LAUSR

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 w N ( T ) as an extension of the classical numerical radius based on the Birkhoff–James orthogonality by w N ( T ) = sup ⁡ { | ξ | : ξ ∈ C , I ⊥ B N ( T − ξ I ) } , 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 w N ( ⋅ ) be a norm on B ( H ) . When w N ( ⋅ ) is a norm, then the geometry of normed space ( B ( H ) , w N ( ⋅ ) ) is also investigated.