On orbits of truncated convolution operators

TL;DR

This paper investigates the dynamics of truncated convolution operators on L^p spaces, proving non-supercyclicity for semigroups generated by finitely many such operators and identifying operators with irregular vectors.

Contribution

It establishes the non-supercyclicity of semigroups generated by finitely many truncated convolution operators and demonstrates the existence of operators with irregular vectors.

Findings

Semigroups generated by finitely many truncated convolution operators are non-supercyclic.

Existence of a truncated convolution operator with irregular vectors.

Provides insights into the dynamical properties of convolution operators.

Abstract

We prove that a semigroup generated by a finitely many truncated convolution operators on $L^p[0,1]$ with $1\leq p<\infty$ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.