Numerical radius in Hilbert C✻-modules

Abstract

. Utilizing the linking algebra of a Hilbert C ∗ -module (cid:2) V , (cid:3)·(cid:3) (cid:3) , we introduce Ω ( x ) as a de fi nition of numerical radius for an element x ∈ V and then show that Ω ( · ) is a norm on V such that 12 (cid:3) x (cid:3) (cid:2) Ω ( x ) (cid:2) (cid:3) x (cid:3) . In addition, we obtain an equivalent condition for Ω ( x ) = 12 (cid:3) x (cid:3) . Moreover, we present a re fi nement of the triangle inequality for the norm Ω ( · ) . Some other related results are also discussed.