On convex-cyclic operators

TL;DR

This paper characterizes convex-cyclic operators using Hahn-Banach, provides examples and conditions for convex-cyclicity, and explores properties of specific classes of operators like m-isometries and diagonalizable normal operators.

Contribution

It offers a Hahn-Banach characterization of convex-cyclicity, constructs examples addressing open questions, and characterizes convex-cyclic properties for various operator classes.

Findings

Constructed an example of a convex-cyclic operator with empty point spectrum of the adjoint.

Proved that m-isometries are not convex-cyclic.

Characterized convex-cyclic diagonalizable normal operators and certain multiplication operators.

Abstract

We give a Hahn-Banach Characterization for convex-cyclicity. We also obtain an example of a bounded linear operator $S$ on a Banach space with $\sigma_{p}(S^*)=\emptyset$ such that $S$ is convex-cyclic, but $S$ is not weakly hypercyclic and $S^2 $ is not convex-cyclic. This solved two questions of Rezaei in \cite{Rezaei} when $\sigma_p(S^*)=\varnothing$. %Recently, Le\'on-Saavedra and Romero de la Rosa \cite{LeRo} provide an example of a convex-cyclic operator $S$ such that the power $S^n$ fails to be convex-cyclic with $\sigma _p(S^*)\neq \varnothing$. In fact they solved tree questions posed by Rezaei in \cite{Rezaei}. Moreover, we prove that $m$-isometries are not convex-cyclic and that $\varepsilon$-hypercyclic operators are convex-cyclic. We also characterize the diagonalizable normal operators that are convex-cyclic and give a condition on the eigenvalues of an arbitrary operator for it to be convex-cyclic. We show that certain adjoint multiplication operators are convex-cyclic and show that some are convex-cyclic but no convex polynomial of the operator is hypercyclic. Also some adjoint multiplication operators are convex-cyclic but not 1-weakly hypercyclic.