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

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.