On spaceability of shifts-like operators on $L^p$

TL;DR

This paper demonstrates the spaceability of hypercyclic vectors for shifts-like operators on $L^p$ spaces, providing new insights into their structure and properties, especially for dissipative composition operators.

Contribution

It establishes the spaceability of hypercyclic vectors for shifts-like operators and characterizes weakly mixing dissipative composition operators of bounded distortion.

Findings

Hypercyclic vectors form a spaceable set for shifts-like operators.

Characterization of weakly mixing dissipative composition operators.

Provides new tools for analyzing composition operators on $L^p$ spaces.

Abstract

We prove the spaceability of the set of hypercyclic vectors for {\em shifts-like operators}. Shift-like operators appear naturally as composition operators on $L^p(X)$, when the underlying space $X$ is dissipative. In the process of proving the main theorem, we provide, among other results of independent interest, a characterization of weakly mixing dissipative composition operators of bounded distortion.