frac{n-m-r}{Q\|\phi\|}.
\end{equation*}The last inequality, combined with (2.21) and (2.24), implies that
\begin{equation*}
\begin{split}
CMQ(r+1)\geq \|x_n\|
\geq \|u_n\|\frac{n-m-r}{Q\|\varphi\|} \gt \frac{n-m-r}{eQ}.
\end{split}
\end{equation*} However, this contradicts the fact that 
$n\geq m+N$ with N satisfying (2.18). Thus, (2.19) holds whenever 
$m\in \mathbb Z^+_0$ and 
$\phi\in\mathcal S(m)$.
Using (2.19), we can easily complete the proof. Choose an integer N satisfying (2.18). Let 
$n\geq m$. Then, 
$n-m=kN+h$ for some 
$k\in \mathbb Z^+_0$ and 
$0\leq h\leq N-1$. From (2.19) and Claim 10, we obtain for 
$\phi \in \mathcal S(m)$,
\begin{equation*}
\begin{split}
\|T(n,m)\phi\| &=\|T(m+kN+h, m)\phi \|\\
&=\|T(m+kN+h,m+kN)T(m+kN,m)\phi\|\\
&\le Q\|T(m+kN, m)\phi \|\\
&\le Qe^{-k}\|\phi \|\\
&\le eQe^{-\frac{n-m}{N}}\|\phi \|.
\end{split}
\end{equation*} Hence, the conclusion of the claim holds with D = eQ and 
$\lambda=1/N$.
Claim 12. There exists 
$Q' \gt 0$ such that
\begin{equation*}
\|T(n,m)\phi \| \le Q'\|\phi \|
\end{equation*} for every 
$n,m\in \mathbb Z^+_0$ with 
$n\leq m$ and 
$\phi \in \mathcal U(m)$.
Proof of claim 12
Given 
$m\in\mathbb Z^+_0$ and 
$\phi \in \mathcal U(m)$, there exists 
$\bar \phi \in \mathcal{U}$ such that 
$\phi=T(m,0)\bar \phi$. Let 
$u\colon\mathbb Z^+_{-r}\to X$ be the solution of Eq. (2.1) such that 
$u_0=\bar \phi$. Consider 
$x\colon \mathbb Z^+_{-r}\to X$ and 
$z\colon \mathbb Z^+_0\to X$ given by
\begin{equation*}
x(k)=\begin{cases}
u(k)& \text{for $-r\leq k\leq m$}\\[10pt]
0 & \text{for $k\geq m+1$,}
\end{cases}
\end{equation*}and
\begin{equation*}
z(k)=x(k+1)-L_k(x_k), \qquad k\in \mathbb Z^+_0,
\end{equation*} so that (2.9) is satisfied. Moreover, since 
$x(k)=0$ for 
$k\geq{m+1}$, it follows that 
$\sup_{k\geq {-r}}|x(k)| \lt \infty$. Furthermore, 
$x_0=u_0\in\mathcal U$ and 
$z(k)=0$ for 
$0\leq k\leq m-1$ and 
$k\geq m+r+1$. Proceeding as in the proof of Claim 10, it can be shown that
\begin{equation*}
\sup_{k\geq 0}|z(k)|\leq 2Me^{\omega r}\| \phi\|.
\end{equation*}From conclusion (2.15) of Claim 7, we conclude that
\begin{equation*}
\sup_{-r\leq k\leq m}|u(k)| \le \sup_{k\geq{-r}} |x(k)| \le C \sup_{k\geq 0}|z(k)|\leq 2CMe^{\omega r}\| \phi\|.
\end{equation*}This implies that the conclusion of the claim holds with
\begin{equation*}
Q':= 2CMe^{\omega r} \gt 0.
\end{equation*}Claim 13. There exist 
$D', \lambda' \gt 0$ such that
\begin{equation*}
\|T(n,m)\phi \| \le D'e^{-\lambda' (m-n)}\|\phi\|
\end{equation*} for every 
$n,m\in \mathbb Z^+_0$ with 
$n\leq m$ and 
$\phi \in \mathcal U(m)$.
Proof of Claim 13
We claim that if
\begin{equation}
N' \gt eCM(Q')^2(r+1)
\end{equation} with Qʹ as in Claim 12, then for every 
$m\geq N'$ and 
$\phi\in\mathcal U(m)$,
\begin{equation}
\|T(n,m)\phi \| \le \frac 1 e \|\phi \|\qquad\text{whenever}~0\leq n\leq m- N^{\prime}.
\end{equation} Suppose, for the sake of contradiction, that (2.25) holds and there exist 
$m\geq N'$ and 
$\phi\in\mathcal U(m)$ such that
\begin{equation}
\|T(n,m)\phi \| \gt \frac{1}{e }\|\phi \|\qquad\text{for some}~n~\text{with}~0\leq n\leq m-N^{\prime}.
\end{equation} Fix n with 
$0\leq n\leq m-N'$ such that the first inequality in (2.27) holds. Evidently, (2.27) implies that 
$\phi\in\mathcal U(m)$ is non-zero. Therefore, there exists a non-zero 
$\bar \phi\in \mathcal{U}$ such that 
$\phi =T(m,0)\bar \phi$. Let u denote the unique solution of Eq. (2.1) with 
$u_0=\bar \phi$ so that 
$u_m=T(m,0)u_0=T(m,0)\bar\phi=\phi$. Since 
$u_n=T(n,m)u_m=T(n,m)\phi$, the first inequality in (2.27) can be written as
\begin{equation}
\|u_{n}\| \gt \frac 1 e \|\phi\|.
