Numerical radius norm and extreme contractions of $L(H)$

TL;DR

This paper characterizes extreme contractions of the space of bounded linear operators on a Hilbert space using the numerical radius norm, revealing differences from the usual operator norm and identifying new classes of extreme contractions.

Contribution

It provides a characterization of extreme contractions under the numerical radius norm, showing that some unitary operators are not extreme and some non-unitary are, unlike in the operator norm case.

Findings

Existence of non-extreme unitary operators under the numerical radius norm

Presence of non-unitary extreme contractions in $L(H)$

Differences from the behavior under the usual operator norm

Abstract

Suppose $L(H)$ is the space of all bounded linear operators on a complex Hilbert space $H.$ This article deals with the problem of characterizing the extreme contractions of $L(H)$ with respect to the numerical radius norm on $L(H).$ In contrast to the usual operator norm, it is proved that there exists a class of unitary operators on $H$ which are not extreme contractions when the numerical radius norm is considered on $L(H).$ Moreover, there are non-unitary operators on $H$ which are extreme contractions as far as the numerical radius norm is concerned.