Numerical radius preserving operators on 𝐵(𝐻)

Abstract

Let H be a Hilbert space over C \mathbb {C} and let B ( H ) B(H) denote the vector space of all bounded linear operators on H. We prove that a linear isomorphism T : B ( H ) → B ( H ) T:B(H) \to B(H) is numerical radius-preserving if and only if it is a multiply of a C ∗ {C^ \ast } -isomorphism by a scalar of modulus one.