Weak Supercyclicity -- An Expository Survey

TL;DR

This survey explores the concepts of supercyclicity and weak supercyclicity in operators on normed spaces, analyzing their relationships with various stability notions and spectral properties to guide future research.

Contribution

It provides a comprehensive overview of weak supercyclicity, clarifies its connections with stability concepts, and describes the spectrum of weakly l-sequentially supercyclic operators.

Findings

Weak supercyclicity relates closely to stability properties.

Spectrum of weakly l-sequentially supercyclic operators is characterized.

The survey synthesizes recent developments in weak supercyclicity research.

Abstract

This is an exposition on supercyclicity and weak supercyclicity, especially designed to advance further developments in weakly supercyclicity, which is a recent research field showing significant momentum during the past two decades. For operators on a normed space, the present paper explores the relationship among supercyclicity and three versions of weak supercyclicity, namely, weak l-sequential, weak sequential, and weak supercyclicities, and their connection with strong stability, weak stability, and weak quasistability. A survey of the literature on weak supercyclicity is followed by an analysis of the interplay between supercyclicity and strong stability, as well as between weak supercyclicity and weak stability. A description of the spectrum of weakly l-sequentially supercyclic operators is also given.