Asymptotic behaviour of Hilbert space operators with applications

TL;DR

This dissertation explores the asymptotic behavior of power-bounded Hilbert space operators, providing characterizations, generalizations of classical theorems, and insights into their structural properties and applications.

Contribution

It offers new characterizations of asymptotic operators, generalizes Sz.-Nagy's similarity theorem, and investigates cyclic properties of weighted shift operators on directed trees.

Findings

Characterization of asymptotic operators

Generalization of Sz.-Nagy's similarity theorem

Results on cyclic properties of weighted shift operators

Abstract

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees.