Topologically transitive sequence of cosine operators on Orlicz spaces
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00088-4 ORIGINAL PAPER
Topologically transitive sequence of cosine operators on Orlicz spaces Ibrahim Akbarbaglu1 · Mohammad Reza Azimi2 · Vishvesh Kumar3 Received: 12 March 2020 / Accepted: 26 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract For a Young function 𝜙 and a locally compact second countable group G, let L𝜙 (G) denote the Orlicz space on G. In this paper, we present a necessary and suf‑ ficient condition for the topological transitivity of a sequence of cosine operators n n )}∞ + Sg,w , defined on L𝜙 (G) . We investigate the conditions {Cn }∞ ∶= { 12 (Tg,w n=1 n=1 for a sequence of cosine operators to be topologically mixing. Further, we go on to prove a similar result for the direct sum of a sequence of cosine operators. Finally, we give an example of topologically transitive sequence of cosine operators. Keywords Hypercyclicity · Topologically transitive · Topologically mixing · Weighted translation operator · Orlicz space · Locally compact group Mathematics Subject Classification 47A16 · 46E30 · 22D05
Communicated by Pedro Tradacete. * Ibrahim Akbarbaglu [email protected]; [email protected] Mohammad Reza Azimi [email protected] Vishvesh Kumar [email protected] 1
Department of Mathematics, Farhangian University, Tehran, Iran
2
Department of Mathematics, Faculty of Sciences, University of Maragheh, Maragheh 55181‑83111, Iran
3
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Ghent, Belgium
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I. Akbarbaglu et al.
1 Introduction and preliminaries A sequence of bounded linear operators {Tk }k∈ℕ0 acting on a Fréchet space X is said to be topologically transitive if for any pair (U, V) consisting of two non-empty open subsets U and V of X, there exists an n ∈ ℕ such that Tn (U) ∩ V ≠ � . A bounded linear operator T is said to be topologically transitive if the sequence {Sk }k∈ℕ0 with Sk ∶= T k , the k-iterates of T with the convention that T 0 = I , the identity operator, is topologi‑ cally transitive as a sequence of bounded linear operators. A bounded linear operator T is called hypercyclic if there exists a vector x ∈ X, called hypercyclic vector, such that the orbit {T k x ∶ k = 0, 1, 2, ...} of x is dense in X, where T 0 is the identity operator on X. It is worth mentioning that these two notions, namely, topological transitivity and hypercyclicity of an operator are more likely equivalent on a Fréchet space X [3, 8]. An operator T is called topologically mixing whenever for any pair (U, V) of two nonempty open subsets U and V of X, there exists an N ∈ ℕ such that T n (U) ∩ V ≠ � for every n ≥ N . An operator of the form I + B , where B denotes the backward shift opera‑ tor, is an example of topologically mixing operator which is also hypercyclic. We say a bounded linear operator T on a Fréchet space X is weakly mixing if and only if T ⊕ T is hypercyclic on X ⊕ X . Note that we
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