# Generalized hyperbolicity, stability and expansivity
for operators on locally convex spaces

###### Abstract.

We introduce and study the notions of (generalized) hyperbolicity, topological
stability and (uniform) topological expansivity for operators on locally
convex spaces. We prove that every generalized hyperbolic operator on a
locally convex space has the finite shadowing property. Contrary to what
happens in the Banach space setting, hyperbolic operators on Fréchet spaces
may fail to have the shadowing property, but we find additional conditions
that ensure the validity of the shadowing property. Assuming that the space is
sequentially complete, we prove that generalized hyperbolicity implies the
strict periodic shadowing property, but we also show that the hypothesis of
sequential completeness is essential. We show that operators with the periodic
shadowing property on topological vector spaces have other interesting
dynamical behaviors, including the fact that the restriction of such an
operator to its chain recurrent set is topologically mixing and Devaney
chaotic. We prove that topologically stable operators on locally convex spaces
have the finite shadowing property and the strict periodic shadowing property.
As a consequence, topologically stable operators on Banach spaces have the
shadowing property. Moreover, we prove that generalized hyperbolicity implies
topological stability for operators on Banach spaces. We prove that uniformly
topologically expansive operators on locally convex spaces are neither Li-
Yorke chaotic nor topologically transitive. Finally, we characterize the
notion of topological expansivity for weighted shifts on Fréchet sequence
spaces. Several examples are provided.

###### Key words and phrases:

Generalized hyperbolicity, shadowing property, topological stability,
expansivity, Li-Yorke chaos, weighted shifts

###### 2020 Mathematics Subject Classification:

Primary 47A16, 37B65, 37B25; Secondary 37B05, 37C50, 37D20.

Nilson C. Bernardes Jr., Blas M. Caraballo, Udayan B. Darji,
Vinícius V. Fávaro and Alfred Peris

##  1\. Introduction

The shadowing property is one of the most important concepts in the modern
theory of dynamical systems. It originated with works by Anosov, Bowen and
Sina$\breve{\text{\i}}$ from the late 1960s and early 1970s, leading to the
famous shadowing lemma in differentiable dynamics, which asserts that a
diffeomorphism has the shadowing property in a neighborhood of its hyperbolic
set [2, 18, 46]. In the setting of linear dynamics, it is well known that
every invertible hyperbolic operator on a Banach space has the shadowing
property in the full space [36, 37]. Moreover, the converse holds in the
finite dimensional setting [36, 37] and for invertible normal operators on
Hilbert spaces [33]. It remained open for a while whether or not this converse
is always true. The solution was given by means of the following theorem
(where $r(T)$ denotes the spectral radius of $T$):  Let $T$ be an invertible
operator on a Banach space $X$. Suppose that $X=M\oplus N$, where $M$ and $N$
are closed subspaces of $X$ with $T(M)\subset M$ and $T^{-1}(N)\subset N$. If
$r(T|_{M})<1$ and $r(T^{-1}|_{N})<1$, then $T$ has the shadowing property [7,
Theorem A].

The class of operators considered in the above theorem clearly contains all
invertible hyperbolic operators, but it also enabled the construction of the
first examples of operators that have the shadowing property but are not
hyperbolic [7, Theorem B], thereby solving the above-mentioned problem in the
negative. The operators considered in the above theorem were named generalized
hyperbolic in [20], where additional dynamical properties of these operators
where investigated. The class of generalized hyperbolic operators is also
closely related to another fundamental concept in dynamical systems, namely:
structural stability. A classical theorem in linear dynamics, often called
Hartman’s theorem, asserts that every invertible hyperbolic operator on a
Banach space is structurally stable [26, 39, 42]. It was soon realized that
the converse holds in the finite dimensional setting [43], but whether or not
the converse of Hartman’s theorem is always true remained open for more than
50 years. This problem was finally settled in [9], where it was obtained a
class of weighted shifts on classical Banach sequence spaces that are
structurally stable but are not hyperbolic [9, Theorem 9]. It turns out that
these weighted shifts are generalized hyperbolic. A little later, it was
proved that every generalized hyperbolic operator is structurally stable [10,
Theorem 1], which enabled the proof of a generalized Grobman-Hartman theorem
[10, Theorem 3].

Another important concept in dynamical systems is that of topological
stability, which was introduced by Walters [48]. A classical result due to
Walters [49] asserts that:  Every topologically stable homeomorphism $h:X\to
X$, where $X$ is a closed topological manifold of dimension at least two, has
the shadowing property (see also [3, Theorem 2.4.9]). Recall that a closed
topological manifold is a compact connected metrizable topological manifold
without boundary [3, Page 28]. Motivated by this result, we will begin a study
of the concept of topological stability in the setting of linear dynamics,
with emphasis on its connections with generalized hyperbolicity and the
shadowing property.

Yet another fundamental concept in dynamical systems is that of expansivity,
which was introduced by Utz [47]. Expansive and uniformly expansive operators
on Banach spaces were studied in [1, 7, 9, 22, 27, 33], for instance. Let us
mention three of the main results obtained so far:

  * •

An invertible operator on a Banach space is uniformly expansive if and only if
its approximate point spectrum does not intersect the unit circle [22, 27].

  * •

An invertible operator on a Banach space is hyperbolic if and only if it is
expansive and has the shadowing property [9, 20].

  * •

A uniformly expansive operator on a Banach space is never Li-Yorke chaotic
[7].

Moreover, complete characterizations of expansive and uniformly expansive
weighted shifts on classical Banach sequence spaces were obtained in [7].

All the works on linear dynamics mentioned above deal with operators on Banach
spaces, but the first results on linear dynamical systems go back to Birkhoff
[12] for the translation operator on the (non-normable) Fréchet space
$H(\mathbb{C})$ of all entire functions, and to MacLane [32] for the
differentiation operator on $H(\mathbb{C})$, which are the classical examples
of chaotic operators. Moreover, even the dynamics of operators on non-
metrizable topological vector spaces has attracted the interest of many
researchers and experienced a great development in recent years (see, e.g.,
Chapter 12 in [25], the articles [13, 14, 15, 16, 19, 21, 24, 40, 44], and
references therein). Our main objective in the present work is to propose a
concept of (generalized) hyperbolicity, a notion of topological stability and
a concept of (uniform) expansivity for operators on locally convex spaces,
with the purpose to initiate investigations on these notions.

In Section 2 we introduce a concept of hyperbolicity and a concept of
generalized hyperbolicity for operators on an arbitrary locally convex space
(Definition 1). We prove that every generalized hyperbolic operator on a
locally convex space has the finite shadowing property (Theorem 2), but the
shadowing property may fail even for hyperbolic operators on the Fréchet space
$H(\mathbb{C})$ of all entire functions (Remark 9). Nevertheless, we give
additional conditions to ensure the validity of the shadowing property
(Theorem 6). For sequentially complete locally convex spaces, we prove that
generalized hyperbolicity implies the strict periodic shadowing property
(Theorem 2), but we also show that the hypothesis of sequential completeness
is essential for the validity of this result (Remark 5). We show that
operators with the periodic shadowing property on topological vector spaces
have other interesting dynamical behaviors (Theorem 3), including the fact
that the restriction of such an operator to its chain recurrent set is
topologically mixing and Devaney chaotic. Some examples illustrating the
theorems are also presented.

In Section 3 we prove that every topologically stable operator on a locally
convex space has the finite shadowing property and the strict periodic
shadowing property (Theorem 12) and that every topologically stable operator
on a Banach space has the shadowing property (Corollary 14). For this purpose,
we show that open convex sets in locally convex spaces of dimension greater
than two have a certain multihomogeneity property (Theorem 17) and we
establish a very general version of the previously mentioned theorem of
Walters [49] in the setting of uniform spaces (Theorem 22). Moreover, we show
that every invertible generalized hyperbolic operator on a sequentially
complete locally convex space has a certain stability property (Theorem 25).
As a consequence, we obtain that every invertible generalized hyperbolic
operator on a Banach space is topologically stable (Corollary 26).

In Section 4 we introduce the notions of topological expansivity and uniform
topological expansivity for invertible operators on an arbitrary locally
convex space (Definition 32). These concepts generalize the well-known notions
of expansivity and uniform expansivity for operators on normed spaces, but
they seem to be more adequate when we go beyond the normed space setting. For
invertible generalized hyperbolic operators on locally convex spaces, the
notions of topological expansivity, uniform topological expansivity and
hyperbolicity coincide (Theorem 35). In particular, every invertible
hyperbolic operator on a locally convex space is uniformly topologically
expansive (Corollary 36). Our main result in this section asserts that
uniformly topologically expansive operators on locally convex spaces are
neither Li-Yorke chaotic nor topologically transitive (Theorem 38).

In Section 5 we characterize the notion of topological expansivity for
weighted shifts on Fréchet sequence spaces (Theorem 41) and, in particular, on
Köthe sequence spaces (Corollary 43). Several concrete examples are presented.

In our final Section 6 we propose some open problems.

##  2\. Generalized hyperbolic operators on locally convex spaces

Throughout $\mathbb{K}$ denotes either the field $\mathbb{R}$ of real numbers
or the field $\mathbb{C}$ of complex numbers, $\mathbb{N}$ denotes the set of
all positive integers and $\mathbb{N}_{0}\\!:=\mathbb{N}\cup\\{0\\}$.

Given a topological vector space $X$ over $\mathbb{K}$ (all topological vector
spaces will be assumed to be Hausdorff), we denote by $L(X)$ the set of all
continuous linear operators on $X$ and by $GL(X)$ the set of those operators
that have a continuous inverse. Recall that $X$ is the topological direct sum
of the subspaces $M_{1},\ldots,M_{n}$ if $X$ is the algebraic direct sum of
$M_{1},\ldots,M_{n}$ and the canonical algebraic isomorphism

$(y_{1},\ldots,y_{n})\mapsto y_{1}+\cdots+y_{n}$

is a homeomorphism from the product space $M_{1}\times\cdots\times M_{n}$ onto
$X$. Recall also that a family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of
seminorms on $X$ is said to be directed if for every $\alpha,\beta\in I$,
there exists $\gamma\in I$ such that
$\|\cdot\|_{\alpha}\leq\|\cdot\|_{\gamma}$ and
$\|\cdot\|_{\beta}\leq\|\cdot\|_{\gamma}$.

If $T\in L(X)$ and $U$ is a neighborhood of $0$ in $X$, recall that a
$U$-pseudotrajectory of $T$ is a finite or infinite sequence $(x_{j})_{i<j<k}$
in $X$ ($-\infty\leq i<k\leq\infty$), consisting of at least two terms, such
that

$Tx_{j}-x_{j+1}\in U\ \ \text{ for all }i<j<k-1.$

A finite $U$-pseudotrajectory of the form $(x_{j})_{j=0}^{k}$ is also called a
$U$-chain for $T$ (from $x_{0}$ to $x_{k}$). If, in addition, $x_{k}=x_{0}$,
then we say that $(x_{j})_{j=0}^{k}$ is a $U$-cycle for $T$. Recall that $T$
has the finite shadowing property (resp. the positive shadowing property) if
for every neighborhood $V$ of $0$ in $X$, there is a neighborhood $U$ of $0$
in $X$ such that every $U$-chain $(x_{j})_{j=0}^{k}$ (resp. every
$U$-pseudotrajectory $(x_{j})_{j\in\mathbb{N}_{0}}$) of $T$ is $V$-shadowed by
the trajectory of some $x\in X$, in the sense that

$x_{j}-T^{j}x\in V\ \ \text{ for all }j\in\\{0,\ldots,k\\}\ \text{ (resp.\ for
all }j\in\mathbb{N}_{0}).$

If $T\in GL(X)$, then the shadowing property is defined by replacing the set
$\mathbb{N}_{0}$ by the set $\mathbb{Z}$ in the definition of positive
shadowing. Recall also that $T$ has the periodic shadowing property [30, 38]
if for every neighborhood $V$ of $0$ in $X$, there is a neighborhood $U$ of
$0$ in $X$ such that every periodic $U$-pseudotrajectory
$(x_{j})_{j\in\mathbb{N}_{0}}$ of $T$ is $V$-shadowed by some
$x\in\operatorname{Per}(T)$ (the set of all periodic points of $T$). By adding
the condition that the periodic point $x$ can be chosen to satisfy $T^{k}x=x$
if $(x_{j})_{j\in\mathbb{N}_{0}}$ has period $k$, then we obtain the strict
periodic shadowing property [29].

We shall now define a notion of generalized hyperbolicity for (not necessarily
invertible) operators on locally convex spaces.

######  Definition 1.

Let $X$ be a locally convex space over $\mathbb{K}$ whose topology is induced
by a directed family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms. We say
that an operator $T\in L(X)$ is generalized hyperbolic if there is a
topological direct sum decomposition

(1) $X=M\oplus N$

with the following properties:

  * (GH1)

$T(M)\subset M$.

  * (GH2)

$T(N)\supset N$ and $T|_{N}:N\to T(N)$ is an isomorphism.

  * (GH3)

For every $\alpha\in I$, there exist $\beta\in I$, $c>0$ and $t\in(0,1)$ such
that

(2) $\|T^{n}y\|_{\alpha}\leq c\,t^{n}\|y\|_{\beta}\ \text{ and }\
\|S^{n}z\|_{\alpha}\leq c\,t^{n}\|z\|_{\beta}\ \text{ whenever }y\in M,z\in
N,n\in\mathbb{N}_{0},$

where $S\\!:=(T|_{N})^{-1}|_{N}\in L(N)$.

If $T\in GL(X)$, then condition (GH2) is equivalent to

  * (GH2’)

$T^{-1}(N)\subset N$,

and the second inequality in (2) can be rewritten as

$\|T^{-n}z\|_{\alpha}\leq c\,t^{n}\|z\|_{\beta}.$

If $M$ and $N$ are $T$-invariant, then we say that the operator $T$ is
hyperbolic. If, in addition, $M=\\{0\\}$ or $N=\\{0\\}$, then we say that the
hyperbolic operator $T$ has trivial splitting.

Note that the above notions are independent of the choice of the directed
family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms inducing the topology
of $X$. Moreover, they generalize to operators on arbitrary locally convex
spaces the corresponding notions for operators on Banach spaces.

As we mentioned in the Introduction, [7, Theorem A] asserts that every
invertible generalized hyperbolic operator on a Banach space has the shadowing
property. Surprisingly enough, there exist invertible hyperbolic operators on
the Fréchet space $H(\mathbb{C})$ of all entire functions that do not have the
shadowing property (Remark 9). Nevertheless, we will prove below that we can
always guarantee the finite shadowing property for generalized hyperbolic
operators on locally convex spaces. Also, it was proved in [11, Theorem 18]
that generalized hyperbolic operators on Banach spaces have the strict
periodic shadowing property. We will see below that the arguments in [11] can
be adapted to extend this result to arbitrary sequentially complete locally
convex spaces, but we will also give a counterexample showing that the
hypothesis of sequential completeness is essential for the validity of this
result (Remark 5). Recall that a topological vector space $X$ is said to be
sequentially complete if every Cauchy sequence in $X$ converges to a point in
$X$, where a sequence $(x_{n})_{n\in\mathbb{N}}$ in $X$ is a Cauchy sequence
if for every neighborhood $V$ of $0$ in $X$, there exists $n_{0}\in\mathbb{N}$
such that $x_{n}-x_{m}\in V$ whenever $n,m\geq n_{0}$.

######  Theorem 2.

Every generalized hyperbolic operator $T$ on a locally convex space $X$ has
the finite shadowing property. If, in addition, $X$ is sequentially complete,
then $T$ also has the strict periodic shadowing property.

###### Proof.

Choose a directed family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms
inducing the topology of $X$ and let $M$, $N$ and $S$ be as in the definition
of generalized hyperbolicity. Let $P_{M}:X\to M$ and $P_{N}:X\to N$ be the
canonical projections associated to the direct sum decomposition (1). Given a
neighborhood $V$ of $0$ in $X$, there exist $\alpha\in I$ and $\varepsilon>0$
such that

$\\{x\in X:\|x\|_{\alpha}<\varepsilon\\}\subset V.$

Let $\beta\in I$, $c>0$ and $t\in(0,1)$ be such that (2) holds. Since the
projections $P_{M}$ and $P_{N}$ are continuous, there exist $\gamma\in I$ and
$d>0$ such that

(3) $\|P_{M}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \ \text{ and }\ \
\|P_{N}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \ \text{ for all }x\in X.$

Let $\delta\\!:=\frac{(1-t)\varepsilon}{3\,c\,d}>0$ and $U\\!:=\\{x\in
X:\|x\|_{\gamma}<\delta\\}$, which is a neighborhood of $0$ in $X$. Let
$(x_{j})_{j=0}^{p}$ be a $U$-chain for $T$ and define
$y_{j}\\!:=x_{j+1}-Tx_{j}$ for each $j\in\\{0,\ldots,p-1\\}$. Then

(4) $\|y_{j}\|_{\gamma}<\delta\ \ \text{ for all }j\in\\{0,\ldots,p-1\\}.$

Consider the vector

(5) $x\\!:=x_{0}+\sum_{j=1}^{p}S^{j}P_{N}y_{j-1}.$

We claim that

(6)
$x_{m}-T^{m}x=\sum_{j=0}^{m-1}T^{j}P_{M}y_{m-j-1}-\sum_{j=1}^{p-m}S^{j}P_{N}y_{m+j-1}\
\ \text{ for all }m\in\\{0,\ldots,p\\},$

where sums of the form $\sum_{j=0}^{-1}a_{j}$ and $\sum_{j=1}^{0}a_{j}$ are
assumed to be $0$. Clearly, (6) holds for $m=0$. Assume that it holds for a
certain $m\in\\{0,\ldots,p-1\\}$. Then

$\displaystyle x_{m+1}-T^{m+1}x$ $\displaystyle=y_{m}+T(x_{m}-T^{m}x)$
$\displaystyle=P_{M}y_{m}+P_{N}y_{m}+\sum_{j=0}^{m-1}T^{j+1}P_{M}y_{m-j-1}-\sum_{j=1}^{p-m}S^{j-1}P_{N}y_{m+j-1}$
$\displaystyle=\sum_{j=0}^{m}T^{j}P_{M}y_{m-j}-\sum_{j=1}^{p-m-1}S^{j}P_{N}y_{m+j}.$

Hence, by induction, we obtain (6). By (2), (3), (4) and (6), for each
$m\in\\{0,\ldots,p\\}$,

$\displaystyle\|x_{m}-T^{m}x\|_{\alpha}$
$\displaystyle\leq\sum_{j=0}^{m-1}\|T^{j}P_{M}y_{m-j-1}\|_{\alpha}+\sum_{j=1}^{p-m}\|S^{j}P_{N}y_{m+j-1}\|_{\alpha}$
$\displaystyle\leq\sum_{j=0}^{m-1}c\,t^{j}\|P_{M}y_{m-j-1}\|_{\beta}+\sum_{j=1}^{p-m}c\,t^{j}\|P_{N}y_{m+j-1}\|_{\beta}$
$\displaystyle\leq\sum_{j=0}^{m-1}c\,d\,t^{j}\|y_{m-j-1}\|_{\gamma}+\sum_{j=1}^{p-m}c\,d\,t^{j}\|y_{m+j-1}\|_{\gamma}$
$\displaystyle<\frac{2\,c\,d\,\delta}{1-t}<\varepsilon.$

This shows that the $U$-chain $(x_{j})_{j=0}^{p}$ is $V$-shadowed by
$(T^{j}x)_{j=0}^{p}$, proving that $T$ has the finite shadowing property.

