Ten questions in linear dynamics

Abstract

Linear dynamical systems are systems of the form (X, T), where X is an infinitedimensional separable Banach space and T ∈ B(X) is a bounded linear operator on X. We present and motivate ten questions concerning these systems, which bear on both topological and ergodic-theoretic aspects of the theory.

Mixing operators with prescribed unimodular eigenvalues

“…Since Hilbert spaces are of cotype 2, the main theorem from [4] implies that, in the situation of Theorem 1.1, for countable Z the Taylor shift T is not ergodic in the Gaussian sense. So, as already mentioned above, Question 3 from [10] can be answered in the negative in so far as the existence of a spanning set of eigenvectors corresponding to a rationally independent set of unimodular eigenvalues does not always imply ergodicity in the Gaussian sense. We are left with the open question whether T is ergodic with respect to some measure of full support or (upper) frequently hypercyclic.…”

“…In this paper, we give examples of topologically mixing operators on Hilbert spaces which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to an arbitrary prescribed closed, countable subset Z of the unit circle T. In particular, such operators cannot be ergodic in the Gaussian sense. Choosing Z as a rationally independent set, Question 3 from [10] can be answered in the negative, at least in the weak form that ergodicity in the Gaussian sense does not always follow from the existence of a spanning set of eigenvectors corresponding to a rationally independent set of unimodular eigenvalues.…”

“…(2) The statement (and proof) of Theorem 2.6 can be modified in such a way that * ∩ T ⊂ Z, that is, the spectrum intersects T only in Z (cf. [10,Question 3]).…”

Linear Dynamical Systems

“…Most of the questions already appear in the literature and questions 1-3 were kindly suggested by Sophie Grivaux. Further open questions can also be found in [88] and [96].…”

Frequently hypercyclic operators with irregularly visiting orbits

“…It follows by considering a countable basis (U p ) p≥1 of non-empty open subsets of X that m-almost every vector x ∈ X is frequently hypercyclic for T , and satisfies dens N T (x, U p ) = m(U p ) for every p ≥ 1. Hence the following question, which was posed in a first version, dating from 2013, of the survey paper [11], arises naturally:…”