Topological transitive sequence of cosine operators on Orlicz space

TL;DR

This paper characterizes when sequences of cosine operators on Orlicz spaces are topologically transitive or mixing, providing necessary and sufficient conditions, and includes an example illustrating these properties.

Contribution

It establishes new criteria for topological transitivity and mixing of cosine operators on Orlicz spaces, extending operator theory in this context.

Findings

Derived necessary and sufficient conditions for topological transitivity.

Identified conditions for topological mixing of cosine operators.

Provided an explicit example of a topologically transitive cosine operator sequence.

Abstract

For a Young function $\phi$ and a locally compact second countable group $G,$ let $L^\phi(G)$ denote the Orlicz space on $G.$ In this article, we present a necessary and sufficient condition for the topological transitivity of a sequence of cosine operators $\{C_n\}_{n=1}^{\infty}:=\{\frac{1}{2}(T^n_{g,w}+S^n_{g,w})\}_{n=1}^{\infty}$, defined on $L^{\phi}(G)$. We investigate the conditions for a sequence of cosine operators to be topological mixing. Moreover, we go on to prove the similar results for the direct sum of a sequence of the cosine operators. At the last, an example of a topological transitive sequence of cosine operators is given.