Now, assume that $X$ is sequentially complete. Assume also that
$(x_{j})_{j=0}^{p}$ is a $U$-cycle for $T$. We extend this $U$-cycle to a
periodic $U$-pseudotrajectory of $T$ by defining

$(x_{j})_{j\in\mathbb{N}_{0}}\\!:=(x_{0},x_{1},\ldots,x_{p},x_{1},\ldots,x_{p},x_{1},\ldots,x_{p},\ldots).$

Let $y_{j}\\!:=x_{j+1}-Tx_{j}$ for each $j\in\mathbb{N}_{0}$. Note that the
sequence $(y_{j})_{j\in\mathbb{N}_{0}}$ is also periodic with period $p$ and
that

$\|y_{j}\|_{\gamma}<\delta\ \ \text{ for all }j\in\mathbb{N}_{0}.$

The vector $x$ defined in (5) may fail to be periodic. However, we can follow
the proof of [11, Theorem 18] and replace (5) by the following definition:

(7)
$x\\!:=x_{0}+\sum_{j=1}^{\infty}S^{j}P_{N}y_{j-1}-\sum_{j=0}^{p-1}\sum_{k=0}^{\infty}T^{kp+j}P_{M}y_{p-j-1}.$

Since the sequence $(y_{j})_{j\in\mathbb{N}_{0}}$ is bounded, the estimates in
(2) imply that the sequence of partial sums of each infinite series in (7) is
a Cauchy sequence. Since we are assuming that $X$ is sequentially complete,
the vector $x$ is well-defined. The computations done in the proof of [11,
Theorem 18] show that

$T^{p}x=x$

and

$x_{m}-T^{m}x=\sum_{j=0}^{m-1}T^{j}P_{M}y_{m-j-1}-\sum_{j=1}^{\infty}S^{j}P_{N}y_{m+j-1}+\sum_{j=0}^{p-1}\sum_{k=0}^{\infty}T^{kp+j+m}P_{M}y_{p-j-1},$

for all $m\in\\{0,\ldots,p\\}$. Hence, $x$ is a periodic point of $T$ with
period $p$ and estimates like to the ones we made before give

$\|x_{m}-T^{m}x\|_{\alpha}<\frac{3\,c\,d\,\delta}{1-t}=\varepsilon\ \ \text{
for all }m\in\\{0,\ldots,p\\}.$

This proves that $T$ has the strict periodic shadowing property. ∎

Let $X$ be a topological vector space. Given $T\in L(X)$, recall that $x\in X$
is a chain recurrent point of $T$ if for every neighborhood $U$ of $0$ in $X$,
there is a $U$-chain (actually a $U$-cycle) for $T$ from $x$ to itself. The
set $\operatorname{CR}(T)$ of all chain recurrent points of $T$ is called the
chain recurrent set of $T$ and $T$ is chain recurrent if
$\operatorname{CR}(T)=X$. Clearly,
$\operatorname{Per}(T)\subset\operatorname{CR}(T)$. Recall also that $T$ is
topologically transitive (resp. topologically mixing) if for any pair $A,B$ of
nonempty open sets in $X$, there exists $n\in\mathbb{N}_{0}$ (resp.
$n_{0}\in\mathbb{N}_{0}$) such that $T^{n}(A)\cap B\neq\emptyset$ (resp. for
all $n\geq n_{0}$). Finally, recall that $T$ is Devaney chaotic if $T$ is
transitive and $\operatorname{Per}(T)$ is dense in $X$.

Operators with the periodic shadowing property have some additional
interesting dynamical properties, which are described in the next theorem.

######  Theorem 3.

Let $X$ be a topological vector space. If $T\in L(X)$ has the (strict)
periodic shadowing property, then the following properties hold:

  * (a)

$\operatorname{Per}(T)$ is dense in $\operatorname{CR}(T)$.

  * (b)

$T|_{\operatorname{CR}(T)}$ has the finite shadowing property and the (strict)
periodic shadowing property.

  * (c)

$T|_{\operatorname{CR}(T)}$ is topologically mixing and Devaney chaotic.

###### Proof.

By [11, Proposition 26], $Y\\!:=\operatorname{CR}(T)$ is a closed
$T$-invariant subspace of $X$. Hence, $S\\!:=T|_{Y}$ is a well-defined
continuous linear operator on $Y$. Given a symmetric neighborhood $V$ of $0$
in $X$, let $U$ be a neighborhood of $0$ in $X$ associated to $V$ according to
the fact that $T$ has the (strict) periodic shadowing property.

Step 1. $\operatorname{Per}(T)$ is dense in $\operatorname{CR}(T)$:

Given $x\in\operatorname{CR}(T)$, there exists a $U$-chain for $T$ from $x$ to
itself. By periodic shadowing, this $U$-cycle for $T$ must be $V$-shadowed by
the trajectory under $T$ of some $y\in\operatorname{Per}(T)$. In particular,
$y\in x+V$.

Step 2. $S$ has the finite shadowing property:

Let $(x_{j})_{j=0}^{k}$ be a $(U\cap Y)$-chain for $S$. Since
$x_{0},x_{k}\in\operatorname{CR}(T)$, [11, Proposition 25] says that there is
a $U$-chain $(x_{k},x_{k+1},\ldots,x_{p})$ for $T$ from $x_{k}$ to $x_{0}$.
Hence, $(x_{j})_{j=0}^{p}$ is a $U$-cycle for $T$. By periodic shadowing, it
must be $V$-shadowed by the trajectory under $T$ of some
$x\in\operatorname{Per}(T)$. In particular,

$x\in Y\ \ \text{ and }\ \ x_{j}-S^{j}x\in V\cap Y\text{ for all
}j\in\\{0,\ldots,k\\}.$

Step 3. $S$ has the (strict) periodic shadowing property:

If $(x_{j})_{j=0}^{k}$ is a $(U\cap Y)$-cycle for $S$, then it is also a
$U$-cycle for $T$, and so it must be $V$-shadowed by the trajectory under $T$
of some $x\in\operatorname{Per}(T)$. Then, $x\in Y$ and $(x_{j})_{j=0}^{k}$ is
$(V\cap Y)$-shadowed by the trajectory under $S$ of $x$. In the strict case,
$x$ can be chosen so that $T^{k}x=x$, that is, $S^{k}x=x$.

Step 4. $S$ is topologically mixing:

By Step 1,

(8)
$\operatorname{CR}(S)\subset\operatorname{CR}(T)=\overline{\operatorname{Per}(T)^{\,X}}=\overline{\operatorname{Per}(S)^{\,X}}=\overline{\operatorname{Per}(S)^{\,Y}}\subset\operatorname{CR}(S).$

Hence, $\operatorname{CR}(S)=\operatorname{CR}(T)=Y$, proving that $S$ is a
chain recurrent operator. Since the concepts of chain recurrence and
topological mixing coincide for operators with the finite shadowing property
[11, Theorem 7], we conclude from Step 2 that $S$ is topologically mixing.

Step 5. $S$ is Devaney chaotic:

By Step 4, $S$ is topologically transitive, and by (8),
$\operatorname{Per}(S)$ is dense in $Y$. ∎

By combining Theorems 2 and 3, we obtain the following result.

######  Corollary 4.

Let $X$ be a sequentially complete locally convex space. If $T\in L(X)$ is
generalized hyperbolic, then the following properties hold:

  * (a)

$\operatorname{Per}(T)$ is dense in $\operatorname{CR}(T)$.

  * (b)

$T|_{\operatorname{CR}(T)}$ has the finite shadowing property and the strict
periodic shadowing property.

  * (c)

$T|_{\operatorname{CR}(T)}$ is topologically mixing and Devaney chaotic.

For invertible generalized hyperbolic operators on Banach spaces, it was
proved in [20] that $T|_{\Omega(T)}$ is topologically mixing and Devaney
chaotic, where $\Omega(T)$ denotes the set of all nonwandering points of $T$.
Recall that $x\in X$ is a nonwandering point of $T$ if for every neighborhood
$A$ of $x$ in $X$, there exists $n\in\mathbb{N}$ such that $T^{n}(A)\cap
A\neq\emptyset$. This result from [20] can be seen as a particular case of the
above corollary, since the denseness of $\operatorname{Per}(T)$ in
$\operatorname{CR}(T)$ implies that $\Omega(T)=\operatorname{CR}(T)$.

######  Remark 5.

The hypothesis of sequential completeness is essential for the validity of the
second assertion in Theorem 2 and for the validity of Corollary 4, even in the
case of normed spaces. As a counterexample, let $X$ be the vector space of all
sequences $(x_{n})_{n\in\mathbb{Z}}$ of scalars with finite support endowed
with any $\ell_{p}$-norm ($1\leq p\leq\infty$) and let $T\in GL(X)$ be the
generalized hyperbolic operator given by

$T((x_{n})_{n\in\mathbb{Z}})\\!:=(w_{n+1}x_{n+1})_{n\in\mathbb{Z}},$

where $w_{n}\\!:=1/2$ if $n\leq 0$ and $w_{n}\\!:=2$ if $n\geq 1$. Given any
$\delta>0$, choose $n\in\mathbb{N}$ such that $2^{n}\delta>1$. Then,

$(0,\delta e_{n},2\delta e_{n-1},\ldots,2^{n-1}\delta e_{1},2^{n}\delta
e_{0},2^{n-1}\delta e_{-1},\ldots,2\delta e_{-n+1},\delta e_{-n},0)$

is a $\delta$-cycle for $T$ that cannot be $1$-shadowed by a periodic point of
$T$, since $\operatorname{Per}(T)=\\{0\\}$. Moreover,
$\operatorname{CR}(T)=X$, that is, $T$ is chain recurrent, but $T$ is not
Devaney chaotic.

Our next goal is to present additional conditions to guarantee the validity of
the shadowing property for generalized hyperbolic operators on locally convex
spaces.

Given a seminorm $\|\cdot\|$ on a vector space $X$, we define the kernel of
$\|\cdot\|$ by

$\operatorname{Ker}(\|\cdot\|)\\!:=\\{x\in X:\|x\|=0\\}.$

A sequence $(x_{n})_{n\in\mathbb{N}}$ in $X$ is said to be a Cauchy sequence
with respect to $\|\cdot\|$ if $\|x_{n}-x_{m}\|\to 0$ as $n,m\to\infty$. The
seminorm $\|\cdot\|$ is said to be complete if every Cauchy sequence
$(x_{n})_{n\in\mathbb{N}}$ with respect to $\|\cdot\|$ has a limit $x\in X$,
in the sense that $\|x_{n}-x\|\to 0$ as $n\to\infty$. In this case, note that
the set of all limits of the sequence $(x_{n})_{n\in\mathbb{N}}$ is exactly
$x+\operatorname{Ker}(\|\cdot\|)$.

######  Theorem 6.

Suppose that the topology of a locally convex space $X$ is induced by a
directed family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of complete seminorms.

  * (a)

If $T\in GL(X)$ is generalized hyperbolic and
$T(\operatorname{Ker}(\|\cdot\|_{\alpha}))=\operatorname{Ker}(\|\cdot\|_{\alpha})$
for all $\alpha\in I$, then $T$ has the shadowing property.

  * (b)

If $T\in L(X)$ is generalized hyperbolic and
$T(\operatorname{Ker}(\|\cdot\|_{\alpha}))\subset\operatorname{Ker}(\|\cdot\|_{\alpha})$
for all $\alpha\in I$, then $T$ has the positive shadowing property.

###### Proof.

(a): Let $X=M\oplus N$ be the topological direct sum decomposition given by
the generalized hyperbolicity of $T$ and let $P_{M}:X\to M$ and $P_{N}:X\to N$
be the canonical projections. Given a neighborhood $V$ of $0$ in $X$, there
exist $\theta\in I$ and $\varepsilon>0$ such that

$\\{x\in X:\|x\|_{\theta}<\varepsilon\\}\subset V.$

Choose $\alpha\in I$ and $a>0$ such that

(9) $\|x\|_{\theta}\leq\|x\|_{\alpha},\ \ \|Tx\|_{\theta}\leq a\|x\|_{\alpha}\
\text{ and }\ \|T^{-1}x\|_{\theta}\leq a\|x\|_{\alpha}\ \text{ for all }x\in
X.$

Let $\beta\in I$, $c>0$ and $t\in(0,1)$ be such that

$\|T^{n}y\|_{\alpha}\leq c\,t^{n}\|y\|_{\beta}\ \text{ and }\
\|T^{-n}z\|_{\alpha}\leq c\,t^{n}\|z\|_{\beta}\ \text{ for all }y\in M,z\in
N,n\in\mathbb{N}_{0}.$

Choose $\gamma\in I$ and $d>0$ such that

$\|P_{M}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \text{ and }\
\|P_{N}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \text{ for all }x\in X.$

Let $\eta\in I$ and $b\geq 1$ satisfy

$\|x\|_{\gamma}\leq\|x\|_{\eta}\ \text{ and }\ \|T^{-1}x\|_{\gamma}\leq
b\,\|x\|_{\eta}\ \text{ for all }x\in X.$

Put $\delta\\!:=\frac{(1-t)\varepsilon}{3\,b\,c\,d}>0$ and $U\\!:=\\{x\in
X:\|x\|_{\eta}<\delta\\}$. Let $(x_{j})_{j\in\mathbb{Z}}$ be a
$U$-pseudotrajectory of $T$ and define $y_{j}\\!:=x_{j+1}-Tx_{j}$ and
$z_{j}\\!:=x_{-j-1}-T^{-1}x_{-j}$ for each $j\in\mathbb{N}_{0}$. Then

$\|y_{j}\|_{\gamma}<\delta\ \text{ and }\ \|z_{j}\|_{\gamma}<b\,\delta\ \text{
for all }j\in\mathbb{N}_{0}.$

For each $k\in\mathbb{N}$, let

$u_{k}\\!:=\sum_{j=1}^{k}T^{-j}P_{N}y_{j-1}\ \ \text{ and }\ \
v_{k}\\!:=\sum_{j=1}^{k}T^{j}P_{M}z_{j-1}.$

For $m\in\mathbb{Z}$ and $j\in\mathbb{N}$, we have that

$\|T^{m-j}P_{N}y_{j-1}\|_{\alpha}\leq c\,t^{j-m}\|P_{N}y_{j-1}\|_{\beta}\leq
c\,d\,t^{j-m}\|y_{j-1}\|_{\gamma}<c\,d\,\delta\,t^{j-m}\ \text{ if }j\geq m,$

and

$\|T^{m+j}P_{M}z_{j-1}\|_{\alpha}\leq c\,t^{j+m}\|P_{M}z_{j-1}\|_{\beta}\leq
c\,d\,t^{j+m}\|z_{j-1}\|_{\gamma}<b\,c\,d\,\delta\,t^{j+m}\ \text{ if
}j\geq-m.$

This implies that the sequences $(T^{m}u_{k})_{k\in\mathbb{N}}$ and
$(T^{m}v_{k})_{k\in\mathbb{N}}$ are Cauchy with respect to
$\|\cdot\|_{\alpha}$. Since we are assuming that $\|\cdot\|_{\alpha}$ is
complete, there exist $p_{m},q_{m}\in X$ such that

$\|p_{m}-T^{m}u_{k}\|_{\alpha}\to 0\ \text{ and }\
\|q_{m}-T^{m}v_{k}\|_{\alpha}\to 0\ \text{ as }k\to\infty\ \ \ \
(m\in\mathbb{Z}).$

Let us prove that

(10) $\|T^{m}p_{0}-T^{m}u_{k}\|_{\theta}\to 0\ \text{ as }k\to\infty,\ \text{
for all }m\in\mathbb{Z}.$

The case $m=0$ is clear. Suppose that (10) holds for a certain
$m\in\mathbb{Z}$. Since

$\|p_{m-1}-T^{m-1}u_{k}\|_{\alpha}\to 0\ \text{ and }\
\|p_{m+1}-T^{m+1}u_{k}\|_{\alpha}\to 0\ \text{ as }k\to\infty,$

we obtain from (9) that

$\|Tp_{m-1}-T^{m}u_{k}\|_{\theta}\to 0\ \text{ and }\
\|T^{-1}p_{m+1}-T^{m}u_{k}\|_{\theta}\to 0\ \text{ as }k\to\infty.$

The induction hypothesis tell us that $\|T^{m}p_{0}-T^{m}u_{k}\|_{\theta}\to
0$ as $k\to\infty$. Therefore,

$Tp_{m-1}-T^{m}p_{0}\in\operatorname{Ker}(\|\cdot\|_{\theta})\ \ \text{ and }\
\ T^{-1}p_{m+1}-T^{m}p_{0}\in\operatorname{Ker}(\|\cdot\|_{\theta}).$

Since
$T(\operatorname{Ker}(\|\cdot\|_{\theta}))=\operatorname{Ker}(\|\cdot\|_{\theta})$,
we conclude that

$p_{m-1}-T^{m-1}p_{0}\in\operatorname{Ker}(\|\cdot\|_{\theta})\ \ \text{ and
}\ \ p_{m+1}-T^{m+1}p_{0}\in\operatorname{Ker}(\|\cdot\|_{\theta}),$

which implies that (10) holds with $m-1$ and with $m+1$ in the place of $m$.