\end{equation} Choose a sequence 
$\psi\colon\mathbb Z^+_0\to[0,1]$ such that
\begin{equation}
\psi(j)=1\quad\text{for}~0\leq j\leq m\qquad\text{and}\qquad\psi(j)=0\quad\text{for}~j\geq m+1.
\end{equation} By Claim 8, 
$0\neq\bar\phi\in\mathcal U$ implies that 
$u_j=T(j,0)\bar\phi\neq0$ for 
$j\geq0$. Therefore, we can define a function 
$x\colon \mathbb Z^+_{-r}\to X$ by
\begin{equation*}
x(k)=\chi(k)u(k) \qquad \text{for}~k\geq{-r},
\end{equation*}where
\begin{equation*}
\chi(k)=\begin{cases}
\displaystyle \sum_{j=0}^{\infty} \psi(j)\|u_j\|^{-1}\qquad&\text{for $-r\leq k\leq0$}, \\[15pt]
\displaystyle\sum_{j=k}^{\infty} \psi(j)\|u_j\|^{-1}
\qquad&\text{for $k\geq1$}.
\end{cases}
\end{equation*}Note that
\begin{equation*}x_0=cu_0=c\bar\phi\in\mathcal U,\qquad\text{where $c=\chi(0)=\sum_{j=0}^{m}\psi(j) \|u_j\|^{-1}$.}
\end{equation*} Since 
$\psi(k)=0$ and hence 
$x(k)=0$ for 
$k\geq m+1$, we have that 
$\sup_{k\geq {-r}}|x(k)| \lt \infty $. Moreover, x satisfies (2.9) with 
$z\colon \mathbb Z^+_0\to X$ defined by
\begin{equation}
z(k)=x(k+1)-L_k(x_k),\qquad k\geq 0.
\end{equation} Since 
$x_k=0$ for 
$k\geq m+r+1$, it follows that 
$z(k)=0$ for 
$k\geq{m+r+1}$. In particular, z is bounded on 
$\mathbb Z^+_0$. From (2.30) and Claim 7, we conclude that
\begin{equation}
\sup_{k\geq{-r}}|x(k)|\leq C\sup_{k\geq{0}}|z(k)|.
\end{equation} Let 
$k\geq0$ be arbitrary. By the same calculations as in the proof of Claim 10, we have
\begin{equation*}
\begin{split}
z(k)&=L_k(\chi(k+1)u_k-\chi_ku_k).
\end{split}
\end{equation*}From this and (2.2), we find that
\begin{equation}
|z(k)|\leq M\|\chi(k+1)u_k-\chi_ku_k\|.
\end{equation} Let 
$\theta\in[-r,0]\cap\mathbb Z$. Then,
\begin{equation*}
\begin{split}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|&=|\left(\chi(k+1)-\chi (k+\theta)\right) u(k+\theta)|.
\end{split}
\end{equation*} From this and the definition of χ, taking into account that 
$0\leq\psi\leq1$ on 
$\mathbb Z^+_0$, we conclude that
\begin{equation}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|\leq \sum_{j=0}^{k} \|u_j\|^{-1}|u(k+\theta)| \qquad \text{whenever}~k+\theta\leq 0.
\end{equation}and
\begin{equation}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|\leq \sum_{j=k+\theta}^{k} \|u_j\|^{-1}|u(k+\theta)| \qquad \text{whenever}~k+\theta \gt 0.
\end{equation} By Claim 8, 
$u_0=\bar\phi\in\mathcal U$ implies that 
$u_j=T(j,0)u_0\in\mathcal U(j)$ for 
$j\geq 0$. Therefore, according to Claim 12, if 
$k+\theta \lt 0$, then
\begin{equation*}
|u(k+\theta)|\leq\|u_{0}\|=\|T(0,j)u_j\|\leq Q'\|u_j\| ,
\end{equation*} and hence, 
$\|u_j\|^{-1}|u(k+\theta)|\leq Q'$ for 
$j\geq0$. If 
$k+\theta\leq 0$ so that 
$k\leq-\theta\leq r$, then from the last inequality and (2.33), we obtain that
\begin{equation}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|\leq Q'(r+1) \qquad \text{whenever}~k+\theta\leq0.
\end{equation} It follows by similar arguments that if 
$k+\theta \gt 0$, then
\begin{equation*}
|u(k+\theta)|\leq\|u_{k+\theta}\|=\|T(k+\theta,j)u_j\|\leq Q'\|u_j\| ,
\end{equation*} and hence, 
$\|u_j\|^{-1}|u(k+\theta)|\leq Q'$ for 
$j\geq k+\theta$. This, together with (2.34), yields
\begin{equation}
|\left(\chi(k+1)u_k-\chi_ku_k\right)(\theta)|\leq Q'(r+1) \qquad \text{whenever}~k+\theta \gt 0.
