$q$-Frequently hypercyclic operators

Abstract

We introduce q-frequently hypercyclic operators and derive a sufficient criterion for a continuous operator to be q-frequently hypercyclic on a locally convex space. Applications are given to obtain q-frequently hypercyclic operators with respect to the norm-, F -norm-and weak*-topologies. Finally, the frequent hypercyclicity of the non-convolution operator T µ defined by T µ (f )(z) = f ′ (µz), |µ| ≥ 1 on the space H(C) of entire functions equipped with the compact-open topology is shown. * Corresponding author. 2010 Mathematics Subject Classification. Primary 47A16; Secondary 46A45. Key words and phrases. q-frequently hypercyclic operator, q-frequent hypercyclicity criterion, unconditional convergence, symmetric Schauder basis, backward shift operator.

F-hypercyclic operators on Fréchet spaces

“…(iii) The lower (m n )-density of A, denoted by d mn (A), is defined through: The notion in (i)-(ii) has been considered recently by Gupta and Mundayadan [20] in the case that q ∈ N, while this notion seems to be new provided that q ∈ [1, ∞) \ N. To the best knowledge of the author, the notion in (iii)-(iv) has not been considered elsewhere.…”

q-FREQUENT HYPERCYCLICITY IN AN ALGEBRA OF OPERATORS

“…The q-frequent hypercyclicity introduced in [11] is to control the frequency of the T -orbit intersecting with each open set. Definition 2.4.…”

“…Analogous to the frequent hypercyclicity criterion, we have a criterion for q-frequently hypercyclic operators and its proof is given in [11] and [12]. Theorem 2.5 (q-Frequent Hypercyclicity Criterion).…”

“…The q-frequent hypercyclicity in a locally convex space was first introduced in [11] and also can be seen as a particular type of (m k )-hypercyclicity appeared in [3]. In terms of frequency, one can see that the q-frequent hypercyclicity lies between hypercyclicity and frequent hypercyclicity.…”

Dynamics of non-convolution operators and holomorphy types

“…In [4], Aron or variants of it, has also been studied in [22,28,31,36,40]. In [36] we extended the work of [4] analyzing the hypercyclic behavior of some non-convolution operators on the space H(C N ).…”