Analogously,

(11) $\|T^{m}q_{0}-T^{m}v_{k}\|_{\theta}\to 0\ \text{ as }k\to\infty,\ \text{
for all }m\in\mathbb{Z}.$

Now, by arguing as in the proof of (6), we obtain

$x_{m}-T^{m}(x_{0}+u_{k})=\sum_{j=0}^{m-1}T^{j}P_{M}y_{m-j-1}-\sum_{j=1}^{k-m}T^{-j}P_{N}y_{m+j-1}$

and

$x_{-m}-T^{-m}(x_{0}+v_{k})=\sum_{j=0}^{m-1}T^{-j}P_{N}z_{m-j-1}-\sum_{j=1}^{k-m}T^{j}P_{M}z_{m+j-1},$

whenever $0\leq m\leq k$. Hence,

(12) $\displaystyle\|x_{m}-T^{m}(x_{0}+u_{k}+v_{k})\|_{\alpha}$
$\displaystyle\leq\|x_{m}-T^{m}(x_{0}+u_{k})\|_{\alpha}+\|T^{m}v_{k}\|_{\alpha}$
$\displaystyle\leq
c\,d\,\delta\Bigg{(}\sum_{j=0}^{m-1}t^{j}+\sum_{j=1}^{k-m}t^{j}\Bigg{)}+b\,c\,d\,\delta\sum_{j=1}^{k}t^{m+j}$

and

(13) $\displaystyle\|x_{-m}-T^{-m}(x_{0}+u_{k}+v_{k})\|_{\alpha}$
$\displaystyle\leq\|x_{-m}-T^{-m}(x_{0}+v_{k})\|_{\alpha}+\|T^{-m}u_{k}\|_{\alpha}$
$\displaystyle\leq
b\,c\,d\,\delta\Bigg{(}\sum_{j=0}^{m-1}t^{j}+\sum_{j=1}^{k-m}t^{j}\Bigg{)}+c\,d\,\delta\sum_{j=1}^{k}t^{m+j},$

whenever $0\leq m\leq k$. Consider the vector

$x\\!:=x_{0}+p_{0}+q_{0}.$

By fixing $m\in\mathbb{N}_{0}$ and letting $k\to\infty$, it follows from (10),
(11), (12) and (13) that

$\|x_{m}-T^{m}x\|_{\theta}\leq
b\,c\,d\,\delta\Bigg{(}\sum_{j=0}^{m-1}t^{j}+2\sum_{j=1}^{\infty}t^{j}\Bigg{)}<\varepsilon$

and

$\|x_{-m}-T^{-m}x\|_{\theta}\leq
b\,c\,d\,\delta\Bigg{(}\sum_{j=0}^{m-1}t^{j}+2\sum_{j=1}^{\infty}t^{j}\Bigg{)}<\varepsilon.$

Thus, the $U$-pseudotrajecotry $(x_{j})_{j\in\mathbb{Z}}$ of $T$ is
$V$-shadowed by the trajectory of $x$ under $T$, proving that $T$ has the
shadowing property.

(b): The proof is analogous (but slightly simpler) and is left to the reader.
∎

######  Remark 7.

As an immediate consequence of Theorem 6, we obtain the following known
results:

  * •

Every invertible generalized hyperbolic operator on a Banach space has the
shadowing property [7, Theorem A].

  * •

Every generalized hyperbolic operator on a Banach space has the positive
shadowing property [8, Proposition 19].

These results can also be derived directly from Theorem 2, since it was proved
in [11, Theorem 1] that (positive) shadowing and finite shadowing coincide for
operators on Banach spaces.

######  Example 8.

Let $H(\mathbb{C})$ be the Fréchet space of all entire functions endowed with
the compact-open topology. Given a zero-free entire function $\phi$, consider
the multiplication operator

$M_{\phi}:f\in H(\mathbb{C})\mapsto\phi\cdot f\in H(\mathbb{C}).$

We have that the following assertions are equivalent:

  * (i)

$M_{\phi}$ has the finite shadowing property;

  * (ii)

$M_{\phi}$ is generalized hyperbolic;

  * (iii)

$M_{\phi}$ is hyperbolic with trivial splitting;

  * (iv)

$\phi$ is a constant function with modulus $\neq 1$.

Indeed, (iv) $\Rightarrow$ (iii) is easy, (iii) $\Rightarrow$ (ii) is obvious,
and (ii) $\Rightarrow$ (i) follows from Theorem 2. It remains to prove that
(i) $\Rightarrow$ (iv). For this purpose, suppose that (i) is true and (iv) is
false. Then, there exists $z_{0}\in\mathbb{C}$ with $|\phi(z_{0})|=1$ and
there exist $H\subset\mathbb{C}$ compact and $\delta>0$ such that the
condition

(14) $|(M_{\phi}(f_{j}))(z)-f_{j+1}(z)|\leq\delta\ \ \text{ for all }z\in
H\text{ and }j\in\\{0,\ldots,k-1\\},$

where $k\in\mathbb{N}$ and $f_{0},\ldots,f_{k}\in H(\mathbb{C})$, implies the
existence of an $f\in H(\mathbb{C})$ with

(15) $|f_{j}(z_{0})-((M_{\phi})^{j}(f))(z_{0})|<1\ \ \text{ for all
}j\in\\{0,\ldots,k\\}.$

Note that (15) implies that

(16) $|f_{k}(z_{0})|<2+|f_{0}(z_{0})|.$

However, by defining $f_{0}\\!:=0$ and $f_{j}\\!:=\phi\cdot
f_{j-1}+\phi(z_{0})^{j-1}\cdot\delta$ for $j\geq 1$, we have that (14) holds
for every $k\in\mathbb{N}$, but (16) fails for $k$ large enough. This
contradiction completes the proof.

######  Remark 9.

We have just seen that the invertible multiplication operators on
$H(\mathbb{C})$ that have the finite shadowing property are exactly those of
the form

$M_{\lambda}:f\in H(\mathbb{C})\mapsto\lambda f\in H(\mathbb{C}),$

where $\lambda\in\mathbb{C}$ and $|\lambda|\not\in\\{0,1\\}$. Nevertheless, it
was proved in [11, Theorem 2] that none of these operators has the shadowing
property. More precisely, $M_{\lambda}$ has the finite shadowing property but
does not have the positive shadowing property whenever $|\lambda|>1$, and
$M_{\lambda}$ has the positive shadowing property but does not have the
shadowing property whenever $0<|\lambda|<1$ (see [11, Remark 3]). This shows
that, in general, an invertible hyperbolic operator may fail to have the
(positive) shadowing property, even in the Fréchet space setting. This also
shows that the hypothesis of completeness of the seminorms
$\|\cdot\|_{\alpha}$ is essential for the validity of Theorem 6.

In strong contrast to the case of the space $H(\mathbb{C})$ of entire
functions, we will now see that shadowing and finite shadowing coincide for
invertible multiplication operators on $C(\Omega)$ spaces.

######  Example 10.

Let $C(\Omega)$ be the complete locally convex space of all continuous maps
$f:\Omega\to\mathbb{K}$ endowed with the compact-open topology, where $\Omega$
is a locally compact Hausdorff space. Note that $C(\Omega)$ is a Fréchet space
if $\Omega$ is $\sigma$-compact. Given a zero-free continuous map
$\phi:\Omega\to\mathbb{K}$, consider the multiplication operator

$M_{\phi}:f\in C(\Omega)\mapsto\phi\cdot f\in C(\Omega).$

We have that the following assertions are equivalent:

  * (i)

$M_{\phi}$ has the finite shadowing property;

  * (ii)

$M_{\phi}$ has the shadowing property;

  * (iii)

$M_{\phi}$ is generalized hyperbolic;

  * (iv)

$M_{\phi}$ is hyperbolic;

  * (v)

$|\phi(x)|\neq 1$ for every $x\in\Omega$.

Indeed, (iv) $\Rightarrow$ (iii) is obvious, (iii) $\Rightarrow$ (ii) follows
from Theorem 6, (ii) $\Rightarrow$ (i) is obvious, and (i) $\Rightarrow$ (v)
is analogous to the proof of (i) $\Rightarrow$ (iv) in the previous example.
It remains to prove that (v) $\Rightarrow$ (iv). For this purpose, assume (v)
and define

$A\\!:=\\{x\in\Omega:|\phi(x)|<1\\}\ \ \text{ and }\ \ \
B\\!:=\\{x\in\Omega:|\phi(x)|>1\\}.$

In view of (v), we have that $\Omega=A\cup B$ and both $A$ and $B$ are
simultaneously open and closed in $\Omega$. Let

$M\\!:=\\{f\in C(\Omega):f=0\text{ on }B\\}\ \ \text{ and }\ \ N\\!:=\\{f\in
C(\Omega):f=0\text{ on }A\\},$

which are closed $M_{\phi}$-invariant subspaces of $C(\Omega)$. It is easy to
check that $C(\Omega)=M\oplus N$ as a topological direct sum. In order to
prove (GH3), consider a seminorm

$\|f\|_{K}\\!:=\max_{x\in K}|f(x)|\ \ \text{ for }f\in C(\Omega),$

where $K$ is a nonempty compact subset of $\Omega$, define

$t\\!:=\max\Big{\\{}\max_{x\in K\cap A}|\phi(x)|,\max_{x\in K\cap
B}\frac{1}{|\phi(x)|}\Big{\\}}<1,$

and note that

$\|(M_{\phi})^{n}f\|_{K}\leq t^{n}\|f\|_{K}\ \text{ and }\
\|(M_{\phi})^{-n}g\|_{K}\leq t^{n}\|g\|_{K},$

whenever $f\in M$, $g\in N$ and $n\in\mathbb{N}_{0}$. Hence, (iv) holds.

It follows immediately from the definition that the inverse $T^{-1}$, the
rotations $\lambda T$, $|\lambda|=1$, and the powers $T^{n}$,
$n\in\mathbb{N}$, of a (generalized) hyperbolic operator $T\in GL(X)$ are
(generalized) hyperbolic.

######  Proposition 11.

Suppose that $X$ is the product of a family $(X_{\ell})_{\ell\in L}$ of
locally convex spaces, $T_{\ell}\in GL(X_{\ell})$ for each $\ell\in L$, and
$T\in GL(X)$ is the product operator given by

$T((x_{\ell})_{\ell\in L})\\!:=(T_{\ell}x_{\ell})_{\ell\in L}.$

If each $T_{\ell}$ is (generalized) hyperbolic, then so is $T$.

###### Proof.

For each $\ell\in L$, let $(\|\cdot\|_{\alpha_{\ell}})_{\alpha_{\ell}\in
I_{\ell}}$ be a directed family of seminorms that induces the topology of
$X_{\ell}$. Recall that the product topology on $X$ is induced by the family
of seminorms given by

$\|(x_{\ell})_{\ell\in
L}\|_{\alpha_{\ell_{1}},\ldots,\alpha_{\ell_{k}}}\\!:=\max\\{\|x_{\ell_{1}}\|_{\alpha_{\ell_{1}}},\ldots,\|x_{\ell_{k}}\|_{\alpha_{\ell_{k}}}\\},$

for $k\in\mathbb{N}$, $\ell_{1},\ldots,\ell_{k}\in L$ and
$\alpha_{\ell_{1}}\in I_{\ell_{1}},\ldots,\alpha_{\ell_{k}}\in I_{\ell_{k}}$.

Suppose that each $T_{\ell}$ is generalized hyperbolic and let
$X_{\ell}=M_{\ell}\oplus N_{\ell}$ be the topological direct sum decomposition
given by the definition of generalized hyperbolicity. Consider the subspaces
$M\\!:=\prod_{\ell\in L}M_{\ell}$ and $N\\!:=\prod_{\ell\in L}N_{\ell}$ of
$X$. It is routine to verify that $X=M\oplus N$ as a topological direct sum
decomposition and that properties (GH1), (GH2’) and (GH3) hold.

If each $T_{\ell}$ is hyperbolic, then it is clear that $T$ is hyperbolic. ∎

Concrete examples of generalized hyperbolic operators on Banach (and Hilbert)
spaces that are not hyperbolic can be found in [7, 9]. In Section 5 we will
exhibit some examples of this type on the (non-normable) Fréchet space
$s(\mathbb{Z})$ of rapidly decreasing sequences on $\mathbb{Z}$.

##  3\. Topological stability for operators

Our main goal in the present section is to establish the following result.

######  Theorem 12.

Let $X$ be a locally convex space. If $T\in GL(X)$ is topologically stable,
then $T$ has the finite shadowing property and the strict periodic shadowing
property.

In view of Theorem 3, we obtain the following result.

######  Corollary 13.

Let $X$ be a locally convex space. If $T\in GL(X)$ is topologically stable,
then the following properties hold:

  * (a)

$\operatorname{Per}(T)$ is dense in $\operatorname{CR}(T)$.

  * (b)

$T|_{\operatorname{CR}(T)}$ has the finite shadowing property and the strict
periodic shadowing property.

  * (c)

$T|_{\operatorname{CR}(T)}$ is topologically mixing and Devaney chaotic.

Since the notions of shadowing and finite shadowing coincide for operators on
Banach spaces [11, Theorem 1], the above theorem implies the following result.

######  Corollary 14.

Let $X$ be a Banach space. If $T\in GL(X)$ is topologically stable, then $T$
has the shadowing property and the strict periodic shadowing property.

The precise definition of topological stability in the present context will be
given later. In order to prove Theorem 12, we shall introduce a certain
concept of multihomogeneity for uniform spaces. So, let us begin by fixing
some notations regarding uniform spaces (see [17, Chapter II] for the basics
on uniform spaces). Consider a uniform space $X$ with uniformity
$\mathcal{U}$. Given $A\subset X$ and $U\in\mathcal{U}$, the set

$U(A)\\!:=\\{y\in X:(x,y)\in U\text{ for some }x\in A\\}$

is called the $U$-neighborhood of $A$ in $X$. If $A$ is a singleton, say
$A=\\{x\\}$, it is usual to write $U(x)$ instead of $U(\\{x\\})$. Recall that,
for each $x\in X$, the family $\\{U(x):U\in\mathcal{U}\\}$ constitutes a
fundamental system of neighborhoods of $x$ in the topology induced by the
uniformity $\mathcal{U}$. Given $U,V\in\mathcal{U}$, we define

$U\circ V\\!:=\\{(x,y)\in X\times X:(x,z)\in V\text{ and }(z,y)\in U\text{ for
some }z\in X\\}.$

Finally, $U\in\mathcal{U}$ is said to be symmetric if $(y,x)\in U$ whenever
$(x,y)\in U$.

Now, recall that a topological space $Y$ is said to be homogeneous if for any
$a,b\in Y$, there is a homeomorphism $h:Y\to Y$ with $h(a)=b$. This concept
was introduced by Sierpiński [45] in 1920. Since then, many variations,
including several notions of local homogeneity, have been introduced and
investigated by several authors. The definition given below can be seen as a
“multihomogeneous” version of Ford’s concept [23] of a strongly locally
homogeneous space.

######  Definition 15.

A uniform space $X$ with uniformity $\mathcal{U}$ is said to be strongly
locally multihomogeneous if for every $V\in\mathcal{U}$, there exists
$U\in\mathcal{U}$ such that $U\subset V$ and for any integer $k\geq 1$, any
pairwise distinct points $a_{1},\ldots,a_{k}\in X$ and any pairwise distinct
points $b_{1},\ldots,b_{k}\in X$ with $(a_{j},b_{j})\in U$ for all
$j\in\\{1,\ldots,k\\}$, there is a homeomorphism $h:X\to X$ satisfying the
following conditions:

  * (a)

$h(a_{j})=b_{j}$ for all $j\in\\{1,\ldots,k\\}$;

  * (b)

$h(U(a_{j}))\subset V(a_{j})$ for all $j\in\\{1,\ldots,k\\}$;

  * (c)

$h(x)=x$ for all $x\in X\backslash(U(a_{1})\cup\ldots\cup U(a_{k}))$.

######  Remark 16.

If $X$ is strongly locally multihomogeneous, then the following property
holds:

(P) For each $V\in\mathcal{U}$, there exists $U\in\mathcal{U}$ such that for
any integer $k\geq 1$, any pairwise distinct points $a_{1},\ldots,a_{k}\in X$
and any pairwise distinct points $b_{1},\ldots,b_{k}\in X$ with
$(a_{j},b_{j})\in U$ for all $j\in\\{1,\ldots,k\\}$, there is a homeomorphism
$h:X\to X$ such that

  * •

$h(a_{j})=b_{j}$ for all $j\in\\{1,\ldots,k\\}$;

  * •

$(x,h(x))\in V$ for all $x\in X$.

Property (P) was called property∗ in [29] in the setting of compact metric
spaces. It is well-known that closed topological manifolds of dimension at
least two have property (P) (see [3, Lemma 2.4.11], for instance) and it was
proved in [29] that any Cantor space $(X,d)$ has property (P). In the sequel
we shall prove that open convex sets in locally convex spaces of dimension at
least two also have property (P).

######  Theorem 17.

Every open convex set in a locally convex space of dimension at least three is
strongly locally multihomogeneous.

For the proof we will need the lemma below.

######  Lemma 18.

Let $C$ be an open connected set in a locally convex space $X$. For any
$a,b\in C$, there is a homeomorphism $h:X\to X$ such that

$h(a)=b\ \ \text{ and }\ \ h(x)=x\text{ for all }x\in X\backslash C.$

###### Proof.

We shall divide the proof in five steps.