\end{equation} Since 
$\theta\in [-r,0]\cap\mathbb Z$ was arbitrary, (2.35) and (2.36) imply that
\begin{equation*}
\begin{split}
\|\chi(k+1)u_k-\chi_ku_k\|\leq Q'(r+1).
\end{split}
\end{equation*} Since 
$k\geq 0$ was arbitrary, the last inequality, combined with (2.32), implies that
\begin{equation*}
\sup_{k\geq0}|z(k)|\leq MQ'(r+1).
\end{equation*}This, together with (2.31), yields
\begin{equation}
\sup_{k\geq-r}|x(k)|\leq CMQ'(r+1).
\end{equation} Since χ is non-increasing on 
$\mathbb Z^+_{-r}$, we have for 
$\theta\in[-r,0]\cap\mathbb Z$,
\begin{equation*}
|x(n+\theta)|=\chi(n+\theta)|u(n+\theta)|\geq\chi(n)|u(n+\theta)|=|u(n+\theta)|\sum_{j=n}^m \|u_j\|^{-1},
\end{equation*}the last equality being a consequence of (2.29). Hence,
\begin{equation*}
\|x_n\|\geq\|u_n\|\sum_{j=n}^m \|u_j\|^{-1}.
\end{equation*} Since 
$u_m=\phi\in\mathcal U(m)$, by Claim 12, we have that 
$\|u_j\|=\|T(j,m)\phi\|\leq Q'\|\phi\|$ for 
$0\leq j\leq m$. This, together with the previous inequality, gives
\begin{equation*}
\|x_n\|\geq\|u_n\|\frac{m-n+1}{Q'\|\phi\|}.
\end{equation*}This, combined with (2.28) and (2.37), yields
\begin{equation*}
\begin{split}
CMQ'(r+1)\geq \|x_n\|\geq\|u_n\|\frac{m-n+1}{Q'\|\phi\|} \gt \frac{m-n+1}{eQ'}.
\end{split}
\end{equation*} The last inequality contradicts the fact that 
$n-m\geq N'$ with N ʹ satisfying (2.25). Thus, (2.26) holds whenever 
$m\geq N'$ and 
$\phi\in\mathcal U(m)$.
Now, using (2.26), we can easily complete the proof. Let 
$0\leq n\leq m$ and 
$\phi \in \mathcal U(m)$. Choose an integer N ʹ satisfying (2.25). Then, 
$m-n=kN'+h$ for some 
$k\in \mathbb Z^+_0$ and 
$0\leq h\leq N'-1$. From (2.26) and Claim 12, we obtain
\begin{equation*}
\begin{split}
\|T(n,m)\phi\| &=\|T(n, n+kN'+h)\phi \|\\
&=\|T(n, n+kN')T(n+kN',n+kN'+h)\phi \| \\
&\le e^{-k}\|T(n+kN',n+kN'+h)\phi \|\\
&\le Q'e^{-k}\|\phi \|\\
&\le eQ'e^{-\frac{m-n}{N'}}\|\phi \|.
\end{split}
\end{equation*} Thus, the conclusion of the claim holds with 
$D'=eQ'$ and 
$\lambda'=1/N'$.
For each 
$n\in \mathbb Z^+_0$, let Pn denote the projection of 
$\mathcal B_r$ onto 
$\mathcal S(n)$ along 
$\mathcal U(n)$ associated with the decomposition (2.17).
Claim 14. The projections Pn, 
$n\in \mathbb Z^+_0$, are uniformly bounded, i.e.,
\begin{equation*}
\sup_{n\geq 0}\|P_n\| \lt \infty.
\end{equation*}Proof of claim 14
Since 
$\operatorname{ker}P_n=\mathcal U(n)$ and 
$\operatorname{im}P_n=\mathcal S(n)$ for 
$n\in\mathbb Z^+_0$, Claims 11 and 13 show that the hypotheses of Lemma 3.1 of Huy and Van Minh [Reference Huy and Van Minh13] are satisfied with 
$X=\mathcal B_r$ and 
$A_n=T(n+1,n)$. Therefore, the desired conclusion follows from [Reference Huy and Van Minh13, Lemma 3.1].
Now we can complete the proof of Theorem 2.3. Let 
$\phi \in \mathcal B_r$ and 
$n,m\in \mathbb Z^+_0$ with 
$n\geq m$ be fixed. From 
$P_m\phi\in\mathcal S(m)$ and (2.16), we have that 
$T(n,m)P_m\phi\in\mathcal S(n)$. Hence,
\begin{equation*}
P_n T(n,m)P_m\phi=T(n,m)P_m\phi.
\end{equation*} Similarly, considering 
$Q_m=I-P_m$, 
$Q_m\phi\in\mathcal U(m)$ and (2.16) imply that 
$T(n,m)Q_m\phi\in\mathcal U(n)$. Hence,
\begin{equation*}
P_n T(n,m)Q_m\phi=0.
\end{equation*} From the above relations, taking into account that 
$\phi=P_m\phi+Q_m\phi$, we conclude that
\begin{equation*}
P_nT(n,m)\phi=P_nT(n,m)P_m\phi+P_nT(n,m)Q_m\phi=T(n,m)P_m\phi.
\end{equation*} Since 
$\phi\in\mathcal B_r$ was arbitrary, this proves (2.6).
