Asymptotic limits of operators similar to normal operators

TL;DR

This paper generalizes Sz.-Nagy's theorem to operators similar to normal operators and characterizes the strong operator topology limits of their self-adjoint iterates, strengthening the understanding of contractions similar to unitaries.

Contribution

It extends Sz.-Nagy's theorem to a broader class of operators and characterizes their asymptotic behavior in the strong operator topology.

Findings

Generalization of the necessity part of Sz.-Nagy's theorem for normal-like operators.

Characterization of strong operator topology limits of self-adjoint iterates.

Strengthening of Sz.-Nagy's theorem for contractions on infinite-dimensional spaces.

Abstract

Sz.-Nagy's famous theorem states that a bounded operator $T$ which acts on a complex Hilbert space $\mathcal{H}$ is similar to a unitary operator if and only if $T$ is invertible and both $T$ and $T^{-1}$ are power bounded. There is an equivalent reformulation of that result which considers the self-adjoint iterates of $T$ and uses a Banach limit $L$. In this paper first we present a generalization of the necessity part in Sz.-Nagy's result concerning operators that are similar to normal operators. In the second part we provide characterization of all possible strong operator topology limits of the self-adjoint iterates of those contractions which are similar to unitary operators and act on a separable infinite-dimensional Hilbert space. This strengthens Sz.-Nagy's theorem for contractions.