Step 1. Let $U\\!:=\\{x\in X:\|x-x_{0}\|<r\\}$, where $\|\cdot\|$ is a
continuous seminorm on $X$, $x_{0}\in X$ and $r>0$. For any $a,b\in U$ with
$\|a-x_{0}\|>0$ and $b-x_{0}=\lambda(a-x_{0})$ for some $\lambda>0$, there is
a homeomorphism $h:X\to X$ such that

$h(a)=b\ \ \textit{ and }\ \ h(x)=x\textit{ for all }x\in(X\backslash
U)\cup(x_{0}+\operatorname{Ker}\|\cdot\|).$

It is enough to consider the case where $x_{0}=0$ and $r=1$. Choose
$k\in\mathbb{N}$ such that

$1<\lambda(1+k\|a\|)<1+k.$

Let $\phi:[1,\infty)\to[1,\infty)$ be the map that linearly interpolates the
points

$(1,1),\ \ (1+k\|a\|,\lambda(1+k\|a\|))\ \text{ and }\ (1+k,1+k),$

and equals the identity on $[1+k,\infty)$. We define $h:X\to X$ by

$h(x)\\!:=\frac{\phi(1+k\|x\|)}{1+k\|x\|}\,x.$

Note that $h$ is continuous, $h(a)=b$ and $h(x)=x$ whenever $\|x\|=0$ or
$\|x\|\geq 1$. In order to prove that $h$ is bijective, we consider the
auxiliary function

$\psi:t\in[0,\infty)\mapsto\frac{\phi(1+kt)\,t}{1+kt}\in[0,\infty),$

which is a homeomorphism. We define $g:X\to X$ by

$g(x)\\!:=x\ \text{ if }\|x\|=0,\ \ \
g(x)\\!:=\frac{\psi^{-1}(\|x\|)}{\|x\|}\,x\ \text{ if }\|x\|>0.$

Simple computations show that $h(g(x))=g(h(x))=x$ for all $x\in X$, that is,
$h$ is bijective and $g$ is its inverse. It remains to prove that $g$ is
continuous. For this purpose, suppose that a net $(x_{\alpha})_{\alpha\in I}$
in $X\backslash\operatorname{Ker}\|\cdot\|$ converges to a point
$x\in\operatorname{Ker}\|\cdot\|$ in the topology of $X$. We have to prove
that $(g(x_{\alpha}))_{\alpha\in I}$ converges to $x$ in the topology of $X$.
Since

$g(x_{\alpha})=\frac{\psi^{-1}(\|x_{\alpha}\|)}{\|x_{\alpha}\|}\,x_{\alpha}\ \
\text{ for all }\alpha\in I,$

it is enough to show that

$\lim_{t\to 0^{+}}\frac{\psi^{-1}(t)}{t}=1,$

which can be obtained from an application of L’Hôpital’s rule.

Step 2. Let $U\\!:=\\{x\in X:\|x-x_{0}\|<r\\}$, where $\|\cdot\|$ is a
continuous seminorm on $X$, $x_{0}\in X$ and $r>0$. For any $a\in U$ with
$\|a-x_{0}\|>0$, there is a homeomorphism $h:X\to X$ such that

$h(a)=x_{0}\ \ \textit{ and }\ \ h(x)=x\textit{ for all }x\in X\backslash U.$

Indeed, let

$t\\!:=\frac{r-\|a-x_{0}\|}{3}>0\ \ \text{ and }\ \
r^{\prime}\\!:=r-t=\frac{2r+\|a-x_{0}\|}{3}>0.$

Let

$x^{\prime}_{0}\\!:=x_{0}-t\cdot\frac{a-x_{0}}{\|a-x_{0}\|}\ \ \text{ and }\ \
U^{\prime}\\!:=\\{x\in X:\|x-x^{\prime}_{0}\|<r^{\prime}\\}.$

Simple computations show that $x_{0}\in U^{\prime}$ and $U^{\prime}\subset U$.
Moreover,

$a-x^{\prime}_{0}=\frac{\|a-x_{0}\|+t}{\|a-x_{0}\|}\,(a-x_{0})=\frac{\|a-x_{0}\|+t}{t}\,(x_{0}-x^{\prime}_{0}),$

which implies that $a\in U^{\prime}$, $\|a-x^{\prime}_{0}\|>0$ and
$x_{0}-x^{\prime}_{0}=\lambda(a-x^{\prime}_{0})$, where

$\lambda\\!:=\frac{t}{\|a-x_{0}\|+t}>0.$

Hence, by Step 1, there is a homeomorphism $h:X\to X$ such that $h(a)=x_{0}$
and $h(x)=x$ for all $x\in X\backslash U^{\prime}$. Since $U^{\prime}\subset
U$, we are done.

Step 3. Let $U\\!:=\\{x\in X:\|x-x_{0}\|<r\\}$, where $\|\cdot\|$ is a
continuous seminorm on $X$, $x_{0}\in X$ and $r>0$. For any $a,b\in U$, there
is a homeomorphism $h:X\to X$ such that

$h(a)=b\ \ \textit{ and }\ \ h(x)=x\textit{ for all }x\in X\backslash U.$

We may assume that $\|\cdot\|$ is not identically zero. Hence, for any
$\varepsilon>0$, there exists $x^{\prime}_{0}\in X$ such that
$\|x^{\prime}_{0}-x_{0}\|<\varepsilon$, $\|a-x^{\prime}_{0}\|>0$ and
$\|b-x^{\prime}_{0}\|>0$. In particular,

$U^{\prime}\\!:=\\{x\in X:\|x-x^{\prime}_{0}\|<r-\varepsilon\\}\subset U.$

Moreover, by choosing $\varepsilon>0$ small enough, we have that $a,b\in
U^{\prime}$. Thus, by Step 2, there are homeomorphisms $f:X\to X$ and $g:X\to
X$ such that

$f(a)=x^{\prime}_{0},\ \ g(b)=x^{\prime}_{0}\ \text{ and }\ f(x)=g(x)=x\text{
for all }x\in X\backslash U^{\prime}.$

Hence, $h\\!:=g^{-1}\circ f$ does the job.

Step 4. Proof of the case where $C$ is convex.

Let $L$ be the line segment joining $a$ and $b$, that is,

$L\\!:=\\{(1-t)a+tb:t\in[0,1]\\}.$

Since we are assuming that $C$ is convex, $L\subset C$. Since $L$ is compact,
there is a continuous seminorm $\|\cdot\|$ on $X$ such that the open convex
neighborhood

$V\\!:=\\{x\in X:\|x\|<1\\}$

of $0$ in $X$ satisfies

$L+V\subset C.$

By compactness, there are finitely many points $c_{1},\ldots,c_{k}\in L$ such
that

$L\subset(c_{1}+V)\cup\ldots\cup(c_{k}+V).$

Moreover, we may assume that $a\in c_{1}+V$, $b\in c_{k}+V$ and, for each
$1\leq j\leq k-1$, we can choose a point

$d_{j}\in(c_{j}+V)\cap(c_{j+1}+V)\cap L.$

By Step 3, there are homeomorphisms $h_{1},\ldots,h_{k}$ from $X$ onto itself
such that

$h_{1}(a)=d_{1},\ h_{2}(d_{1})=d_{2},\ \ldots,\ h_{k-1}(d_{k-2})=d_{k-1},\
h_{k}(d_{k-1})=b,\text{ and}$ $h_{j}(x)=x\ \ \text{ for all }x\in
X\backslash(c_{j}+V)\ \ \ (1\leq j\leq k).$

Consequently, the homeomorphism $h\\!:=h_{k}\circ\cdots\circ h_{1}$ has the
desired properties.

Step 5. Proof of the general case.

Since $C$ is open and connected, there is a piecewise linear path
$\psi:[0,1]\to X$ such that

$\psi(0)=a,\ \ \psi(1)=b\ \text{ and }\ P\\!:=\psi([0,1])\subset C.$

We can write $P=L_{1}\cup\ldots\cup L_{k}$, where each $L_{j}$ is a line
segment, $a$ is the initial point of $L_{1}$, $b$ is the final point of
$L_{k}$ and, for each $j\in\\{1,\ldots,k-1\\}$, the final point $d_{j}$ of
$L_{j}$ coincides with the initial point of $L_{j+1}$. Let $V$ be an open
convex neighborhood of $0$ in $X$ such that $P+V\subset C$. Since each set
$L_{j}+V$ is open and convex, Step 4 gives us homeomorphisms
$h_{1},\ldots,h_{k}$ from $X$ onto itself such that

$h_{1}(a)=d_{1},\ h_{2}(d_{1})=d_{2},\ \ldots,\ h_{k-1}(d_{k-2})=d_{k-1},\
h_{k}(d_{k-1})=b,\text{ and}$ $h_{j}(x)=x\ \ \text{ for all }x\in
X\backslash(L_{j}+V)\ \ \ (1\leq j\leq k).$

Hence, $h\\!:=h_{k}\circ\cdots\circ h_{1}$ is the homeomorphism we were
looking for. ∎

We are now in position to prove Theorem 17.

###### Proof.

Let $C$ be an open convex set in a locally convex space $X$ of dimension at
least three. Recall that a basis for the uniformity of $C$ (induced by that of
$X$) is given by the sets

$\widetilde{V}\\!:=\\{(x,y)\in C\times C:y-x\in V\\},$

as $V$ runs through the set of all neighborhoods of $0$ in $X$. Note that

$\widetilde{V}(x)=(x+V)\cap C\ \ \text{ for all }x\in C.$

Given a neighborhood $V$ of $0$ in $X$, choose a neighborhood $W$ of $0$ in
$X$ with $W+W\subset V$. Let $U$ be an open absolutely convex neighborhood of
$0$ in $X$ with $U+U+U+U\subset W$. Take $k\geq 1$, $a_{1},\ldots,a_{k}\in C$
pairwise distinct and $b_{1},\ldots,b_{k}\in C$ pairwise distinct with
$(a_{j},b_{j})\in\widetilde{U}$ for all $j\in\\{1,\ldots,k\\}$. We have to
find a homeomorphism $h:C\to C$ with the following properties:

  * (a)

$h(a_{j})=b_{j}$ for all $j\in\\{1,\ldots,k\\}$;

  * (b)

$h(\widetilde{U}(a_{j}))\subset\widetilde{V}(a_{j})$ for all
$j\in\\{1,\ldots,k\\}$;

  * (c)

$h(x)=x$ for all $x\in
C\backslash\big{(}\widetilde{U}(a_{1})\cup\ldots\cup\widetilde{U}(a_{k})\big{)}$.

Let $U^{\prime}$ be an open absolutely convex neighborhood of $0$ in $X$ such
that $U^{\prime}\subset U$,

$b_{j}+U^{\prime}\subset\widetilde{U}(a_{j})\ \ \text{ for all
}j\in\\{1,\ldots,k\\},$

and the sets $b_{1}+U^{\prime},\ldots,b_{k}+U^{\prime}$ are pairwise disjoint.
Choose points $b^{\prime}_{j}\in b_{j}+U^{\prime}$, $j\in\\{1,\ldots,k\\}$, so
that the points $a_{1},\ldots,a_{k},b^{\prime}_{1},\ldots,b^{\prime}_{k}$ are
pairwise distinct. Since $X$ has dimension at least three and each
$\widetilde{U}(a_{j})$ is open and convex, there exist piecewise linear paths

$\psi_{j}:[0,1]\to X,\ \ \ j\in\\{1,\ldots,k\\},$

such that

$\psi_{j}(0)=a_{j},\ \ \psi_{j}(1)=b^{\prime}_{j},\ \
P_{j}\\!:=\psi_{j}([0,1])\subset\widetilde{U}(a_{j})\ \ \
(j\in\\{1,\ldots,k\\})$

and the sets $P_{1},\ldots,P_{k}$ are pairwise disjoint. Let $Z$ be an open
convex neighborhood of $0$ in $X$ such that

$C_{j}\\!:=P_{j}+Z\subset\widetilde{U}(a_{j})\ \ \text{ for all
}j\in\\{1,\ldots,k\\},$

and the sets $C_{1},\ldots,C_{k}$ are pairwise disjoint. Since $C_{j}$ is open
and connected, Lemma 18 guarantees that there is a homeomorphism $g_{j}:X\to
X$ such that

$g_{j}(a_{j})=b^{\prime}_{j}\ \ \text{ and }\ \ g_{j}(x)=x\text{ for all }x\in
X\backslash C_{j}.$

Consider the homeomorphism $g\\!:=g_{k}\circ\cdots\circ g_{1}:X\to X$. Also by
Lemma 18, there is a homeomorphism $f_{j}:X\to X$ such that

$f_{j}(b^{\prime}_{j})=b_{j}\ \ \text{ and }\ \ f_{j}(x)=x\text{ for all }x\in
X\backslash(b_{j}+U^{\prime}).$

Consider the homeomorphism $f\\!:=f_{k}\circ\cdots\circ f_{1}:X\to X$. Since
the homeomorphism $f\circ g:X\to X$ maps $C$ onto $C$, it induces a
homeomorphism $h:C\to C$. It is clear that properties (a) and (c) hold. For
each $x\in C$, since $g(x)\in x+U+U$ and $f(x)\in x+U^{\prime}+U^{\prime}$, we
have that

$h(x)=f(g(x))\in(x+U+U+U+U)\cap C\subset\widetilde{W}(x).$

This implies property (b) and completes the proof. ∎

######  Corollary 19.

Every open convex set in a locally convex space of dimension at least two has
property (P).

###### Proof.

Let $C$ be an open convex set in a locally convex space $X$ of dimension at
least two. If $\dim X\geq 3$, then the result follows from the previous
theorem. If $\dim X=2$, then we may assume that $X=\mathbb{R}^{2}$ endowed
with the max norm. Given $\delta>0$, consider the following collections of
pairwise disjoint open convex sets:

$\displaystyle\mathcal{C}_{1}\\!$
$\displaystyle:=\big{\\{}\big{(}\,]2n\delta,(2n+2)\delta[\ \times\
]2m\delta,(2m+2)\delta[\,\big{)}\cap C:n,m\in\mathbb{Z}\big{\\}},$
$\displaystyle\mathcal{C}_{2}\\!$
$\displaystyle:=\big{\\{}\big{(}\,](2n+1)\delta,(2n+3)\delta[\ \times\
]2m\delta,(2m+2)\delta[\,\big{)}\cap C:n,m\in\mathbb{Z}\big{\\}},$
$\displaystyle\mathcal{C}_{3}\\!$
$\displaystyle:=\big{\\{}\big{(}\,]2n\delta,(2n+2)\delta[\ \times\
](2m+1)\delta,(2m+3)\delta[\,\big{)}\cap C:n,m\in\mathbb{Z}\big{\\}},$
$\displaystyle\mathcal{C}_{4}\\!$
$\displaystyle:=\big{\\{}\big{(}\,](2n+1)\delta,(2n+3)\delta[\ \times\
](2m+1)\delta,(2m+3)\delta[\,\big{)}\cap C:n,m\in\mathbb{Z}\big{\\}}.$

Take $k\geq 1$, $a_{1},\ldots,a_{k}\in C$ pairwise distinct and
$b_{1},\ldots,b_{k}\in C$ pairwise distinct with $\|a_{j}-b_{j}\|<\delta$ for
all $j\in\\{1,\ldots,k\\}$. An argument used in the previous proof shows that
it is enough to consider the case in which
$a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}$ are pairwise distinct. Let

$\displaystyle I_{1}\\!$ $\displaystyle:=\\{j\in\\{1,\ldots,k\\}:\text{some
}A\in\mathcal{C}_{1}\text{ contains both }a_{j}\text{ and }b_{j}\\},$
$\displaystyle I_{2}\\!$ $\displaystyle:=\\{j\in\\{1,\ldots,k\\}\backslash
I_{1}:\text{some }A\in\mathcal{C}_{2}\text{ contains both }a_{j}\text{ and
}b_{j}\\},$ $\displaystyle I_{3}\\!$
$\displaystyle:=\\{j\in\\{1,\ldots,k\\}\backslash(I_{1}\cup I_{2}):\text{some
}A\in\mathcal{C}_{3}\text{ contains both }a_{j}\text{ and }b_{j}\\},$
$\displaystyle I_{4}\\!$
$\displaystyle:=\\{j\in\\{1,\ldots,k\\}\backslash(I_{1}\cup I_{2}\cup
I_{3}):\text{some }A\in\mathcal{C}_{4}\text{ contains both }a_{j}\text{ and
}b_{j}\\}.$

Then $I_{1}\cup\ldots\cup I_{4}=\\{1,\ldots,k\\}$. Now, for each
$\ell\in\\{1,\ldots,4\\}$, we consider a homeomorphism $h_{\ell}:C\to C$ in
the following way: if $A\in\mathcal{C}_{\ell}$ does not contain any pair
$a_{j},b_{j}$ with $j\in I_{\ell}$, then $h_{\ell}$ is the identity map on
$\overline{A}$ (closure relative to $C$); if $A\in\mathcal{C}_{\ell}$ contains
such a pair $a_{j},b_{j}$, then $h_{\ell}$ maps $\overline{A}$ onto itself,
maps $a_{j}$ to $b_{j}$ for each $a_{j},b_{j}\in A$ with $j\in I_{\ell}$ and
equals the identity on the boundary of $A$ (relative to $C$) and at each of
the other $a_{i}$:s and $b_{i}$:s. Then, the homeomorphism
$h\\!:=h_{4}\circ\cdots\circ h_{1}:C\to C$ maps $a_{j}$ to $b_{j}$ for all $j$
and satisfies $\|h(x)-x\|<8\delta$ for all $x\in C$. This proves that $C$ has
property (P). ∎

######  Remark 20.

In the case of complex scalars, Corollary 19 holds without any restriction on
the dimension of the space. But in the case of real scalars, the result is
clearly false if the dimension of the space is one.

Let us now define the concepts of finite shadowing and strict periodic
shadowing in the setting of uniform spaces. Let $X$ be a uniform space with
uniformity $\mathcal{U}$ and let $f:X\to X$ be a continuous map. Given
$U\in\mathcal{U}$, a finite sequence $(x_{j})_{j=0}^{k}$ is said to be a
$U$-chain (resp. a $U$-cycle) for $f$ if

$(f(x_{j}),x_{j+1})\in U\ \ \text{ for all }j\in\\{0,\ldots,k-1\\}\ \ \
(\text{resp.\ and }x_{k}=x_{0}).$

The map $f$ has the finite shadowing property (resp. the strict periodic
shadowing property) if for every $V\in\mathcal{U}$, there exists
$U\in\mathcal{U}$ such that for any $U$-chain (resp. $U$-cycle)
$(x_{j})_{j=0}^{k}$ for $f$, there exists $a\in X$ with

$(x_{j},f^{j}(a))\in V\ \ \text{ for all }j\in\\{0,\ldots,k\\}\ \ \
(\text{resp.\ and }f^{k}(a)=a).$

The classical notion of topological stability for homeomorphisms on compact
metric spaces can be extended to the uniform space setting in the following
natural way.