Evidently, 
$\operatorname{ker}P_m=\mathcal U(m)$ for 
$m\in\mathbb Z^+$. Therefore, from Claim 8 and (2.16), it follows that the restriction 
$T(n,m)\rvert_{\operatorname{ker} P(m)} \colon \operatorname{ker} P(m)\to \operatorname{ker} P(n)$ is invertible and onto. Furthermore, by Claim 14, the projections Pn, 
$n\in \mathbb Z^+_0$, are uniformly bounded. Combining this fact with Claims 11 and 13, we conclude that the exponential estimates (2.7) and (2.8) are also satisfied. Thus, (2.1) admits an exponential dichotomy.
Remark 2.22. In the proof of Theorem 2.3, we have shown that the Perron property (see Claim 1) implies the existence of an exponential dichotomy for Eq. (1.3). Results of this type have a long history that goes back to the pioneering works of Perron [Reference Perron21] for ordinary differential equations and Li [Reference Li15] for difference equations. Subsequent important contributions are due to Massera and Schäffer [Reference Massera and Schäffer16, Reference Massera and Schäffer17], Daleckiı and Kreın [Reference Daleckiı and Kreın9], Coppel [Reference Coppel8] and Henry [Reference Henry11], who was the first to consider the case of non-invertible dynamics. For more recent contributions, we refer to [Reference Huy12–Reference Latushkin, Randolph and Schnaubelt14, Reference Sasu and Sasu24–Reference Sasu and Sasu26, Reference Van Minh, Räbiger and Schnaubelt29] and the references therein. A comprehensive overview of the relationship between hyperbolicity and the Perron property is given in [Reference Barreira, Dragičević and Valls3].
3. Shadowing of linear Volterra difference equations with infinite delay
In this section, we are interested in the shadowing of the Volterra difference equation with infinite delay (1.5), where the kernel A satisfies condition (1.6). The phase space for Eq. (1.5) is the Banach space 
$(\mathcal B_\gamma,\|\cdot\|)$ given by
\begin{equation*}
\mathcal B_\gamma=\biggl\{\,\phi\colon\mathbb Z^-_0\to\mathbb C^d:\sup_{\theta\in\mathbb Z^-_0}|\phi(\theta)|e^{\gamma\theta} \lt \infty\,\biggr\},
\qquad \|\phi\|=\sup_{\theta\in\mathbb Z^-_0}|\phi(\theta)| e^{\gamma\theta},\qquad\phi\in\mathcal B_\gamma.
\end{equation*}Under condition (1.6), Eq. (1.5) can be written equivalently in the form
\begin{equation}
x(n+1)=L(x_n),
\end{equation} where 
$x_n\in\mathcal B_\gamma$ is defined by 
$x_n(\theta)=x(n+\theta)$ for 
$\theta\in\mathbb Z^-_0$ and 
$L\colon\mathcal B_\gamma\to\mathbb C^d$ is a bounded linear functional defined by
\begin{equation*}
L(\phi)=\sum_{j=0}^\infty A(j)\phi(-j),\qquad\phi\in\mathcal B_\gamma.
\end{equation*} It is known (see, e.g., [Reference Matsunaga, Murakami, Nagabuchi and Nakano18], [Reference Nagabuchi20]) that if (1.6) holds, then for every 
$\phi\in\mathcal B_\gamma$, there exists a unique function 
$x\colon\mathbb Z\to\mathbb C^d$ satisfying Eq. (1.5) (equivalently, Eq. (3.1)) such that 
$x(\theta)=\phi(\theta)$ for 
$\theta\in\mathbb Z^-_0$. We shall call x the solution of Eq. (1.5) (or (3.1)) on 
$\mathbb Z^+_0$ with initial value 
$x_0=\phi$. By a solution of Eq. (1.5) on 
$\mathbb Z^+_0$, we mean a solution x with initial value 
$x_0=\phi$ for some 
$\phi\in\mathcal B_\gamma$.
For Eq. (1.5), the definition of shadowing can be modified as follows.
Definition 3.1. We say that Eq. (1.5) is shadowable on 
$\mathbb Z^+_0$ if, for each ϵ > 0, there exists δ > 0 such that for every function 
$y\colon \mathbb Z\to \mathbb C^d$ satisfying
\begin{equation*}
\sup_{n\geq 0}|y(n+1)-L(y_n)| \le \delta,
\end{equation*} there exists a solution x of (1.5) on 
$\mathbb Z^+_0$ such that
\begin{equation*}
\sup_{n\geq 0} \|x_n-y_n\| \le \epsilon.
\end{equation*} The main result of this section is the following theorem, which shows that, under condition (1.6), Eq. (1.5) is shadowable on 
$\mathbb Z^+_0$ if and only if it is hyperbolic.
Theorem 3.2 Suppose that (1.6) holds. Then, the following statements are equivalent.
(i) Eq. (1.5) is shadowable on
$\mathbb Z^+_0$;(ii) The characteristic equation (1.7) has no root on the unit circle
$|\lambda|=1$.
Before we give a proof of Theorem 3.2, we summarize some facts from the spectral theory of linear Volterra difference equations with infinite delay ([Reference Matsunaga, Murakami, Nagabuchi and Nakano18], [Reference Murakami19], [Reference Nagabuchi20]).
