Boundedly Spaced Subsequences and Weak Dynamics

TL;DR

This paper characterizes weak supercyclicity for Hilbert-space contractions and unitary operators, linking it to boundedly spaced subsequences and exploring conditions for weak stability and supercyclicity.

Contribution

It provides a new characterization of weak supercyclicity for unitary operators using boundedly spaced subsequences, addressing an open question in operator theory.

Findings

Weak supercyclicity for Hilbert-space contractions is equivalent to that for unitary operators.

Characterization of weakly supercyclic unitary operators via boundedly spaced subsequences.

Weak supercyclicity for weakly unstable operators relates to their spectral properties.

Abstract

The purpose of this paper is to characterize weak supercyclicity for Hilbert-space contractions, which is shown to be equivalent to characterizing weak supercyclicity for unitary operators$.$ This is naturally motivated by an open question that asks whether every weakly supercyclic power bounded operator is weakly stable (which in turn is naturally motivated by a result that asserts that every supercyclic power bounded operator is strongly stable)$.$ Precisely, weakly supercyclicity is investigated in light of boundedly spaced subsequences as discussed in Lemma 3.1$.$ The main result in Theorem 4.1 characterizes weakly l-sequentially supercyclic unitary operators $U\!$ that are weakly unstable in terms of boundedly spaced subsequences of the power sequence $\{U^n\}.$ Remark 4.1 shows that characterizing any form of weak super\-cyclicity for weakly unstable unitary operators is equivalent to characterizing any form of weak supercyclicity for weakly unstable contractions after the Nagy--Foia\c s--Langer decomposition.