######  Definition 21.

Let $X$ be a uniform space with uniformity $\mathcal{U}$. A homeomorphism
$h:X\to X$ is said to be topologically stable if for every $V\in\mathcal{U}$,
there exists $U\in\mathcal{U}$ such that for any homeomorphism $g:X\to X$ with

$(h(x),g(x))\in U\ \ \text{ for all }x\in X,$

there is a continuous map $\phi:X\to X$ satisfying

$h\circ\phi=\phi\circ g\ \ \ \text{ and }\ \ \ (x,\phi(x))\in V\text{ for all
}x\in X.$

The concept of topological stability was introduced by Walters [48] in 1970.
Later Walters [49] established the following result:  If $X$ is a closed
topological manifold of dimension at least two and $h:X\to X$ is a
topologically stable homeomorphism, then $h$ has the shadowing property.

Recently, Kawaguchi [29] observed that Walters’ original arguments can be used
to prove the following more general result:  If $X$ is a perfect compact
metric space with property (P) and $h:X\to X$ is a topologically stable
homeomorphism, then $h$ has the shadowing property and the strict periodic
shadowing property.

It is an amazing fact that Walters’ original arguments can actually be adapted
to establish the following very general result.

######  Theorem 22.

Let $X$ be a Hausdorff uniform space with uniformity $\mathcal{U}$. If $X$ has
property (P) and no isolated point, then every topologically stable
homeomorphism $h:X\to X$ has the finite shadowing property and the strict
periodic shadowing property.

Recall that the notions of shadowing and finite shadowing coincide for
homeomorphisms on compact metric spaces [41, Lemma 1.1.1]. With this basic
result in hand, we see that Theorem 22 generalizes the above-mentioned
results.

###### Proof.

Fix $V\in\mathcal{U}$ and choose $V^{\prime}\in\mathcal{U}$ with
$V^{\prime}\circ V^{\prime}\subset V$. Take a symmetric
$W^{\prime}\in\mathcal{U}$ associated to $V^{\prime}$ according to the
topological stability of $h$. Let $U^{\prime}\in\mathcal{U}$ be associated to
$W^{\prime}$ according to the fact that $X$ has property (P) and take a
symmetric $U\in\mathcal{U}$ with $U\circ U\circ U\subset U^{\prime}$. Given a
$U$-chain $(x_{j})_{j=0}^{k}$ for $h$, we will find a point $a\in X$ such that

$(x_{j},h^{j}(a))\in V\ \ \text{ for all }j\in\\{0,\ldots,k\\}.$

By the continuity of $h$, there exists $Z\in\mathcal{U}$, $Z\subset U\cap
V^{\prime}$, such that

$h(Z(x_{j}))\subset U(h(x_{j}))\ \ \text{ for all }j\in\\{0,\ldots,k\\}.$

Since $X$ is a Hausdorff space without isolated points, we can choose points
$x^{\prime}_{j}\in Z(x_{j})$, $j\in\\{0,\ldots,k\\}$, so that
$x^{\prime}_{0},\ldots,x^{\prime}_{k}$ are pairwise distinct. Since

$(h(x^{\prime}_{j}),x^{\prime}_{j+1})\in U^{\prime}\ \ \text{ for all
}j\in\\{0,\ldots,k-1\\},$

property (P) gives us a homeomorphism $f:X\to X$ such that

$f(h(x^{\prime}_{j}))=x^{\prime}_{j+1}\ \ \text{ for all
}j\in\\{0,\ldots,k-1\\}$

and

$(x,f(x))\in W^{\prime}\ \ \text{ for all }x\in X.$

Consider the homeomorphism $g\\!:=f\circ h$. By topological stability, there
is a continuous map $\phi:X\to X$ such that

$h\circ\phi=\phi\circ g\ \ \ \text{ and }\ \ \ (x,\phi(x))\in V^{\prime}\text{
for all }x\in X.$

Put $a\\!:=\phi(x^{\prime}_{0})\in X$. Since
$g^{j}(x^{\prime}_{0})=x^{\prime}_{j}$, we get

$(x_{j},h^{j}(a))=(x_{j},\phi(g^{j}(x^{\prime}_{0})))=(x_{j},\phi(x^{\prime}_{j}))\in
V\ \ \ (j\in\\{0,\ldots,k\\}).$

This proves that $h$ has the finite shadowing property.

In the case in which $(x_{j})_{j=0}^{k}$ is a $U$-cycle for $h$, the points
$x^{\prime}_{j}\in Z(x_{j})$, $j\in\\{0,\ldots,k\\}$, can be chosen so that
$x^{\prime}_{0},\ldots,x^{\prime}_{k-1}$ are pairwise distinct and
$x^{\prime}_{k}=x^{\prime}_{0}$. With this choice, we have that the point $a$
has the following additional property:

$h^{k}(a)=h^{k}(\phi(x^{\prime}_{0}))=\phi(g^{k}(x^{\prime}_{0}))=\phi(x^{\prime}_{k})=a.$

This proves that $h$ has the strict periodic shadowing property. ∎

In view of Corollary 19 and Theorem 22, we obtain the following result.

######  Corollary 23.

Let $C$ be an open convex set in a locally convex space $X$ of dimension at
least two. If a homeomorphism $h:C\to C$ is topologically stable, then $h$ has
the finite shadowing property and the strict periodic shadowing property.

Theorem 12 is a special case of the above corollary in the case of spaces of
dimension greater than $1$. But in the case of the real line $\mathbb{R}$, it
is easy to see that if $T\in GL(\mathbb{R})$ is topologically stable, then $T$
must be hyperbolic, and so it has the shadowing property and the strict
periodic shadowing property. This completes the proof of Theorem 12.

Our next goal is to investigate the topological stability of generalized
hyperbolic operators. For this purpose, recall that a subset $E$ of a locally
convex space $X$ is said to be bounded if for each neighborhood $V$ of $0$ in
$X$, there exists $r>0$ such that $E\subset\lambda V$ whenever $|\lambda|\geq
r$. If $(\|\cdot\|_{\alpha})_{\alpha\in I}$ is a family of seminorms inducing
the topology of $X$, then this is equivalent to say that $E$ is bounded with
respect to $\|\cdot\|_{\alpha}$ for every $\alpha\in I$.

Given locally convex spaces $X$ and $Y$, we denote by $F_{b}(X;Y)$ (resp.
$C_{b}(X;Y)$, $U_{b}(X;Y)$) the vector space of all bounded maps (resp. all
bounded continuous maps, all bounded uniformly continuous maps) from $X$ into
$Y$. The next lemma generalizes Step 1 of the proof of [10, Theorem 1]. The
proof given below is based on the original proof from [10].

######  Lemma 24.

Let $X$ be a sequentially complete locally convex space. Let $T\in GL(X)$ be a
generalized hyperbolic operator, let

$X=M\oplus N$

be the direct sum decomposition given by the definition of generalized
hyperbolicity, and let $P_{M}:X\to M$ and $P_{N}:X\to N$ be the canonical
projections. Consider the closed subspace $Y\\!:=M+T^{-1}(N)$ of $X$. For any
bijective map $R:X\to X$, the linear map

$\Psi:\varphi\in F_{b}(X;Y)\mapsto\varphi\circ R-T\circ\varphi\in F_{b}(X;X)$

is bijective and its inverse is given by

(17)
$\Psi^{-1}(\phi)(x)=\sum_{k=0}^{\infty}T^{k}P_{M}(\phi(R^{-k-1}x))-\sum_{k=1}^{\infty}T^{-k}P_{N}(\phi(R^{k-1}x)).$

Moreover, if $R$ is a homeomorphism (resp. a uniform homeomorphism), then

$\Psi(C_{b}(X;Y))=C_{b}(X;X)\ \ \ (\text{resp.}\
\Psi(U_{b}(X;Y))=U_{b}(X;X)).$

###### Proof.

Let $(\|\cdot\|_{\alpha})_{\alpha\in I}$ be a directed family of seminorms
inducing the topology of $X$. By the definition of generalized hyperbolicity,
for each $\alpha\in I$, there exist $\beta_{\alpha}\in I$, $c_{\alpha}>0$ and
$t_{\alpha}\in(0,1)$ such that

$\|T^{n}y\|_{\alpha}\leq c_{\alpha}\,t_{\alpha}^{n}\,\|y\|_{\beta_{\alpha}}\ \
\text{ and }\ \ \|T^{-n}z\|_{\alpha}\leq
c_{\alpha}\,t_{\alpha}^{n}\,\|z\|_{\beta_{\alpha}},$

whenever $y\in M$, $z\in N$ and $n\in\mathbb{N}_{0}$. We shall now divide the
proof in three steps.

Step 1. $\Psi$ is injective.

Suppose that $\varphi\in F_{b}(X;Y)$ and $\Psi(\varphi)=0$. Then $\varphi\circ
R^{n}=T^{n}\circ\varphi$ for all $n\in\mathbb{N}$, which gives the following
formulas

(18) $\varphi(x)=T^{-n}(\varphi(R^{n}x))\ \ \text{ and }\ \
\varphi(x)=T^{n}(\varphi(R^{-n}x))\ \ \ \ \ (x\in X,n\in\mathbb{N}).$

By the first of these equalities, $\varphi(x)=y_{n}(x)+z_{n}(x)$, where

$y_{n}(x)=T^{-n}P_{M}(\varphi(R^{n}x))\ \ \text{ and }\ \
z_{n}(x)=T^{-n}P_{N}(\varphi(R^{n}x)).$

Clearly, $z_{n}(x)\in T^{-1}(N)$ for all $x\in X$ and $n\in\mathbb{N}$. Write

$y_{n}(x)=a_{n}(x)+b_{n}(x)\ \text{ with }a_{n}(x)\in M\text{ and }b_{n}(x)\in
N.$

We claim that $y_{n}(x)\in M$ (i.e., $b_{n}(x)=0$) for all $x$ and $n$. For
$n=1$,

$T(b_{1}(x))=T(y_{1}(x))-T(a_{1}(x))=P_{M}(\varphi(Rx))-T(a_{1}(x))\in M.$

Since $\varphi(x)=a_{1}(x)+(b_{1}(x)+z_{1}(x))$, $a_{1}(x)\in M$ and
$b_{1}(x)+z_{1}(x)\in N$, the fact that $\varphi(x)\in Y$ implies that
$b_{1}(x)\in T^{-1}(M)\cap T^{-1}(N)=\\{0\\}$. Suppose that, for a certain
$n\geq 1$, $b_{n}(x)=0$ for all $x$. Then,

$T(b_{n+1}(x))=T(y_{n+1}(x))-T(a_{n+1}(x))=y_{n}(Rx)-T(a_{n+1}(x))\in M.$

By arguing as above, we conclude that $b_{n+1}(x)=0$. By induction, our claim
is proved. This implies that $P_{N}(\varphi(x))=z_{n}(x)$ for all
$n\in\mathbb{N}$. Hence, for each $\alpha\in I$,

$\|P_{N}(\varphi(x))\|_{\alpha}\leq
c_{\alpha}t_{\alpha}^{n}\|P_{N}(\varphi(R^{n}x))\|_{\beta_{\alpha}}\to 0\
\text{ as }n\to\infty,$

since $P_{N}(\varphi(X))$ is a bounded subset of $X$. Thus,

(19) $P_{N}(\varphi(x))=0\ \ \text{ for all }x\in X.$

By the second equality in (18),
$\varphi(x)=y^{\prime}_{n}(x)+z^{\prime}_{n}(x)$, where

$y^{\prime}_{n}(x)=T^{n}P_{M}(\varphi(R^{-n}x))\ \ \text{ and }\ \
z^{\prime}_{n}(x)=T^{n}P_{N}(\varphi(R^{-n}x)).$

Clearly, $y^{\prime}_{n}(x)\in M$ for all $x$ and $n$. Since
$\varphi(R^{-1}x)\in Y$, we have that $P_{N}(\varphi(R^{-1}x))\in T^{-1}(N)$,
that is, $z^{\prime}_{1}(x)=TP_{N}(\varphi(R^{-1}x))\in N$. Hence,
$z^{\prime}_{1}(x)\in T^{-1}(N)$. If $n\geq 1$ and $z^{\prime}_{n}(x)\in
T^{-1}(N)$ for all $x$, then
$z^{\prime}_{n+1}(x)=T(z^{\prime}_{n}(R^{-1}x))\in N$, and so
$z^{\prime}_{n+1}(x)\in T^{-1}(N)$. By induction, $z^{\prime}_{n}(x)\in
T^{-1}(N)$ for all $x$ and $n$. Thus, $P_{M}(\varphi(x))=y^{\prime}_{n}(x)$
for all $n\in\mathbb{N}$. As in the proof of (19), we obtain

(20) $P_{M}(\varphi(x))=0\ \ \text{ for all }x\in X.$

By (19) and (20), $\varphi=0$, proving the injectivity of $\Psi$.

Step 2. $\Psi$ is surjective (hence bijective) and its inverse is given by
(17).

Take $\phi\in F_{b}(X;X)$. Since $P_{M}(\phi(X))$ and $P_{N}(\phi(X))$ are
bounded subsets of $X$,

$\Big{\|}\sum_{k=i}^{j}T^{k}P_{M}(\phi(R^{-k-1}x))\Big{\|}_{\alpha}\leq
c_{\alpha}\sum_{k=i}^{j}t_{\alpha}^{k}\|P_{M}(\phi(R^{-k-1}x))\|_{\beta_{\alpha}}\to
0\ \text{ as }i,j\to\infty$

and

$\Big{\|}\sum_{k=i}^{j}T^{-k}P_{N}(\phi(R^{k-1}x))\Big{\|}_{\alpha}\leq
c_{\alpha}\sum_{k=i}^{j}t_{\alpha}^{k}\|P_{N}(\phi(R^{k-1}x))\|_{\beta_{\alpha}}\to
0\ \text{ as }i,j\to\infty.$

Hence, by the sequential completeness of $X$, we can define

$\varphi(x)\\!:=\sum_{k=0}^{\infty}T^{k}P_{M}(\phi(R^{-k-1}x))-\sum_{k=1}^{\infty}T^{-k}P_{N}(\phi(R^{k-1}x))\
\ \ \ (x\in X).$

Note that the first series lies in $M$ and the second one lies in $T^{-1}(N)$.
Moreover, the above estimates imply that $\varphi(X)$ is a bounded subset of
$X$. So, $\varphi\in F_{b}(X;Y)$. Finally,

$\displaystyle\Psi(\varphi)(x)$ $\displaystyle=\varphi(Rx)-T(\varphi(x))$
$\displaystyle=\big{(}P_{M}(\varphi(Rx))-TP_{M}(\varphi(x))\big{)}+\big{(}P_{N}(\varphi(Rx))-TP_{N}(\varphi(x))\big{)}$
$\displaystyle=\Big{(}\sum_{k=0}^{\infty}T^{k}P_{M}(\phi(R^{-k}x))-\sum_{k=0}^{\infty}T^{k+1}P_{M}(\phi(R^{-k-1}x))\Big{)}$
$\displaystyle\ \ \ \
+\Big{(}-\sum_{k=1}^{\infty}T^{-k}P_{N}(\phi(R^{k}x))+\sum_{k=1}^{\infty}T^{-k+1}P_{N}(\phi(R^{k-1}x))\Big{)}$
$\displaystyle=P_{M}(\phi(x))+P_{N}(\phi(x))$ $\displaystyle=\phi(x).$

Thus, $\Psi$ is surjective (hence bijective) and (17) holds.

Step 3. The last assertion in the theorem holds.

This follows from the definition of $\Psi$ and the fact that both series in
(17) are uniformly convergent on $X$ (in view of the estimates obtained in
Step 2). ∎

######  Theorem 25.

Let $T\in GL(X)$ be a generalized hyperbolic operator, where $X$ is a
sequentially complete locally convex space. For every neighborhood $V$ of $0$
in $X$, there exists a neighborhood $U$ of $0$ in $X$ such that for any
homeomorphism (resp. any uniform homeomorphism) $S:X\to X$ satisfying

(21) $T-S\text{ is a bounded map}\ \ \ \text{ and }\ \ \ Tx-Sx\in U\text{ for
all }x\in X,$

there is a continuous map (resp. a uniformly continuous map) $\phi:X\to X$
with

$T\circ\phi=\phi\circ S\ \ \ \text{ and }\ \ \ \phi(x)-x\in V\text{ for all
}x\in X.$

###### Proof.

Let $(\|\cdot\|_{\alpha})_{\alpha\in I}$ be a directed family of seminorms
inducing the topology of $X$. Let $X=M\oplus N$ be the direct sum
decomposition given by the generalized hyperbolicity of $T$ and let
$P_{M}:X\to M$ and $P_{N}:X\to N$ be the canonical projections. Given a
neighborhood $V$ of $0$ in $X$, there exist $\alpha\in I$ and $\varepsilon>0$
such that

$\\{x\in X:\|x\|_{\alpha}<\varepsilon\\}\subset V.$

There exist $\beta\in I$, $c>0$ and $t\in(0,1)$ such that

(22) $\|T^{n}y\|_{\alpha}\leq c\,t^{n}\|y\|_{\beta}\ \ \text{ and }\ \
\|T^{-n}z\|_{\alpha}\leq c\,t^{n}\|z\|_{\beta},$

whenever $y\in M$, $z\in N$ and $n\in\mathbb{N}_{0}$. By the continuity of the
projections $P_{M}$ and $P_{N}$, there exist $\gamma\in I$ and $d>0$ such that

(23) $\|P_{M}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \ \text{ and }\ \
\|P_{N}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \ \ \ \ (x\in X).$

Let $\delta\\!:=\frac{(1-t)\varepsilon}{2cd}$ and consider the neighborhood
$U\\!:=\\{x\in X:\|x\|_{\gamma}<\delta\\}$ of $0$ in $X$. Let $S:X\to X$ be a
homeomorphism such that $T-S$ is a bounded map and

(24) $Tx-Sx\in U\ \ \text{ for all }x\in X.$

By Lemma 24, the linear map

$\Psi:\varphi\in C_{b}(X;Y)\mapsto\varphi\circ S-T\circ\varphi\in C_{b}(X;X)$

is bijective, where $Y\\!:=M+T^{-1}(N)$. Let

$\phi\\!:=I+\Psi^{-1}(T-S),$

which is a continuous map from $X$ into $X$. Since

$T-S=\Psi(\phi-I)=\Psi(\phi)-\Psi(I)=\phi\circ S-T\circ\phi+T-S,$

we get

$T\circ\phi=\phi\circ S.$

Now, by (17),

$\phi(x)-x=\sum_{k=0}^{\infty}T^{k}P_{M}((T-S)(S^{-k-1}x))-\sum_{k=1}^{\infty}T^{-k}P_{N}((T-S)(S^{k-1}x)),$

for every $x\in X$. Hence, simple computations using (22), (23) and (24) give

$\|\phi(x)-x\|_{\alpha}<\varepsilon\ \ \text{ for all }x\in X,$

which implies that $\phi(x)-x\in V$ for all $x\in X$.

