Some improvements of the Katznelson-Tzafriri theorem on Hilbert space

TL;DR

This paper advances the Katznelson-Tzafriri theorem on Hilbert spaces by extending its applicability to broader classes of semigroups and representations, including unbounded cases.

Contribution

It introduces new versions of the theorem for bounded and unbounded representations of abelian semigroups, enhancing its scope and quantitative aspects.

Findings

Extended the theorem to bounded representations of large abelian semigroups

Provided a quantified version for contractive representations

Outlined an improved theorem for individual orbits, including unbounded cases

Abstract

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations.