On isometric asymptotes of operators quasisimilar to isometries

TL;DR

This paper constructs operators quasisimilar to isometries that lack isometric asymptotes, expanding understanding of the asymptotic behavior of such operators and challenging previous assumptions about their structure.

Contribution

It introduces new examples of operators quasisimilar to isometries without isometric asymptotes, including a contraction related to the unilateral shift.

Findings

Operators quasisimilar to isometries can lack isometric asymptotes.

Constructed a contraction quasisimilar to the unilateral shift with a non-zero unitary summand in its asymptote.

Extended the theory of asymptotic behavior of power bounded operators.

Abstract

The notion of isometric and unitary asymptotes was introduced for power bounded operators in 1989 and was generalized in 2016--2019 by K\'erchy. In particular, it was shown that there exist operators without unitary asymptote. In this paper operators are constructed which are quasisimilar to isometries and do not have isometric asymptotes. Also a contraction is constructed which is quasisimilar to the unilateral shift of infinite multiplicity and whose isometric asymptote contains a (non-zero) unitary summand.