If $S$ is a uniform homeomorphism, then we can consider $\Psi$ as a bijection
from $U_{b}(X;Y)$ onto $U_{b}(X;X)$; hence, the map $\phi$ obtained above is
uniformly continuous. ∎

If $X$ is a Banach space, then the first condition in (21) is superfluous, and
so we obtain the following result.

######  Corollary 26.

Every invertible generalized hyperbolic operator on a Banach space is
topologically stable.

A famous result known as Walters’ stability theorem asserts that:

> If a homeomorphism of a compact metric space is expansive and has the
> shadowing property, then it is topologically stable [49, Theorem 4].

In linear dynamics, the following more precise result is known:

> For invertible operators on Banach spaces, expansivity plus the shadowing
> property is equivalent to hyperbolicity [9, Theorem 1].

In particular, expansive operators with the shadowing property on Banach
spaces are topologically stable. Moreover, by Corollaries 14 and 26, we also
obtain the following characterization of hyperbolicity.

######  Corollary 27.

An invertible operator on a Banach space is hyperbolic if and only if it is
expansive and topologically stable.

Since hyperbolicity and the shadowing property coincide for invertible
operators in the finite-dimensional setting, Corollaries 14 and 26 also imply
the following fact.

######  Corollary 28.

For invertible operators on finite-dimensional normed spaces, the concepts of
hyperbolicity and topological stability coincide.

######  Remark 29.

Let $X$, $T$, $M$, $N$ and $Y$ be as in the previous theorem and its proof.
The proof of the theorem shows that: For any homeomorphism $S:X\to X$ such
that

$T-S\ \text{ is a bounded map},$

there exists a continuous map $\phi:X\to X$ such that

(25) $T\circ\phi=\phi\circ S\ \ \ \text{ and }\ \ \ \phi-I\text{ is a bounded
map}.$

Moreover, $\phi$ is unique as long as we require that $(\phi-I)(X)\subset Y$.
If $T$ is hyperbolic, then $Y=X$ and we have the uniqueness of the continuous
map $\phi:X\to X$ satisfying (25). However, in the nonhyperbolic case, such a
map $\phi:X\to X$ is never unique. Indeed, if $T$ is not hyperbolic, then we
can choose a nonzero vector $y\in T^{-1}(M)\cap N$. Hence,

$z\\!:=\sum_{n=-\infty}^{\infty}T^{n}y$

defines a nontrivial fixed point of $T$ and the continuous map $\psi:X\to X$
given by $\psi(x)\\!:=\phi(x)+z$ ($x\in X$) also satisfies (25).

######  Remark 30.

Let $T$ be the invertible generalized hyperbolic operator defined in Remark 5.
Since $T$ does not have the periodic shadowing property, it follows from
Theorem 12 that $T$ is not topologically stable. This shows that the
hypothesis of sequential completeness (resp. of completeness) is essential for
the validity of Theorem 25 (resp. of Corollary 26).

##  4\. Expansivity and uniform expansivity for operators

Given a seminorm $\|\cdot\|$ on a vector space $X$, we define the unit sphere
of $\|\cdot\|$ by

$S_{\|\cdot\|}\\!:=\\{x\in X:\|x\|=1\\}.$

If $X$ is a normed space with norm $\|\cdot\|$, we also write $S_{X}$ instead
of $S_{\|\cdot\|}$.

Given a metric space $M$ with metric $d$, recall that a homeomorphism $h:M\to
M$ is said to be expansive if there is a constant $c>0$ (an expansivity
constant for $h$) such that, for every $x,y\in M$ with $x\neq y$, there exists
$n\in\mathbb{Z}$ with $d(h^{n}(x),h^{n}(y))\geq c$.

If $X$ is a normed space and $T\in GL(X)$, then it is well known that $T$ is
expansive if and only if the following property holds:

  * •

For every $x\in S_{X}$, there exists $n\in\mathbb{Z}$ such that
$\|T^{n}x\|\geq 2$.

Actually, this property is often used as the definition of expansive operators
on normed spaces and motivated the concept of uniform expansivity. Recall that
the operator $T$ is said to be uniformly expansive if the following property
holds:

  * •

There exists $n\in\mathbb{N}$ such that, for every $x\in S_{X}$,
$\|T^{n}x\|\geq 2$ or $\|T^{-n}x\|\geq 2$.

We refer the reader to [1, 7, 9, 22, 27, 33] for the study of expansive and
uniformly expansive operators on Banach spaces.

Let us now consider the following simple example in the Fréchet space setting.

######  Example 31.

Let $C(\mathbb{C})$ be the Fréchet space of all continuous maps
$f:\mathbb{C}\to\mathbb{C}$ endowed with the compact-open topology, which is
induced by the seminorms

$\|f\|_{k}\\!:=\max_{|z|\leq k}|f(z)|\ \ \ \ (k\in\mathbb{N}).$

A canonical compatible invariant metric on $C(\mathbb{C})$ is given by

$d(f,g)\\!:=\sum_{k=1}^{\infty}\frac{1}{2^{k}}\min\\{1,\|f-g\|_{k}\\}.$

Let $T\\!:=2I$ be twice the identity operator on $C(\mathbb{C})$. Since
$d(T^{n}f,T^{n}g)=d(2^{n}f,2^{n}g)\leq\frac{1}{2^{k}}$ for all
$n\in\mathbb{Z}$, whenever $f,g\in C(\mathbb{C})$ have supports disjoint from
the disk $\\{z\in\mathbb{C}:|z|\leq k\\}$, it follows that $T$ is not
expansive.

Intuitively speaking, an operator of the form $2I$ should be expansive. Hence,
the above example shows that the metric notion of expansivity does not behave
so well in the Fréchet space setting as it does in the Banach space setting.

Below we will propose an alternative notion of (uniform) expansivity for
operators on locally convex spaces, which is motivated by the following simple
characterizations in the case of invertible operators on normed spaces, as
observed in [7, Proposition 19]:

  * •

$T$ is expansive $\Leftrightarrow$ $\sup_{n\in\mathbb{Z}}\|T^{n}x\|=\infty$
for every nonzero $x\in X$.

  * •

$T$ is uniformly expansive $\Leftrightarrow$ $S_{X}=A\cup B$ where
$\lim_{n\to\infty}\|T^{n}x\|=\infty$ uniformly on $A$ and
$\lim_{n\to\infty}\|T^{-n}x\|=\infty$ uniformly on $B$.

Our definition is also motivated by the fact that the important notion of
bounded set in functional analysis is the topological one: a subset $E$ of a
topological vector space $X$ is said to be bounded if for each neighborhood
$V$ of $0$ in $X$, there exists $r>0$ such that $E\subset\lambda V$ whenever
$|\lambda|\geq r$. If $X$ is a normed space, then the set $E$ is bounded if
and only if it is bounded in the metric sense, i.e., it has finite diameter.
Nevertheless, the metric notion of bounded set becomes essentially useless
when we go beyond the normed space setting, while the topological notion plays
a central role in all of functional analysis. As we have seen above, in normed
spaces, expansivity means that the orbit of each nonzero vector is an
unbounded set. These comments together with Example 31 suggest that the metric
notion of expansivity may not be the most appropriate one in more general
settings.

######  Definition 32.

Let $X$ be a locally convex space over $\mathbb{K}$ whose topology is induced
by a directed family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms. We say
that an operator $T\in GL(X)$ is topologically expansive if the following
condition holds:

  * (E)

For each nonzero $x\in X$, there exists $\alpha\in I$ such that
$\sup_{n\in\mathbb{Z}}\|T^{n}x\|_{\alpha}=\infty$.

We say that the operator $T$ is uniformly topologically expansive if:

  * (UE)

For every $\alpha\in I$, there exists $\beta\in I$ such that we can write
$S_{\|\cdot\|_{\alpha}}=A_{\alpha}\cup B_{\alpha}$, where

$\|T^{n}x\|_{\beta}\to\infty\text{ uniformly on }A_{\alpha}\text{ as
}n\to\infty$

and

$\|T^{-n}x\|_{\beta}\to\infty\text{ uniformly on }B_{\alpha}\text{ as
}n\to\infty.$

Note that the above notions are independent of the choice of the directed
family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms inducing the topology
of $X$. Moreover, condition (E) means that the orbit
$\operatorname{Orb}(x,T)\\!:=\\{T^{n}x:n\in\mathbb{Z}\\}$ is an unbounded set
in $X$ for every nonzero $x\in X$.

######  Remark 33.

If $X$ is a metrizable locally convex space, then its topology is induced by
an increasing sequence $(\|\cdot\|_{k})_{k\in\mathbb{N}}$ of seminorms and a
canonical compatible invariant metric on $X$ is given by

$d(x,y)\\!:=\sum_{k=1}^{\infty}\frac{1}{2^{k}}\min\\{1,\|x-y\|_{k}\\}.$

In this case, it is easy to show that every expansive operator on $X$ is
topologically expansive. However, the converse is not true in general, since
operators of the form $\lambda I$, with $|\lambda|\not\in\\{0,1\\}$, are
always uniformly topologically expansive but may fail to be expansive (Example
31).

######  Proposition 34.

Let $X$ be a locally convex space whose topology is induced by a directed
family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms. An operator $T\in
GL(X)$ is topologically expansive if and only if for every $x\in
X\backslash\\{0\\}$, there exists $\alpha\in I$ such that $\|x\|_{\alpha}\neq
0$ and, for each $y\in\operatorname{Orb}(x,T)$, there exists $n\in\mathbb{Z}$
with $\|T^{n}y\|_{\alpha}\geq 2\|y\|_{\alpha}$.

###### Proof.

($\Rightarrow$): Let $x\in X\backslash\\{0\\}$ and let $\alpha\in I$ be as in
(E). Without loss of generality, we may assume that $\|x\|_{\alpha}\neq 0$.
Given $y\in\operatorname{Orb}(x,T)$, since the set
$\\{\|T^{n}y\|_{\alpha}:n\in\mathbb{Z}\\}$ is unbounded in $\mathbb{R}$, there
exists $n\in\mathbb{Z}$ with $\|T^{n}y\|_{\alpha}\geq 2\|y\|_{\alpha}$.

($\Leftarrow$): Given $x\in X\backslash\\{0\\}$, let $\alpha\in I$ be as in
the hypothesis. By induction, we obtain a sequence $(n_{j})_{j\in\mathbb{N}}$
in $\mathbb{Z}$ such that

$\|T^{n_{1}+\cdots+n_{j}}x\|_{\alpha}\geq 2^{j}\|x\|_{\alpha}\ \ \text{ for
all }j\in\mathbb{N}.$

Thus, $\\{\|T^{n}x\|_{\alpha}:n\in\mathbb{Z}\\}$ is unbounded in $\mathbb{R}$,
giving (E). ∎

Note that the constant $2$ can be replaced by any constant $c>1$ in the above
proposition.

######  Theorem 35.

For any invertible generalized hyperbolic operator $T$ on a locally convex
space $X$, the following assertions are equivalent:

  * (i)

$T$ is topologically expansive;

  * (ii)

$T$ is uniformly topologically expansive;

  * (iii)

$T$ is hyperbolic.

###### Proof.

Let $(\|\cdot\|_{\alpha})_{\alpha\in I}$ be a directed family of seminorms
that induces the topology of $X$. Let $X=M\oplus N$ be the topological direct
sum decomposition given by the generalized hyperbolicity of $T$ and let
$P_{M}:X\to M$ and $P_{N}:X\to N$ be the canonical projections.

(iii) $\Rightarrow$ (ii): Given $\alpha\in I$, there exist $\beta\in I$, $c>0$
and $t\in(0,1)$ such that

$\|T^{-n}y\|_{\beta}\geq c^{-1}t^{-n}\|y\|_{\alpha}\ \text{ and }\
\|T^{n}z\|_{\beta}\geq c^{-1}t^{-n}\|z\|_{\alpha}\ \text{ for all }y\in M,z\in
N,n\in\mathbb{N}_{0}.$

Choose $\gamma\in I$ and $d>0$ such that

$\|P_{M}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \text{ and }\
\|P_{N}x\|_{\beta}\leq d\,\|x\|_{\gamma}\ \text{ for all }x\in X.$

Let

$A_{\alpha}\\!:=\\{x\in S_{\|\cdot\|_{\alpha}}:\|P_{N}x\|_{\alpha}\geq 1/2\\}\
\text{ and }\ B_{\alpha}\\!:=\\{x\in
S_{\|\cdot\|_{\alpha}}:\|P_{M}x\|_{\alpha}\geq 1/2\\}.$

Then $S_{\|\cdot\|_{\alpha}}=A_{\alpha}\cup B_{\alpha}$ and

$\|T^{n}x_{1}\|_{\gamma}\geq\frac{1}{2\,c\,d\,t^{n}}\ \text{ and }\
\|T^{-n}x_{2}\|_{\gamma}\geq\frac{1}{2\,c\,d\,t^{n}}\ \text{ for all }x_{1}\in
A_{\alpha},x_{2}\in B_{\alpha},n\in\mathbb{N}_{0},$

which proves that $T$ is uniformly topologically expansive.

(ii) $\Rightarrow$ (i): Clear.

(i) $\Rightarrow$ (iii): Take $x\in M\cap T(N)$ and $\alpha\in I$. Let
$\beta\in I$, $c>0$ and $t\in(0,1)$ be as in (GH3). Since $x\in M$ and
$T^{-1}x\in N$, we have that

$\|T^{n}x\|_{\alpha}\leq c\,t^{n}\|x\|_{\beta}\ \text{ and }\
\|T^{-n}T^{-1}x\|_{\alpha}\leq c\,t^{n}\|T^{-1}x\|_{\beta}\ \text{ for all
}n\in\mathbb{N}_{0}.$

Hence,

$\sup_{n\in\mathbb{Z}}\|T^{n}x\|_{\alpha}\leq
c\,\max\\{\|x\|_{\beta},\|T^{-1}x\|_{\beta}\\}<\infty.$

Since $\alpha\in I$ is arbitrary and $T$ is topologically expansive, we
conclude that $x=0$. This proves that $M\cap T(N)=\\{0\\}$, which implies that
$T$ is hyperbolic. ∎

######  Corollary 36.

Every invertible hyperbolic operator on a locally convex space is uniformly
topologically expansive.

In the next section we will exhibit some examples of uniformly topologically
expansive operators on the Fréchet space $s(\mathbb{Z})$ of rapidly decreasing
sequences on $\mathbb{Z}$ that are not hyperbolic.

We want to finish this section by showing that uniform topological expansivity
avoids the possibility of even the weakest properties related to chaotic
behavior, namely, topological transitivity and Li-Yorke chaos.

Given a metric space $M$ with metric $d$, recall that a continuous map $f:M\to
M$ is said to be Li-Yorke chaotic if there is an uncountable set $S\subset M$
such that each pair $(x,y)$ of distinct points in $S$ is a Li-Yorke pair for
$f$, in the sense that

$\liminf_{n\to\infty}d(f^{n}(x),f^{n}(y))=0\ \ \text{ and }\ \
\limsup_{n\to\infty}d(f^{n}(x),f^{n}(y))>0.$

An extensive study of Li-Yorke chaos in the setting of linear dynamics was
developed in [5, 6]. In particular, the following useful characterizations
were obtained:  For any continuous linear operator $T$ on a Fréchet space $X$,
the following assertions are equivalent:

  * (i)

$T$ is Li-Yorke chaotic;

  * (ii)

$T$ admits a semi-irregular vector, that is, a vector $x\in X$ such that the
sequence $(T^{n}x)_{n\in\mathbb{N}}$ does not converge to zero but has a
subsequence converging to zero;

  * (iii)

$T$ admits an irregular vector, that is, a vector $x\in X$ such that the
sequence $(T^{n}x)_{n\in\mathbb{N}}$ is unbounded but has a subsequence
converging to zero.

The notion of Li-Yorke chaos was extended to group actions on Hausdorff
uniform spaces in [4]. In the case of a continuous linear operator $T$ on a
topological vector space $X$, the definition reads as follows: the operator
$T$ is said to be Li-Yorke chaotic if there is an uncountable set $S\subset X$
such that each pair $(x,y)$ of distinct points in $S$ is a Li-Yorke pair for
$T$, in the sense that the following conditions hold:

  * (LY1)

For every neighborhood $V$ of $0$ in $X$, there exists $n\in\mathbb{N}$ such
that $T^{n}x-T^{n}y\in V$.

  * (LY2)

There exists a neighborhood $U$ of $0$ in $X$ such that $T^{n}x-T^{n}y\not\in
U$ for infinitely many values of $n$.

It was observed in [19] that the equivalence (i) $\Leftrightarrow$ (ii)
remains true in this more general setting, where $x$ semi-irregular for $T$
means that the sequence $(T^{n}x)_{n\in\mathbb{N}}$ does not converge to zero
but has a subnet converging to zero.

In order to show that uniformly topologically expansive operators are neither
topologically transitive nor Li-Yorke chaotic we will need a technical lemma.

######  Lemma 37.

If a continuous linear operator $T$ on a locally convex space $X$, whose
topology is induced by a directed family $(\|\cdot\|_{\alpha})_{\alpha\in I}$
of seminorms, is topologically transitive or Li-Yorke chaotic, then it
satisfies the following property:

$\displaystyle\exists\alpha_{0}\in I,\ \forall\beta\in I,\
\exists\varepsilon>0,\ \forall k\in\mathbb{N},\ \exists m,n\in\mathbb{N},\
\exists x\in X\mbox{ such that}$ (26) $\displaystyle k<m<n-k,\
\|x\|_{\beta}<1,\ \|T^{n}x\|_{\beta}<1,\ \mbox{ and }\
\|T^{m}x\|_{\alpha_{0}}>\varepsilon.$

###### Proof.

Suppose that $T$ is topologically transitive. Select any $\alpha_{0}\in I$
such that $\|\cdot\|_{\alpha_{0}}$ is non-zero. Given $\beta\in I$ arbitrary,
we fix $\varepsilon\\!:=1$, and once $k\in\mathbb{N}$ is fixed, by
transitivity of $T$ we select $m\in\mathbb{N}$, $m>k$, with $T^{m}(A)\cap
B\neq\emptyset$, where

$A\\!:=\\{u\in X:\|u\|_{\beta}<1\\}\ \mbox{ and }\ B\\!:=\\{v\in
X:\|v\|_{\alpha_{0}}>1\\}.$

Thus, $C\\!:=A\cap T^{-m}(B)$ is a non-empty open set, and applying the
transitivity property once again we find $n\in\mathbb{N}$, $n>m+k$, with
$T^{n}(C)\cap A\neq\emptyset$. Selecting any $x\in C\cap T^{-n}(A)$ yields
property (37).

