On Weak Supercyclicity II

TL;DR

This paper explores weak supercyclicity in bounded linear operators, revealing limitations in Hilbert spaces and characterizing properties of normed-space operators, including spectral and compactness conditions.

Contribution

It provides new insights into weak supercyclicity for various classes of operators, including spectral properties and equivalences for compact operators.

Findings

Self-adjoint operators are not weakly supercyclic.

Weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint.

The point spectrum of the adjoint of a power bounded supercyclic operator is limited.

Abstract

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is either empty or is a singleton in the open unit disk, (v) weak l-sequential supercyclicity coincides with supercyclicity for compact operators, and (vi) every compact weakly l-sequentially supercyclic operator is quasinilpotent.