Another generalization of the numerical radius for Hilbert space operators

Abstract

Let B(H) be the C⁎-algebra of all bounded linear operators on a Hilbert space H. Let N(⋅) be an arbitrary norm on B(H) and I stand for the identity operator. For T∈B(H), we introduce the wN(T) as an extension of the classical numerical radius based on the Birkhoff–James orthogonality bywN(T)=sup⁡{|ξ|:ξ∈C,I⊥BN(T−ξI)}, and present some of its essentially properties. Moreover, we give a concrete example of this seminorm. Among other things, we obtain a necessary and sufficient condition that wN(⋅) be a norm on B(H). When wN(⋅) is a norm, then the geometry of normed space (B(H),wN(⋅)) is also investigated.