Notes on subspace-hypercyclic operators

Abstract

a b s t r a c tA bounded linear operator T on a Banach space X is called subspace-hypercyclic for aIn this case we show that p(T ) has a relatively dense range for every real or complex polynomial p, which in turn answers a question posed in Madore and Martinez-Avendano (2011) [8]. As a consequence, the algebraic structure of the set of subspace-hypercyclic vectors can be described.

Some questions about subspace-hypercyclic operators

“…In the present paper we answer negatively the following questions from [7]: Question 1. Let M be a nontrivial subspace of a Banach space X and x ∈ M .…”

“…We show examples that answer some questions posed by H. Rezaei [7]. In particular, we provide an example of an operator T such that Orb(T, x) ∩ M is somewhere dense in M , but it is not everywhere dense in M .…”

“…Rezaei shows in [7] that, if a bounded linear operator T acting on a Banach space X is subspace-hypercyclic for some subspace M of X and p is complex polynomial, then ker(p(T * )) ⊆ M ⊥ , which provides an affirmative answer to question (v) of [6]. Also, he proves as a consequence that, under general additional conditions, a subspace-hypercyclic operator T for a subspace M of X has a dense linear manifold of M consisting entirely, except for zero, of vectors that are subspace-hypercyclic for T .…”

“…Also, he proves as a consequence that, under general additional conditions, a subspace-hypercyclic operator T for a subspace M of X has a dense linear manifold of M consisting entirely, except for zero, of vectors that are subspace-hypercyclic for T . More examples and results of subspace-hypercyclic operators can be found in [5,6,7].…”

The $M$-Hypercyclicity Criterion of $C_{0}$-Semigroups

“…Let a bounded operator T on separable Banach space X and M = {0} a subspace of X, T is called subspace-hypercyclic (or M -hypercyclic) if there is a vector x ∈ X such that Orb(T, x) ∩ M is dense in M . Madore, Martinez-Avendano [15] and Rezaei [18] proved that if T is M −transitive, then T is M −hypercyclic, recall that a operator is M -transitive if for each pair of non-empty open sets of M there is n ∈ N such that T n (U ) ∩ V contains a non-empty relatively open set in M . Madore, Martinez-Avendano [15] proved that the converse is not always true, there are subspace-hpercyclic operators that not subspace-transitive (see [10], [14] and [22]).…”