Related papers: Occasionally attracting compact sets and compact-s…

Let $T$ be a positive operator on a complex Banach lattice. It is a long open problem whether the peripheral spectrum $\sigma_{\operatorname{per}}(T)$ of $T$ is always cyclic. We consider several growth conditions on $T$, involving its…

In this article, we address a problem posed by F. Bayart regarding the existence of an infinite-dimensional closed vector subspace (excluding the null operator) within the set of supercyclic operators on Banach spaces. We resolve this…

For a Banach space $X$ its subset $Y\subseteq X$ is called overcomplete if $|Y|=dens(X)$ and $Z$ is linearly dense in $X$ for every $Z\subseteq Y$ with $|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was introduced…

We give an affirmative answer to a question asked by Faghih-Ahmadi and Hedayatian regarding supercyclic vectors. We show that if $\mathcal X$ is an infinite-dimensional normed linear space and $T$ is a supercyclic operator on $\mathcal X$,…

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem~2 which states that for a locally compact group $G$, $G$ is compact if there…

A tuple $(T_1,\dots,T_n)$ of continuous linear operators on a topological vector space $X$ is called hypercyclic if there is $x\in X$ such that the the orbit of $x$ under the action of the semigroup generated by $T_1,\dots,T_n$ is dense in…

Given a power-bounded linear operator T in a Banach space and a probability F on the non-negative integers, one can form a `subordinated' operator S = \sum_k F(k) T^k. We obtain asymptotic properties of the subordinated discrete semigroup…

Let $X$ be a complex topological vector space with $dim(X)>1$ and $\mathcal{B}(X)$ the set of all continuous linear operators on $X$. The concept of hypercyclicity for a subset of $\mathcal{B}(X)$, was introduced in \cite{AKH}. In this…

A Banach space $X$ has \textit{property $(K)$}, whenever every weak* null sequence in the dual space admits a convex block subsequence $(f_{n})_{n=1}^\infty$ so that $\langle f_{n},x_{n}\rangle\to 0$ as $n\to \infty$ for every weakly null…

In the present paper we prove that a necessary condition for a Banach space $X$ to admit a generating compact Lipschitz retract $K$, which satisfies an additional mild assumption on its shape, is that $X$ enjoys the Bounded Approximation…

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of…

A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact…

An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly…

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions.…

A nonempty closed convex bounded subset $C$ of a Banach space is said to have the weak approximate fixed point property if for every continuous map $f:C\to C$ there is a sequence $\{x_n\}$ in $C$ such that $x_n-f(x_n)$ converge weakly to 0.…

There is no supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}.$ There exist, however, weakly l-sequentially supercyclic unitary operators$.$ We show that if $T$ is a weakly l-sequentially supercyclic power bounded operator…

We prove that a closed convex subset of a Banach space is (super-)weakly compact if and only if it is (super)-ergodic. As a consequence we deduce that super weakly compact sets are characterized by the fixed point property for continuous…

We provide a sufficient condition for an operator $T$ on a non-metrizable and sequentially separable topological vector space $X$ to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition…

Classes of Banach spaces that are finitely, strongly finitely or elementary equivalent are introduced. On sets of these classes topologies are defined in such a way that sets of defined classes become compact totally disconnected…

Letting $E$, $F$ be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator $E\rightarrow F$ is unconditionally converging, then every polynomial from $E$ to $F$ is unconditionally…