In case that $T$ is Li-Yorke chaotic, fix any semi-irregular vector $u\in X$
for $T$. In particular, there exist $\alpha_{0}\in I$, $\delta\in\;]0,1[$ and
an increasing sequence $(m_{j})_{j\in\mathbb{N}}$ of positive integers such
that $\|T^{m_{j}}u\|_{\alpha_{0}}>\delta$ for all $j\in\mathbb{N}$. Given any
$\beta\in I$, by semi-irregularity of $u$ we also find another increasing
sequence $(n_{j})_{j\in\mathbb{N}}$ of positive integers such that
$\|T^{n_{j}}u\|_{\beta}<1$ for all $j\in\mathbb{N}$. We pick
$\lambda\in\;]0,1[$ such that $\|\lambda u\|_{\beta}<1$. If we fix
$\varepsilon\\!:=\delta\lambda$ and $x\\!:=\lambda u$, then $\|x\|_{\beta}<1$
and, for an arbitrary $k\in\mathbb{N}$, we have the existence of $m>k$ with
$\|T^{m}x\|_{\alpha_{0}}>\varepsilon$ and also the existence of $n>m+k$ with
$\|T^{n}x\|_{\beta}<1$, as desired. ∎

Observe that property (37) is trivially satisfied for a unimodular multiple of
the identity operator, so it cannot be identified as a “chaotic” property, per
se.

We are now in a position to show that uniform topological expansivity avoids
the possibility of Li-Yorke chaos or topological transitivity. In particular,
the following result generalizes [7, Theorem C] from Banach spaces to
arbitrary locally convex spaces.

######  Theorem 38.

A uniformly topologically expansive operator on a locally convex space is
neither Li-Yorke chaotic nor topologically transitive.

###### Proof.

Let $T$ be a uniformly topologically expansive operator on a locally convex
space $X$. Let $(\|\cdot\|_{\alpha})_{\alpha\in I}$ be a directed family of
seminorms that induces the topology of $X$. Suppose that $T$ is Li-Yorke
chaotic or topologically transitive. Then it satisfies property (37) of Lemma
37. Let $\alpha\\!:=\alpha_{0}\in I$ be given by this property. By Definition
32, we can decompose $S_{\|\cdot\|_{\alpha}}=A_{\alpha}\cup B_{\alpha}$ and
find $\beta\in I$ (w.l.o.g., $\beta\geq\alpha$) satisfying the conditions of
uniform topological expansivity. Let $\varepsilon>0$ be associated with
$\alpha_{0}$ and $\beta$ in property (37). Since $\|T^{j}u\|_{\beta}\to\infty$
and $\|T^{-j}v\|_{\beta}\to\infty$, uniformly on $u\in A_{\alpha}$ and $v\in
B_{\alpha}$, as $j\to\infty$, we find $k\in\mathbb{N}$ with

$\min\\{\|T^{j}u\|_{\beta},\|T^{-j}v\|_{\beta}\\}>1/\varepsilon$

for each pair $(u,v)\in A_{\alpha}\times B_{\alpha}$ and for any $j\geq k$. By
property (37), there are $m,n\in\mathbb{N}$, $k<m<n-k$, and $x\in X$ so that

$\|x\|_{\beta}<1,\ \|T^{n}x\|_{\beta}<1,\ \mbox{ and }\
\|T^{m}x\|_{\alpha_{0}}>\varepsilon.$

We set $y\\!:=\left(\|T^{m}x\|_{\alpha}^{-1}\right)T^{m}x\in
S_{\|\cdot\|_{\alpha}}$. If $y\in A_{\alpha}$, then

$\frac{1}{\varepsilon}<\|T^{n-m}y\|_{\beta}=\frac{1}{\|T^{m}x\|_{\alpha}}\|T^{n}x\|_{\beta}<\frac{1}{\varepsilon}\,,$

a contradiction. If $y\in B_{\alpha}$, then

$\frac{1}{\varepsilon}<\|T^{-m}y\|_{\beta}=\frac{1}{\|T^{m}x\|_{\alpha}}\|x\|_{\beta}<\frac{1}{\varepsilon}\,,$

which yields a contradiction too, and we conclude that $T$ cannot satisfy
property (37), so $T$ is neither Li-Yorke chaotic nor topologically
transitive. ∎

##  5\. Weighted shifts on Fréchet sequence spaces

Our main goal in this section is to explore the notion of topological
expansivity for weighted shifts on Fréchet sequence spaces, but some weighted
shifts with other dynamical properties will be presented at the end of the
section.

Let $\omega(\mathbb{Z})\\!:=\mathbb{K}^{\mathbb{Z}}$ be the Fréchet space of
all scalar sequences equipped with the product topology. Recall that the
bilateral forward (resp. backward) shift on $\omega(\mathbb{Z})$ is the
continuous linear operator $F:\omega(\mathbb{Z})\to\omega(\mathbb{Z})$ (resp.
$B:\omega(\mathbb{Z})\to\omega(\mathbb{Z})$) given by

$F((x_{n})_{n\in\mathbb{Z}})\\!:=(x_{n-1})_{n\in\mathbb{Z}}\ \ \ \ \
(\text{resp. }B((x_{n})_{n\in\mathbb{Z}})\\!:=(x_{n+1})_{n\in\mathbb{Z}}).$

More generally, given a sequence $w\\!:=(w_{n})_{n\in\mathbb{Z}}$ of nonzero
scalars (called a weight sequence), recall that the bilateral weighted forward
(resp. backward) shift on $\omega(\mathbb{Z})$ is the continuous linear
operator $F_{w}:\omega(\mathbb{Z})\to\omega(\mathbb{Z})$ (resp.
$B_{w}:\omega(\mathbb{Z})\to\omega(\mathbb{Z})$) given by

$F_{w}((x_{n})_{n\in\mathbb{Z}})\\!:=(w_{n-1}x_{n-1})_{n\in\mathbb{Z}}\ \ \ \
\ (\text{resp.
}B_{w}((x_{n})_{n\in\mathbb{Z}})\\!:=(w_{n+1}x_{n+1})_{n\in\mathbb{Z}}).$

Note that the operators $F_{w}$ and $B_{w}$ are invertible on
$\omega(\mathbb{Z})$, and their inverses are given by

$F_{w}^{-1}=B_{v}\ \ \text{ and }\ \ B_{w}^{-1}=F_{u},\text{ where
}v\\!:=\Big{(}\frac{1}{w_{n-1}}\Big{)}_{n\in\mathbb{Z}}\text{ and
}u\\!:=\Big{(}\frac{1}{w_{n+1}}\Big{)}_{n\in\mathbb{Z}}.$

We will also consider these operators on a Fréchet sequence space over
$\mathbb{Z}$, that is, a subspace $X$ of $\omega(\mathbb{Z})$ equipped with a
topology under which $X$ is a Fréchet space and the embedding
$X\hookrightarrow\omega(\mathbb{Z})$ is continuous. This continuity is
equivalent to the continuity of each coordinate functional
$(x_{n})_{n\in\mathbb{Z}}\in X\mapsto x_{j}\in\mathbb{K}$, $j\in\mathbb{Z}$.
It follows from the closed graph theorem that $F_{w}$ (resp. $B_{w}$) induces
a continuous linear operator on $X$ as soon as it maps $X$ into itself. In
this case, by abuse of notation, we also write $F_{w}:X\to X$ (resp.
$B_{w}:X\to X$). The canonical vectors
$e_{n}\\!:=(\delta_{j,n})_{j\in\mathbb{Z}}$, $n\in\mathbb{Z}$, of
$\omega(\mathbb{Z})$ form a basis in $X$ if they belong to $X$ and every
sequence $x\\!:=(x_{n})_{n\in\mathbb{Z}}\in X$ satisfies

$x=\lim_{m,n\to\infty}(\ldots,0,0,x_{-m},x_{-m+1},\ldots,x_{n-1},x_{n},0,0,\ldots).$

This means that each $x\in X$ has a unique representation
$x=\sum_{n\in\mathbb{Z}}x_{n}e_{n}$, with scalars $x_{n}\in\mathbb{K}$,
$n\in\mathbb{Z}$. Clearly $(e_{n})_{n\in\mathbb{Z}}$ is a basis of the
following Fréchet sequence spaces: $\ell^{p}(\mathbb{Z})$, $1\leq p<\infty$,
$c_{0}(\mathbb{Z})$ and $\omega(\mathbb{Z})$.

Our main goal in this section is to characterize topological expansivity for
weighted shifts on Fréchet sequence spaces. In order to state our results, we
will need the following definition.

######  Definition 39.

Let $X$ be a locally convex space over $\mathbb{K}$ whose topology is induced
by a directed family $(\|\cdot\|_{\alpha})_{\alpha\in I}$ of seminorms. We say
that an operator $T\in L(X)$ is topologically positively expansive if the
following condition holds:

  * (PE)

For each nonzero $x\in X$, there exists $\alpha\in I$ such that
$\sup_{n\in\mathbb{N}}\|T^{n}x\|_{\alpha}=\infty$.

######  Theorem 40.

Suppose that $X$ is a Fréchet sequence space over $\mathbb{Z}$ in which the
sequence $(e_{n})_{n\in\mathbb{Z}}$ of canonical vectors is a basis,
$(\|\cdot\|_{k})_{k\in\mathbb{N}}$ is an increasing sequence of seminorms that
induces the topology of $X$, and the bilateral forward shift $F$ is an
invertible operator on $X$. Then the following assertions are equivalent:

  * (i)

$F:X\to X$ is topologically expansive;

  * (ii)

there exists $k\in\mathbb{N}$ such that

    * (a)

$\sup\limits_{n\in\mathbb{N}}\|e_{n}\|_{k}=\infty$ or

    * (b)

$\sup\limits_{n\in\mathbb{N}}\|e_{-n}\|_{k}=\infty$;

  * (iii)

    * (a)

$F:X\to X$ is topologically positively expansive or

    * (b)

$F^{-1}:X\to X$ is topologically positively expansive.

###### Proof.

(i) $\Rightarrow$ (ii): Assume that $F$ is topologically expansive. By
Definition 32, there exists $k\in\mathbb{N}$ such that

$\sup_{n\in\mathbb{N}}\|F^{n}e_{1}\|_{k}=\infty\ \ \ \ \text{or}\ \ \ \
\sup_{n\in\mathbb{N}}\|F^{-n}e_{1}\|_{k}=\infty.$

The first equality means that
$\sup\limits_{n\in\mathbb{N}}\|e_{n+1}\|_{k}=\infty$, whereas the second one
means that $\sup\limits_{n\in\mathbb{N}}\|e_{-n+1}\|_{k}=\infty$.

(ii) $\Rightarrow$ (iii): First, assume that there exists $k\in\mathbb{N}$
such that

(27) $\sup\limits_{n\in\mathbb{N}}\|e_{n}\|_{k}=\infty.$

Since $(e_{n})_{n\in\mathbb{Z}}$ is a basis of $X$, for any
$x\\!:=(x_{n})_{n\in\mathbb{Z}}\in X$, the sequence
$(x_{n}e_{n})_{n\in\mathbb{Z}}$ converges to $0$ in $X$ as $n\to\pm\infty$.
Hence, by the Banach–Steinhaus theorem, the family of operators $T_{n}$ on
$X$, $n\in\mathbb{Z}$, defined by $x\mapsto x_{n}e_{n}$, is equicontinuous.
Hence, there are $C>0$ and $\ell\in\mathbb{N}$ such that

(28) $\|T_{n}x\|_{k}\leq C\|x\|_{\ell}\ \ \text{ for all }x\in X\text{ and
}n\in\mathbb{Z}.$

Let $x\\!:=(x_{n})_{n\in\mathbb{Z}}$ be any nonzero vector in $X$ and choose
$j\in\mathbb{Z}$ such that $x_{j}\neq 0$. Then

$F^{m}x=(x_{n-m})_{n\in\mathbb{Z}}=\sum_{n\in\mathbb{Z}}x_{n-m}e_{n}\ \ \text{
for all }m\in\mathbb{N}.$

By (28), we get

$\|x_{j}e_{m+j}\|_{k}=\|T_{m+j}(F^{m}x)\|_{k}\leq C\|F^{m}x\|_{\ell}\ \ \text{
for all }m\in\mathbb{N}.$

In view of (27), we conclude that

$\sup_{m\in\mathbb{N}}\|F^{m}x\|_{\ell}=\infty.$

Hence, $F$ is topologically positively expansive. Analogously,
$\sup\limits_{n\in\mathbb{N}}\|e_{-n}\|_{k}=\infty$ implies that $F^{-1}$ is
topologically positively expansive.

(iii) $\Rightarrow$ (i): Trivial. ∎

Now we will see that, using a suitable conjugacy, this result can be
generalized to bilateral weighted forward shifts. Let $F_{w}:X\to X$ be an
invertible bilateral weighted forward shift on a Fréchet sequence space $X$
and define the sequence of weights $v\\!:=(v_{n})_{n\in\mathbb{Z}}$ by

$v_{n}\\!:=\left(\prod_{\nu=n}^{0}w_{\nu}\right)^{-1}\ \ \ \textnormal{for}\ \
n\leq 0,\ \ \ v_{1}\\!:=1,\ \ \ v_{n}\\!:=\prod_{\nu=1}^{n-1}w_{\nu}\ \ \
\textnormal{for}\ \ n\geq 2.$

By setting

$X_{v}\\!:=\\{(x_{n})_{n\in\mathbb{Z}}\in\omega(\mathbb{Z}):(v_{n}x_{n})_{n\in\mathbb{Z}}\in
X\\},$

the mapping $\phi_{v}:X_{v}\to X$, given by
$(x_{n})_{n\in\mathbb{Z}}\mapsto(v_{n}x_{n})_{n\in\mathbb{Z}}$, is an
algebraic isomorphism. If, in addition, we consider the topology of $X_{v}$
given by

$\textrm{ a subset }U\textrm{ of }X_{v}\textrm{ is open if and only if
}\phi_{v}(U)\textrm{ is open in }X,$

then $X$ becomes a Fréchet sequence space. If $(e_{n})_{n\in\mathbb{Z}}$ is a
basis in $X$, then it is also a basis in $X_{v}$, and the following diagram
commutes:

$\textstyle{X_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\phi_{v}}$$\textstyle{X_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{v}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{w}}$$\textstyle{X}$

Hence, $F$ and $F_{w}$ are conjugate operators. Moreover, $F$ and $F_{w}$ are
simultaneously invertible operators. The previous construction is similar to
the construction of [25, pp. 96, 101]. Combining these facts and Theorem 40,
we easily obtain the result below.

######  Theorem 41.

Suppose that $X$ is a Fréchet sequence space over $\mathbb{Z}$ in which the
sequence $(e_{n})_{n\in\mathbb{Z}}$ of canonical vectors is a basis,
$(\|\cdot\|_{k})_{k\in\mathbb{N}}$ is an increasing sequence of seminorms that
induces the topology of $X$, and the bilateral weighted forward shift $F_{w}$
is an invertible operator on $X$. Then the following assertions are
equivalent:

  * (i)

$F_{w}:X\to X$ is topologically expansive;

  * (ii)

there exists $k\in\mathbb{N}$ such that

    * (a)

$\sup\limits_{n\in\mathbb{N}}|w_{1}\cdot\ldots\cdot
w_{n}|\|e_{n+1}\|_{k}=\infty$ or

    * (b)

$\sup\limits_{n\in\mathbb{N}}|w_{-n+1}\cdot\ldots\cdot
w_{0}|^{-1}\|e_{-n+1}\|_{k}=\infty$;

  * (iii)

    * (a)

$F_{w}:X\to X$ is topologically positively expansive or

    * (b)

$F_{w}^{-1}:X\to X$ is topologically positively expansive.

######  Remark 42.

(a) Note that, in Theorems 40 and 41, (ii.a) is equivalent to (iii.a), and
(ii.b) is equivalent to (iii.b). In particular, if $X$ is a Fréchet sequence
space over $\mathbb{N}$, then a similar formulation of the above result gives
a characterization for topologically positively expansive weighted forward
shifts.

(b) The study of topological expansiveness for invertible bilateral weighted
backward shifts can be reduced to the corresponding case of forward shifts. In
this case, $B_{w}$ is topologically expansive if and only if $B_{w}$ or
$B_{w}^{-1}$ is topologically positively expansive, and this is equivalent to
the existence of $k\in\mathbb{N}$ such that

$\sup_{n\in\mathbb{N}}|w_{-n+2}\cdot\ldots\cdot
w_{0}w_{1}|\|e_{-n+1}\|_{k}=\infty\ \ \ \textnormal{or}\ \ \
\sup_{n\in\mathbb{N}}|w_{2}\cdot\ldots\cdot
w_{n+1}|^{-1}\|e_{n+1}\|_{k}=\infty,$

respectively.

(c) If $X$ is a Fréchet sequence space over $\mathbb{N}$ which contains the
sequence $a\\!:=(1,0,0,\dots)$, then the unilateral weighted backward shift
$B_{w}$ is not topologically positively expansive, since $B_{w}a=0$.

