Operators commuting with the Volterra operator are not weakly supercyclic

TL;DR

This paper proves that all bounded linear operators on Lp[0,1] commuting with the Volterra operator are not weakly supercyclic, answering a previously open question in operator theory.

Contribution

It establishes that operators commuting with the Volterra operator cannot be weakly supercyclic, providing an algebraic condition that explains this property.

Findings

Operators commuting with V are not weakly supercyclic

Provides an algebraic condition preventing weak supercyclicity

Answers an open question by Le9on-Saavedra and Piqueras-Lerena

Abstract

We prove that any bounded linear operator on $L_p[0,1]$ for $1\leq p<\infty$, commuting with the Volterra operator $V$, is not weakly supercyclic, which answers affirmatively a question raised by L\'eon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic flavored condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with $V$.