A Coanalytic Rank on Super-Ergodic Operators

TL;DR

This paper uses Descriptive Set Theory to analyze the topological complexity of various classes of operators in Banach spaces, introducing a coanalytic rank to characterize super-ergodic operators and related spaces.

Contribution

It introduces a coanalytic rank to classify super-ergodic operators and explores the topological complexity of operator families in infinite-dimensional Banach spaces.

Findings

Families of ergodic, uniform-ergodic, Cesaro-bounded, and power-bounded operators are Borel sets.

The family of super-ergodic operators is either coanalytic or Borel depending on the space.

Coanalytic ranks and trees are used to characterize super-ergodic operators and space structures.

Abstract

Techniques from Descriptive Set Theory are applied in order to study the Topological Complexity of families of operators naturally connected to ergodic operators in infinite dimensional Banach Spaces. The families of ergodic, uniform-ergodic,Cesaro-bounded and power-bounded operators are shown to be Borel sets, while the family of super-ergodic operators is shown to be either coanalytic or Borel according to specific structures of the Space. Moreover, trees and coanalytic ranks are introduced to characterize super-ergodic operators as well as spaces where the above classes of operators do not coincide.