An infinite matrix $A\\!:=(a_{j,k})_{j,k\in\mathbb{N}}$ is said to be a Köthe
matrix if $0\leq a_{j,k}\leq a_{j,k+1}$ for all $j,k\in\mathbb{N}$, and for
each $j\in\mathbb{N}$, there exists $k\in\mathbb{N}$ with $a_{j,k}>0$. For
$1\leq p<\infty$, consider

$\lambda_{p}(A)\\!:=\left\\{x\\!:=(x_{j})_{j=1}^{\infty}\in\omega(\mathbb{N}):\|x\|_{k}\\!:=\left(\sum_{j=1}^{\infty}|x_{j}a_{j,k}|^{p}\right)^{1/p}<\infty\text{
for all }k\in\mathbb{N}\right\\},$

and, for $p=0$,

$\lambda_{0}(A)\\!:=\left\\{x\\!:=(x_{j})_{j=1}^{\infty}\in\omega(\mathbb{N}):\lim_{j\to\infty}x_{j}a_{j,k}=0,\
\|x\|_{k}\\!:=\sup_{j\in\mathbb{N}}|x_{j}|a_{j,k}\text{ for all
}k\in\mathbb{N}\right\\},$

which are the corresponding Köthe echelon spaces (see [28, 31, 35]). These
spaces constitute a natural class of Fréchet sequence spaces (whose topology
is given by the corresponding sequence $(\|\cdot\|_{k})_{k\in\mathbb{N}}$ of
seminorms) in which many authors have studied the dynamical properties of
weighted shifts (see, for instance, [34, 50] and references therein). In a
natural way, the Köthe spaces can be extended to sequence spaces indexed over
$\mathbb{Z}$. In this setting,
$A\\!:=(a_{j,k})_{j\in\mathbb{Z},k\in\mathbb{N}}$ is a Köthe matrix on
$\mathbb{Z}$ if $0\leq a_{j,k}\leq a_{j,k+1}$ for all $j\in\mathbb{Z}$ and
$k\in\mathbb{N}$, and for each $j\in\mathbb{Z}$, there exists $k\in\mathbb{N}$
with $a_{j,k}>0$.

The spaces $\lambda_{p}(A,\mathbb{Z})$ and $\lambda_{0}(A,\mathbb{Z})$ are
defined as in the case of $\mathbb{N}$, with the sums and the supremum taken
over the $\mathbb{Z}$ (see [34, Section 3.3]). A simple computation shows that
$(e_{n})_{n\in\mathbb{Z}}$ is a basis of $\lambda_{1}(A,\mathbb{Z})$.
Moreover, the bilateral weighted forward shift $F_{w}$ defines an operator on
$\lambda_{p}(A,\mathbb{Z})$ if and only if

(29) $\forall\ k\in\mathbb{N},\ \exists\ m\in\mathbb{N},\ m>k:\
\sup_{j\in\mathbb{Z}}|w_{j}|\frac{a_{j+1,k}}{a_{j,m}}<\infty,$

and $F_{w}$ is invertible if and only if

(30) $\forall\ k\in\mathbb{N},\ \exists\ m\in\mathbb{N},\ m>k:\
\sup_{j\in\mathbb{Z}}\frac{a_{j,k}}{|w_{j}|a_{j+1,m}}<\infty,$

where $a_{j,m}=0$ in (29) implies $a_{j+1,k}=0$, and $a_{j+1,m}=0$ in (30)
implies $a_{j,k}=0$. In these cases we adopt $0/0$ as $1$. By applying Theorem
41 to the bilateral weighted shifts on the Köthe spaces, we deduce the
following result.

######  Corollary 43.

Let $A$ be a Köthe matrix on $\mathbb{Z}$, $w$ be a weight sequence satisfying
(29) and (30), and $1\leq p<\infty$ or $p=0$. Suppose that
$(e_{n})_{n\in\mathbb{Z}}$ is a basis of $\lambda_{p}(A,\mathbb{Z})$. Then the
following assertions are equivalent:

  * (i)

$F_{w}:\lambda_{p}(A,\mathbb{Z})\to\lambda_{p}(A,\mathbb{Z})$ is topologically
expansive;

  * (ii)

there exists $k\in\mathbb{N}$ such that

    * (a)

$\sup\limits_{n\in\mathbb{N}}|w_{1}\cdot\ldots\cdot w_{n}|a_{n+1,k}=\infty$ or

    * (b)

$\sup\limits_{n\in\mathbb{N}}|w_{-n+1}\cdot\ldots\cdot
w_{0}|^{-1}a_{-n+1,k}=\infty$;

  * (iii)

    * (a)

$F_{w}:\lambda_{p}(A,\mathbb{Z})\to\lambda_{p}(A,\mathbb{Z})$ is topologically
positively expansive or

    * (b)

$F_{w}^{-1}:\lambda_{p}(A,\mathbb{Z})\to\lambda_{p}(A,\mathbb{Z})$ is
topologically positively expansive.

######  Example 44.

Let $w\\!:=(w_{n})_{n\in\mathbb{Z}}$ be a weight sequence and $1\leq p<\infty$
or $p=0$.

  * (i)

If $a_{j,k}=1$ for all $j\in\mathbb{Z}$ and $k\in\mathbb{N}$, then

$\lambda_{p}(A,\mathbb{Z})=\ell^{p}(\mathbb{Z})\ \ \ \ (\text{resp.
}\lambda_{0}(A,\mathbb{Z})=c_{0}(\mathbb{Z})).$

If, in addition,

$\sup_{j\in\mathbb{Z}}|w_{j}|<\infty\ \ \text{ and }\ \
\inf_{j\in\mathbb{Z}}|w_{j}|>0,$

which means that $F_{w}$ is a well-defined invertible operator on
$\ell^{p}(\mathbb{Z})$ (resp. on $c_{0}(\mathbb{Z})$), then from Corollary 43
we recover [7, Theorem E(1)].

  * (ii)

If the matrix $A$ is such that for each $k\in\mathbb{N}$, there is
$j_{k}\in\mathbb{Z}$ satisfying $a_{j,k}$ = 0 for all $|j|>j_{k}$, then

$\lambda_{p}(A,\mathbb{Z})=\omega(\mathbb{Z}).$

Furthermore, the bilateral weighted shifts on $\omega(\mathbb{Z})$ are never
topologically expansive. Indeed, consider a weighted forward shift $F_{w}$ on
$\omega(\mathbb{Z})$. Since, for each $k\in\mathbb{N}$, we can choose
$j_{k}\in\mathbb{Z}$ such that

$\sup_{n\in\mathbb{N}}|w_{1}\cdot\ldots\cdot w_{n}|a_{n+1,k}=\sup_{1\leq
n\leq|j_{k}|-1}|w_{1}\cdot\ldots\cdot w_{n}|a_{n+1,k}<\infty$

and

$\sup_{n\in\mathbb{N}}|w_{-n+1}\cdot\ldots\cdot
w_{0}|^{-1}a_{-n+1,k}=\sup_{1\leq n\leq|j_{k}|+1}|w_{-n+1}\cdot\ldots\cdot
w_{0}|^{-1}a_{-n+1,k}<\infty,$

the result follows from Corollary 43.

  * (iii)

If $a_{j,k}=(|j|+1)^{k}$ for all $j\in\mathbb{Z}$ and $k\in\mathbb{N}$, then

$\lambda_{p}(A,\mathbb{Z})=\lambda_{1}(A,\mathbb{Z})=s(\mathbb{Z}),$

where $s(\mathbb{Z})$ denotes the the space of all rapidly decreasing
sequences on $\mathbb{Z}$. Assume that $\sup_{j\in\mathbb{Z}}|w_{j}|<\infty$
and $\inf_{j\in\mathbb{Z}}|w_{j}|>0$. Then clearly $F_{w}$ defines an
invertible operator on $s(\mathbb{Z})$. Although these conditions are
sufficient to $F_{w}$ be invertible, they are not necessary, since if
$w_{j}=\sqrt{|j|+1}$, then $F_{w}$ also defines an invertible operator on
$s(\mathbb{Z})$. Under these assumptions we obtain, from Corollary 43, the
following equivalences:

    * (I)

$F_{w}:s(\mathbb{Z})\to s(\mathbb{Z})$ is topologically expansive;

    * (II)

there exists $k\in\mathbb{N}$ such that

      * (a)

$\sup\limits_{n\in\mathbb{N}}|w_{1}\cdot\ldots\cdot w_{n}|n^{k}=\infty$ or

      * (b)

$\sup\limits_{n\in\mathbb{N}}|w_{-n}\cdot\ldots\cdot
w_{-1}|^{-1}n^{k}=\infty$;

    * (III)

      * (a)

$F_{w}:s(\mathbb{Z})\to s(\mathbb{Z})$ is topologically positively expansive
or

      * (b)

$F_{w}^{-1}:s(\mathbb{Z})\to s(\mathbb{Z})$ is topologically positively
expansive.

  * (iv)

Let $l,r>0$. If $w_{j}=(|j|+1)^{r}$ for every $j\geq 1$ and
$w_{j}=(|j|+1)^{-l}$ for every $j\leq 0$, it follows from Corollary 43 that
both operators $F_{w}$ and $F_{w}^{-1}$ on $s(\mathbb{Z})$ are topologically
positively expansive.

  * (v)

If $v=(v_{j})_{j\in\mathbb{Z}}$ is a positive weight sequence and
$a_{j,k}=v_{j}$ for all $j\in\mathbb{Z}$ and $k\in\mathbb{N}$, then

$\lambda_{p}(A,\mathbb{Z})=\ell^{p}(\mathbb{Z},v)\ \ \ \ (\text{resp.
}\lambda_{0}(A,\mathbb{Z})=c_{0}(\mathbb{Z},v)).$

If, in addition,

$\sup_{j\in\mathbb{Z}}\frac{|w_{j}|v_{j+1}}{v_{j}}<\infty\ \ \ \text{ and }\ \
\ \sup_{j\in\mathbb{Z}}\frac{v_{j}}{|w_{j}|v_{j+1}}<\infty,$

which means that $F_{w}$ is a well-defined invertible operator on
$\ell^{p}(\mathbb{Z},v)$ (resp. on $c_{0}(\mathbb{Z},v)$), then Corollary 43
implies that the topologically positively expansive bilateral weighted forward
shifts are characterized by

$\sup\limits_{n\in\mathbb{N}}|w_{1}\cdot\ldots\cdot w_{n}|v_{n+1}=\infty.$

As we promised after the statement of Corollary 36, we will now exhibit some
examples of uniformly topologically expansive operators on $s(\mathbb{Z})$
that are not hyperbolic.

######  Example 45.

Consider $s(\mathbb{Z})$ as being $\lambda_{1}(A,\mathbb{Z})$ (Example
44(iii)), that is, consider $s(\mathbb{Z})$ endowed with the seminorms

$\|x\|_{k}\\!:=\sum_{j=-\infty}^{\infty}(|j|+1)^{k}|x_{j}|,\ \ \
x\\!:=(x_{j})_{j\in\mathbb{Z}}\in s(\mathbb{Z}),\ k\in\mathbb{N}.$

Given $a>1$, consider the weight sequence

$w\\!:=(\ldots,a^{-1},a^{-1},a^{-1},a,a,a,\ldots),$

where the first $a$ appears at position $0$. Then $F_{w}$ is an invertible
operator on $s(\mathbb{Z})$. Let $P_{M}:s(\mathbb{Z})\to M$ and
$P_{N}:s(\mathbb{Z})\to N$ denote the canonical projections associated to the
topological direct sum decomposition $s(\mathbb{Z})=M\oplus N$, where

$M\\!:=\\{x\in s(\mathbb{Z}):x_{j}=0\text{ for all }j\leq-1\\}\ \text{ and }\
N\\!:=\\{x\in s(\mathbb{Z}):x_{j}=0\text{ for all }j\geq 0\\}.$

For each $k\in\mathbb{N}$, let

$A_{k}\\!:=\\{x\in S_{\|\cdot\|_{k}}:\|P_{M}x\|_{k}\geq 1/2\\}\ \text{ and }\
B_{k}\\!:=\\{x\in S_{\|\cdot\|_{k}}:\|P_{N}x\|_{k}\geq 1/2\\}.$

Clearly, $S_{\|\cdot\|_{k}}=A_{k}\cup B_{k}$. Moreover, for each
$n\in\mathbb{N}$,

$\|F_{w}^{n}x\|_{k}\geq\|F_{w}^{n}P_{M}x\|_{k}\geq a^{n}\|P_{M}x\|_{k}\geq
a^{n}/2\ \text{ for all }x\in A_{k}$

and, similarly, $\|F_{w}^{-n}x\|_{k}\geq a^{n}/2$ for all $x\in B_{k}$. This
proves that $F_{w}$ is uniformly topologically expansive. On the other hand,
since $\|F_{w}^{n}x\|_{k}\to+\infty$ as $n\to\pm\infty$, for every nonzero
$x\in s(\mathbb{Z})$, $F_{w}$ is not hyperbolic.

By choosing $a\in(0,1)$ in the above example, we can fulfill the promise we
made at the end of Section 2 and exhibit some examples of generalized
hyperbolic operators on $s(\mathbb{Z})$ that are not hyperbolic.

######  Example 46.

Given $a\in(0,1)$, let $w$, $M$ and $N$ be as in the previous example. For
each $k\in\mathbb{N}$, define

$c_{k}\\!:=\sup_{n\in\mathbb{N}_{0}}\,[(n+1)^{k}(\sqrt{a})^{n}]<\infty.$

For any $y\in M$ and $n\in\mathbb{N}_{0}$,

$\displaystyle\|F_{w}^{n}y\|_{k}$
$\displaystyle=(n+1)^{k}a^{n}|y_{0}|+(n+2)^{k}a^{n}|y_{1}|+(n+3)^{k}a^{n}|y_{2}|+\cdots$
$\displaystyle=a^{n}\Big{(}(n+1)^{k}|y_{0}|+\frac{(n+2)^{k}}{2^{k}}\,2^{k}|y_{1}|+\frac{(n+3)^{k}}{3^{k}}\,3^{k}|y_{2}|+\cdots\Big{)}$
$\displaystyle\leq(n+1)^{k}a^{n}\|y\|_{k}$ $\displaystyle\leq
c_{k}(\sqrt{a})^{n}\|y\|_{k}.$

Analogously, for any $z\in N$ and $n\in\mathbb{N}_{0}$,

$\|F_{w}^{-n}z\|_{k}\leq c_{k}(\sqrt{a})^{n}\|z\|_{k}.$

Thus, $F_{w}$ is generalized hyperbolic. However, $F_{w}$ is not hyperbolic
because it is not topologically expansive. Indeed, for any $k\in\mathbb{N}$,
since

$\|F_{w}^{n}e_{0}\|_{k}\to 0\text{ as }n\to\pm\infty,$

we have that $\sup_{n\in\mathbb{Z}}\|F_{w}^{n}e_{0}\|_{k}<\infty$. Another way
to see that $F_{w}$ is not topologically expansive is to note that the weight
sequence $w$ does not satisfy condition (II) in Example 44(iii).

##  6\. Some open problems

If someone asks for an example of an operator with the shadowing property on a
Banach space $X$, one would certainly mention an operator such as $2I_{X}$ or
$\frac{1}{2}I_{X}$. The fact that these operators do not have the shadowing
property when $X=H(\mathbb{C})$ is somehow surprising and suggests the
following question.

Problem A. Does every Fréchet space (or locally convex space) support an
operator with the shadowing property? How about the Fréchet space
$H(\mathbb{C})$?

Similarly, the simplest examples of expansive operators on a Banach space $X$
are operators such as $2I_{X}$ of $\frac{1}{2}I_{X}$. But these operators are
not expansive when $X=\mathbb{K}^{\mathbb{Z}}$.

Problem B. Does every Fréchet space (or locally convex space) support an
expansive operator? How about the Fréchet space $\mathbb{K}^{\mathbb{Z}}$? (We
emphasize that expansive is meant in the metric sense.)

It was observed in Remark 9 that the hypothesis of completeness of the
seminorms is essential for the validity of Theorem 6, but we do not know if
the other technical hypothesis in Theorem 6 can be removed or not. In other
words, we have the following question.

Problem C. Let $X$ be a locally convex space whose topology is induced by a
directed family of complete seminorms. Is it true that generalized
hyperbolicity implies shadowing for operators on $X$?

We proved that every invertible generalized hyperbolic operator on a Banach
space is topologically stable (Corollary 26). However, in the case of an
arbitrary sequentially complete locally convex space, we proved that
invertible generalized hyperbolic operators have a stability property which
seems to be pretty close to topological stability (Theorem 25), but we had to
require the additional hypothesis that $T-S$ is a bounded map (see (21)) in
order to be able to apply Lemma 24. So, we have the following question.

Problem D. Can we remove the additional hypothesis that $T-S$ is a bounded map
from (21)? In other words, is every invertible generalized hyperbolic operator
on a non-normable sequentially complete locally convex space topologically
stable? How about the case of non-normable Fréchet spaces?

Topologically expansive weighted shifts on Fréchet sequence spaces were
characterized in Theorem 40.

Problem E. Characterize the concept of uniform topological expansivity for
weighted shifts on Fréchet sequence spaces.

As a final problem, let us mention that the following basic question is still
open.

Problem F. If $X$ is a Banach space and $T\in GL(X)$ has the shadowing
property, must $T$ be generalized hyperbolic?

## Acknowledgements

The first author is beneficiary of a grant within the framework of the grants
for the retraining, modality María Zambrano, in the Spanish university system
(Spanish Ministry of Universities, financed by the European Union,
NextGenerationEU) and was also partially supported by CNPq, Project
#308238/2021-4, and by CAPES, Finance Code 001. The fifth and the first
authors were partially supported by the Spanish Ministery
MCIN/AEI/10.13039/501100011033/FEDER, UE, Projects PID2019-105011GB-I00 and
PID2022-139449NB-I00. The fifth and the third authors were partially supported
by Generalitat Valenciana, Project PROMETEU/2021/070. The fourth author was
partially supported by FAPEMIG Grants RED-00133-21 and APQ-01853-23.

## References

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  * [2] D. V. Anosov,  Geodesic flows on closed Riemannian manifolds of negative curvature (Russian), Trudy Mat. Inst. Steklov.  90 (1967), 209 pp.
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Nilson C. Bernardes Jr.

Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica
de València, Edifici 8E, Acces F, 4a Planta, 46022 València, Spain, and
Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro,
Caixa Postal 68530, RJ 21945-970, Brazil.
e-mail address<EMAIL_ADDRESS>

Blas M. Caraballo

ESAP, Escuela Superior de Administración Pública, Cartagena de Indias,
Bolívar, Colombia.
e-mail address<EMAIL_ADDRESS>

Udayan B. Darji

Department of Mathematics, University of Louisville, Louisville, KY
40208-2772, USA.
e-mail address<EMAIL_ADDRESS>

Vinícius V. Fávaro

Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, MG
38400-902, Brazil.
e-mail address<EMAIL_ADDRESS>

Alfred Peris

Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica
de València, Edifici 8E, Acces F, 4a Planta, 46022 València, Spain.
e-mail address<EMAIL_ADDRESS>