Entropy for compact operators and results on entropy and specification

TL;DR

This paper studies the topological entropy of operators in Banach and F-spaces, revealing that compact operators have finite entropy depending on their spectrum, and explores the implications of the specification property.

Contribution

It establishes the relationship between the specification property and entropy for operators on Banach and F-spaces, including new results on the invariance principle.

Findings

Compact operators have finite entropy depending on their point spectrum.

Specification property implies infinite topological entropy.

Operator specification property implies positive entropy.

Abstract

We investigate the topological entropy of operators. More precisely, in the Banach space setting, we show that compact operators have finite entropy, which depends solely on their point spectrum. Moreover, for operators on \(F\)-spaces, we explore the relationship between the specification property and entropy. In particular, we show that the specification property implies infinite topological entropy, while the operator specification property implies positive entropy. We also show that the invariance principle fails for the class of compact operators.