For each 
$n\in\mathbb Z^+_0$, define 
$T(n)\colon\mathcal B_\gamma\to\mathcal B_\gamma$ by 
$T(n)\phi=x_n(\phi)$ for 
$\phi\in\mathcal B_\gamma$, where 
$x(\phi)$ is the unique solution of Eq. (1.5) on 
$\mathbb Z^+_0$ with initial value 
$x_0(\phi)=\phi$. It is well known (see [Reference Matsunaga, Murakami, Nagabuchi and Nakano18, Reference Murakami19]) that T(n) is a bounded linear operator in 
$\mathcal B_\gamma$ which has the semigroup property 
$T(0)=I$, the identity on 
$\mathcal B_\gamma$, and 
$T(n+m)=T(n)T(m)$ for 
$n,m\in\mathbb Z^+_0$. As a consequence, we have that
\begin{equation*}
T(n)=T^n,\qquad n\in\mathbb Z^+_0,\qquad\text{where}\quad T:=T(1).
\end{equation*} From the definition of the solution operator 
$T=T(1)$ and Eq. (1.5), we have that
\begin{equation}
[T(\phi)](\theta)=\begin{cases}
\displaystyle\sum_{j=0}^\infty A(j)\phi(-j)&\qquad\text{for}~\theta=0,\\
\phi(\theta+1)&\qquad\text{for}~\theta\leq-1.
\end{cases}
\end{equation} If (1.6) holds, then the characteristic function 
$\varDelta$ defined by (1.8) is an analytic function of the complex variable λ in the region 
$|\lambda| \gt e^{-\gamma}$. Denote by 
$\varSigma$ the set of characteristic roots of Eq. (1.5),
\begin{equation*}
\varSigma=\{\,\lambda\in\mathbb C:|\lambda| \gt e^{-\gamma},\,\det\varDelta(\lambda)=0\,\},
\end{equation*}and define
\begin{equation*}
\varSigma^{cu}=\{\,\lambda\in\varSigma:|\lambda|\geq1\}.
\end{equation*} It follows from the analyticity of 
$\varDelta$ that 
$\varSigma^{cu}$ is a finite spectral set for T. The corresponding spectral projection 
$\varPi^{cu}$ on 
$\mathcal B_\gamma$ is defined by
\begin{equation*}
\varPi^{cu}=\frac{1}{2\pi i}\int_C (\lambda I-T)^{-1}\,d\lambda,
\end{equation*} where C is any rectifiable Jordan curve, which is disjoint with 
$\varSigma$ and contains 
$\varSigma^{cu}$ in its interior, but no point of 
$\varSigma^s:=\varSigma\setminus\varSigma^{cu}$. The phase space 
$\mathcal B_\gamma$ can be decomposed into the direct sum
\begin{equation*}
\mathcal B_\gamma=\mathcal B_\gamma^{cu}\oplus\mathcal B_\gamma^s
\end{equation*} with 
$\mathcal B_\gamma^{cu}=\varPi^{cu}(\mathcal B_\gamma)$ and 
$\mathcal B_\gamma^{s}=\varPi^{s}(\mathcal B_\gamma)$, where 
$\varPi^{s}=I-\varPi^{cu}$. The subspaces 
$\mathcal B_\gamma^{cu}$ and 
$\mathcal B_\gamma^{s}$ are called the centre-unstable subspace and the stable subspace of 
$\mathcal B_\gamma$, respectively. The spectra of the restrictions 
$T^{cu}:=T|_{\mathcal B_{\gamma}^{cu}}$ and 
$T^s:=T|_{\mathcal B_{\gamma}^{s}}$ satisfy
\begin{equation*}
\sigma(T^{cu})=\varSigma^{cu}\quad \text{and } \quad\sigma(T^s)=\sigma(T)\setminus\varSigma^{cu}=\{\,\lambda\in\sigma(T):|\lambda| \lt 1\,\}.
\end{equation*} If 
$\varSigma^{cu}$ is non-empty, then it consists of finitely many eigenvalues ofT,
\begin{equation*}
\varSigma^{cu}=\{\,\lambda_1,\dots,\lambda_r\,\},
\end{equation*} and 
$\mathcal B_\gamma^{cu}$ can be written as a direct sum of the nullspaces
\begin{equation*}
\mathcal B_\gamma^{cu}=\operatorname{ker}((T-\lambda_1 I)^{p_1})\oplus\cdots\oplus \operatorname{ker}((T-\lambda_r I)^{p_r}),
\end{equation*} where pj is the index (ascent) of λj, 
$j=1,\dots,r$ (see [Reference Matsunaga, Murakami, Nagabuchi and Nakano18, Remark 2.1, p. 62]).
