Related papers: Frequent hypercyclicity, chaos, and unconditional …

We show that if $X$ and $Y$ are Banach spaces, where $Y$ is separable and polyhedral, and if $T:X \to Y$ is a bounded linear operator such that $T^*(Y^*)$ contains a boundary $B$ of $X$, then $X$ is separable and isomorphic to a polyhedral…

We first generalize the results of Le\'on and M\"uller [Studia Math. 175(1) 2006] on hypercyclic subspaces to sequences of operators on Fr\'echet spaces with a continuous norm. Then we study the particular case of iterates of an operator T…

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (…

A linear operator $U$ acting boundedly on an infinite-dimensional separable complex Hilbert space $H$ is universal if every linear bounded operator acting on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its…

It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when }…

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…

We prove that a Tychonoff space $X$ is an Ascoli space (resp., a sequentially Ascoli space) if and only if for each Banach space $E$, every $k$-continuous and almost $k$-compact (resp., almost $k$-sequential) map $T$ form $X$ into the…

We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic.

It is shown that a specific ordering of the Rademacher chaos leads to a basic sequence in a wide class of symmetric spaces on the segment [0,1]. Necessary and sufficient conditions on a such space are found for Rademacher chaos to possess…

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and…

We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $\varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $\varepsilon$ such that $T+F$ has an…

Let $T:X\to X$ be a compact linear (or more generally affine) operator from a Banach space into itself. For each $x\in X$, the sequence of iterates $T^nx, n=0,1,...$ and its averages $\frac{1}{k}\sum_{k=0}^nT^{k-1}x, n=0,1,...$ are either…

A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result…

A sequence $\{T_n\}_{n=1}^{\infty}$ of bounded linear operators between separable Banach spaces $X, Y$ is called diskcyclic if there exists a vector $x\in X$ such that the disk-scaled orbit $\{\alpha T_n x: n\in \mathbb{N}, \alpha…

An operator $T$ acting on a Banach space $X$ is said to be recurrent if for each $U$; a nonempty open subset of $X$, there exists $n\in\mathbb{N}$ such that $T^n(U)\cap U\neq\emptyset.$ In the present work, we generalize this notion from a…

This paper deals with the problem of when, given a collection $\mathcal C$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space $Z$ with a Schauder basis so that every element in…

Let $X$ be a Banach space, $(e_n)_{n=1}^\infty$ be its basis, and $S_\alpha$ be a Schreier family of order alpha. We introduce Condition A which is a weaker version of the Continuum Hypothesis. Granted Condition A, we show that if the basis…

A Banach space $X$ is said to have the Daugavet property if every operator $T: X\to X$ of rank~$1$ satisfies $\|Id+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy…

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially…

We study recurrent operators from a new perspective by introducing the notion of hyper-recurrent operators and establish robust connections with quasi-rigid operators. For example, we prove that a recurrent operator on a separable Banach…