Entropy for compact operators and results on entropy and specification

TL;DR

The paper analyzes topological entropy for linear operators on infinite-dimensional spaces, establishing that compact operators have finite entropy governed by their point spectrum through $h_{top}(T)=\sum_{|\lambda|>1}\log|\lambda|$. It introduces the specification property (SP) and the operator specification property (OSP) in $F$-spaces, proving SP implies infinite entropy while OSP yields positive entropy, and showing that the variational principle can fail in the compact-operator setting. The results are derived via a Riesz-decomposition-based spectral decomposition that reduces entropy to the unstable subspace, with consequences for invariant measures. The paper also provides examples, including weighted shifts with infinite entropy and SP-realized weighted shifts on certain $F$-spaces, clarifying the relationship between chaotic behavior and entropy in infinite dimensions.

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.

Theorems & Definitions (27)

• Theorem A
• Theorem B
• Theorem 2.1: meise1997introduction
• Theorem 2.2: grosse2011linear
• Definition 2.3
• Theorem 2.4: walters2000introduction
• Theorem 2.5: walters2000introduction
• Definition 2.6
• Definition 2.7
• Lemma 3.1