An explicit representation of the spectral projection 
$\varPi^{cu}$ can be given using the duality between Eq. (1.5) and its formal adjoint equation
\begin{equation}
y(n-1)=\sum_{j=0}^\infty y(n+j) A(j),\qquad n\in\mathbb Z^+_0,
\end{equation} where 
$y(n)\in\mathbb C^{d*}$. Here 
$\mathbb C^{d*}$ denotes the d-dimensional space of complex row vectors with a norm 
$|\cdot|$, which is compatible with the given norm on 
$\mathbb C^d$, i.e., 
$|x^*x|\leq|x^*||x|$ for all 
$x\in\mathbb C^d$. The superscript 
${}^*$ indicates the conjugate transpose. The phase space for the formal adjoint Eq. (3.3) is the Banach space 
$(\mathcal B_{\tilde\gamma}^\sharp,\|\cdot\|)$ defined by
\begin{equation*}
\mathcal B_{\tilde\gamma}^\sharp=\biggl\{\,\psi\colon\mathbb Z^+_0\to\mathbb C^{d*}:\sup_{\zeta\in\mathbb Z^+_0}|\psi(\zeta)|e^{-\tilde\gamma\zeta} \lt \infty\,\biggr\},
\quad \|\psi\|=\sup_{\zeta\in\mathbb Z^+_0}|\psi(\zeta)| e^{-\tilde\gamma\zeta},\quad\psi\in\mathcal B_{\tilde\gamma}^\sharp,
\end{equation*} where 
$\tilde\gamma$ is a fixed number such that 
$0 \lt \tilde\gamma \lt \gamma$. The solution operator 
$T^\sharp\colon\mathcal B_{\tilde\gamma}^\sharp\to\mathcal B_{\tilde\gamma}^\sharp$ of Eq. (3.3) is given by
\begin{equation}
[T^\sharp(\psi)](\zeta)=\begin{cases}
\displaystyle\sum_{j=0}^\infty \psi(j)A(j)&\qquad\text{if}\ ~\zeta=0,\\
\psi(\zeta-1)&\qquad\text{if}\ ~\zeta\geq1.
\end{cases}
\end{equation} Define a bilinear form 
$\langle\cdot,\cdot\rangle\colon\mathcal B_{\tilde\gamma}^\sharp\times \mathcal B_\gamma\to\mathbb C$ by
\begin{equation*}
\langle\psi,\phi\rangle=\psi(0)\phi(0)+\sum_{j=1}^\infty\sum_{\zeta=0}^{j-1}\psi(\zeta+1)A(j)\phi(\zeta-j),
\qquad\phi\in\mathcal B_\gamma,\quad\psi\in \mathcal B_{\tilde\gamma}^\sharp.
\end{equation*}As shown in [Reference Matsunaga, Murakami, Nagabuchi and Nakano18, In eq. (3.3), p. 64], this bilinear form is bounded, i.e., there exists K > 0 such that
\begin{equation}
|\langle\psi,\phi\rangle|\leq K\|\psi\|\|\phi\|,
\qquad\phi\in\mathcal B_\gamma,\quad\psi\in \mathcal B_{\tilde\gamma}^\sharp.
\end{equation}Moreover, between Eqs. (1.5) and (3.3), we have the following duality relation (see [Reference Matsunaga, Murakami, Nagabuchi and Nakano18, Lemma 3.1])
\begin{equation}
\langle\psi,T\phi\rangle=\langle T^\sharp\psi,\phi\rangle,\qquad \phi\in\mathcal B_\gamma,\quad\psi\in\mathcal B_{\tilde\gamma}^\sharp.
\end{equation} It is known that T and 
$T^\sharp$ have the same spectrum and the dimension of the subspace
\begin{equation*}
\mathcal N^\sharp:=\operatorname{ker}((T^\sharp-\lambda_1 I)^{p_1})\oplus\cdots\oplus \operatorname{ker}((T^\sharp-\lambda_r I)^{p_r})
\end{equation*} of 
$\mathcal B_{\tilde\gamma}^\sharp$ is the same as the (finite) dimension of the centre-unstable subspace 
$\mathcal B_\gamma^{cu}$, which will be denoted by s. Let 
$\{\,\phi_1,\dots,\phi_s\,\}$ and 
$\{\,\psi_1,\dots,\psi_s\,\}$ be bases for 
$\mathcal B_\gamma^{cu}$ and 
$\mathcal N^\sharp$, respectively. Define 
$\varPhi=(\phi_1,\dots,\phi_s)$ and 
$\varPsi=\operatorname{col}(\psi_1,\dots,\psi_s)$. Then, the s × s matrix 
$\langle\varPsi,\varPhi\rangle$ given by 
$\langle\varPsi,\varPhi\rangle=(\langle\psi_i,\phi_j\rangle)_{i,j=1,\dots,s}$ is non-singular, therefore, by replacing 
$\varPsi$ with 
$\langle\varPsi,\varPhi\rangle^{-1}\varPsi$, we may (and do) assume that 
$\langle\varPsi,\varPhi\rangle=E$, the s × s unit matrix. The projection 
$\varPi^{cu}\colon\mathcal B_\gamma\to\mathcal B_\gamma^{cu}$ can be given explicitly by (see [Reference Matsunaga, Murakami, Nagabuchi and Nakano18, Theorem 3.1])
\begin{equation}
\varPi^{cu}\phi=\varPhi\langle\varPsi,\phi\rangle,\qquad\phi\in\mathcal B_\gamma,
\end{equation} where 
$\langle\varPsi,\phi\rangle$ denotes the column vector 
$\operatorname{col}(\langle\psi_1,\phi\rangle,\dots,\langle\psi_s,\phi\rangle)$.
The subspace 
$\mathcal B_\gamma^{cu}$ is invariant under the solution operator T. If B denotes the representation matrix of the linear transformation 
$T|_{\mathcal B_\gamma^{cu}}$ with respect to the basis 
$\varPhi$ of 
$\mathcal B_\gamma^{cu}$, then
\begin{equation}
T\varPhi=\varPhi B\qquad\text{and}\qquad \sigma(B)=\varSigma^{cu}.
\end{equation}A similar argument yields the existence of a square matrix C such that
\begin{equation}
T^\sharp\varPsi=C\varPsi\qquad\text{and}\qquad \sigma(C)=\varSigma^{cu}.
\end{equation}Now suppose that x is a solution of the non-homogeneous equation
\begin{equation}
x(n+1)=L(x_n)+p(n), \qquad n\in \mathbb Z^+_0.
\end{equation} with initial value 
$x_0=\phi$ for some 
$\phi\in\mathcal B_\gamma$. Then, x satisfies the following representation formula in 
$\mathcal B_\gamma$ (see [Reference Murakami19, Theorem 2.1]), which is called the variation of constants formula for Eq. (3.10) in the phase space,
\begin{equation}
x_n=T(n)\phi+\sum_{j=0}^{n-1}T(n-1-j)\varGamma p(j),\qquad n\in\mathbb Z^+_0,
\end{equation} where the operator 
$\varGamma\colon\mathbb C^d\to\mathcal B_\gamma$ is defined by
\begin{equation}
[\varGamma x](\theta)=\begin{cases}
x&\qquad\text{if}~\theta=0,\\
0&\qquad\text{if}~\theta\leq-1.
\end{cases}
\end{equation}Evidently,
\begin{equation}
\|\varGamma x\|=|x|,\qquad x\in\mathbb C^d.
\end{equation} Finally, let z(n) be the coordinate of the projection 
$\varPi^{cu}x_n$ with respect to the basis 
$\varPhi$, i.e., 
$\varPi^{cu}x_n=\varPhi z(n)$ for 
$n\in\mathbb Z^+_0$. In view of (3.7), z(n) is given explicitly by
\begin{equation}
z(n)=\langle\varPsi,x_n\rangle,\qquad n\in\mathbb Z^+_0.
\end{equation} Moreover, it is known (see [Reference Nagabuchi20, Theorem 3]) that z satisfies the following first order difference equation in 
$\mathbb C^s$,
\begin{equation}
z(n+1)=Bz(n)+\langle\varPsi,\varGamma p(n)\rangle,\qquad n\in\mathbb Z^+_0,
\end{equation}with B as in (3.8).
Now we are in a position to give a proof of Theorem 2.3.
Proof of Theorem 2.3
$(i)\Rightarrow(ii)$. Suppose, for the sake of contradiction, that Eq. (1.5) is shadowable, but (ii) does not hold. The shadowing property of Eq. (1.5) implies the following Perron-type property.
Claim 15.
For every bounded function 
$p\colon \mathbb Z^+_0\to \mathbb C^d$, there exists a function 
$x\colon \mathbb Z\to\mathbb C^d$ satisfying the non-homogeneous Eq. (3.10) with
\begin{equation}
\sup_{n\geq 0}|x(n)| \lt \infty.
\end{equation} The proof of Claim 15 is almost identical with that of Claim 1 in the proof of Theorem 2.3, therefore we omit it. Since (ii) does not hold, there exists a characteristic root 
$\lambda\in\varSigma$ with 
$|\lambda|=1$. Evidently, 
$\lambda\in\varSigma^{cu}$, therefore the second relation in (3.8) implies the existence of a non-zero vector 
$v\in\mathbb C^{s*}$ such that
\begin{equation}
vB=\lambda v.
\end{equation} Define 
$p\colon\mathbb Z^+_0\to\mathbb C^d$ by
\begin{equation}
p(n)=\lambda^{n+1}(v\varPsi(0))^*,\qquad n\in\mathbb Z^+_0.
\end{equation} Since 
$\sup_{n\geq 0}|p(n)|=|(v\varPsi(0))^*| \lt \infty$, by Claim 15, the non-homogeneous Eq. (3.10) has at least one solution x on 
$\mathbb Z^+_0$ such that (3.16) holds. The corresponding coordinate function z defined by (3.14) satisfies the difference Eq. (3.15) From (3.5), (3.13), (3.14), and (3.16), we obtain that z and hence the function 
$u\colon\mathbb Z^+_0\to\mathbb C$ defined by
\begin{equation}
u(n)=vz(n),\qquad n\in\mathbb Z^+_0,
\end{equation} is bounded on 
$\mathbb Z^+_0$. Multiplying Eq. (3.15) from the left by v and using (3.17), we obtain that
\begin{equation}
u(n+1)=\lambda u(n)+v\langle\varPsi,\varGamma p(n)\rangle,\qquad n\in\mathbb Z^+_0.
\end{equation} It follows from (3.12), (3.18), and the definition of the bilinear form 
$\langle\cdot,\cdot\rangle$ that
\begin{equation*}
\langle\varPsi,\varGamma p(n)\rangle=\varPsi(0)p(n)=\lambda^{n+1}\varPsi(0)(v\varPsi(0))^*.
\end{equation*}This, together with (3.20), implies that
\begin{equation}
u(n+1)=\lambda u(n)+c\lambda^{n+1},\qquad n\in\mathbb Z^+_0,
\end{equation}with
\begin{equation}
c=(v\varPsi(0))(v\varPsi(0))^*=|(v\varPsi(0))^*|_2^2,
\end{equation} where 
$|\cdot|_2$ denotes the l 2-norm on 
$\mathbb C^d$. From Eq. (3.21), it follows by the variation of constants formula that
\begin{equation}
u(n)=\lambda^{n}(u(0)+cn),\qquad n\in\mathbb Z^+_0.
\end{equation}It follows from the duality (3.6) that B = C, where B and C have the meaning from (3.8) and (3.9), respectively. Indeed, (3.6) implies that
\begin{equation*}
B=\langle \varPsi,\varPhi\rangle B
=\langle\varPsi,\varPhi B\rangle
=\langle\varPsi,T\varPhi\rangle
=\langle T^\sharp\varPsi,\varPhi\rangle
=\langle C\varPsi,\varPhi\rangle
=C\langle\varPsi,\varPhi\rangle
=C.
\end{equation*} From (3.2) and the relation 
$T\varPhi=\varPhi B$ (see (3.8)), we find that
\begin{equation}
\varPhi(\theta)=\varPhi(0)B^\theta,\qquad\theta\in\mathbb Z^-_0.
\end{equation} Similarly, from (3.4) and the relation 
$T^\sharp\varPsi=B\varPsi$ (see (3.9)), we have that
\begin{equation}
\varPsi(\zeta)=B^{-\zeta}\varPsi(0),\qquad\zeta\in\mathbb Z^+_0.
\end{equation}Next we will show that
\begin{equation}
v\varPsi(0)\neq0.
\end{equation} Suppose, for the sake of contradiction, that 
$v\varPsi(0)=0$. Then, by (3.17) and (3.25), we have
\begin{equation*}
v\varPsi(\zeta)=vB^{-\zeta}\varPsi(0)=\lambda^{-\zeta}v\varPsi(0)=0\qquad\text{for all}~\zeta\in\mathbb Z^+_0.
\end{equation*} Thus, 
$v\varPsi$ is identically zero on 
$\mathbb Z^+_0$. On the other hand, v ≠ 0 implies that 
$v\varPsi$ is a non-trivial linear combination of the basis elements 
$\psi_1,\dots,\psi_s$ of 
$\mathcal N^\sharp$, and hence, it cannot be identically zero on 
$\mathbb Z^+_0$. This contradiction proves that (3.26) holds.
From (3.22) and (3.26), we find that c > 0. From this and (3.23), taking into account that 
$|\lambda|=1$, we obtain that
\begin{equation*}
|u(n)|=|u(0)+cn|\rightarrow\infty,\qquad n\to\infty,
\end{equation*}which contradicts the boundedness of u.
$(ii) \Rightarrow (i)$. Suppose that the characteristic Eq. (1.7) has no root with 
$|\lambda|=1$. Then, the exponential estimates of the solution operator on the stable and unstable subspaces of 
$\mathcal B_\gamma$ (see, e.g., [Reference Nagabuchi20, Theorem 1]) imply that Eq. (1.5) admits an exponential dichotomy on 
$\mathbb Z^+_0$ as defined in [Reference Dragičević and Pituk10] with projections 
$P_n=\varPi^s$ for 
$n\in\mathbb Z^+_0$. By the application of [Reference Dragičević and Pituk10, Theorem 1] with 
$f_n=0$, c = 0 and µ = 1, we conclude that Eq. (1.5) is shadowable on 
$\mathbb Z^+_0$.
Remark 3.4. It is known that certain solutions of Eq. (1.5) can be continued backward in the sense that they satisfy Eq. (1.5) for all 
$n\in\mathbb Z$. Such solutions are sometimes called global solutions or entire solutions. In a recent article [Reference Barreira and Valls4], Barreira and Valls have considered the Ulam–Hyers stability, a special case of shadowing, of the global solutions for difference equations with finite delays. In [Reference Barreira and Valls4, Theorem 5], they have proved the analogue of a recent shadowing theorem [Reference Dragičević and Pituk10, Theorem 1] for global solutions. Moreover, in the special case of linear autonomous equations in finite dimensional spaces, they have shown that the Ulam–Hyers stability of the global solutions is equivalent to the existence of an exponential dichotomy whenever the coefficients are scalar (see [Reference Barreira and Valls4, Theorem 8]) or the Jordan blocks associated with the central directions are diagonal (see [Reference Barreira and Valls4, Theorem 9]). Using a similar argument as in the proof of Theorem 3.2, it can be shown that the assumption about the Jordan blocks in [Reference Barreira and Valls4, Theorem 9] is superfluous. In this sense, Theorem 3.2 may be viewed as an improvement of [Reference Barreira and Valls4, Theorem 9].
Acknowledgements
L.B. was partially supported by a CNPq-Brazil PQ fellowship under grant no. 307633/2021-7. D.D. was supported in part by the University of Rijeka under the project uniri-iskusni-prirod-23-98 3046. M.P. was supported by the Hungarian National Research, Development and Innovation Office under grant no. K139346.
