Numerical Radius Perserving Operators on B(H)

On a Numerical Radius Preserving Onto Isometry onL(X)

“…In paper [6] of Chan, it was shown that if is a numerical radius preserving isomorphism on L(H), then ( ) = for some constant of modulus 1, where is the identity map on H. We show that the same result holds for an onto isometry on L( ) when has uniform convexity and uniform smoothness. Before that, we first see the following.…”

“…Particularly, when H is a complex Hilbert space, it was shown that an isomorphism on L(H) is * -isomorphism if and only if it is numerical range preserving. Later, Chan [6] showed that an isomorphism on L(H) is numerical radius preserving if and only if is a * -isomorphism for some scalar of modulus 1. These results say that for each numerical radius preserving isomorphism on L(H) there exists a scalar of modulus 1 such that is a numerical range preserving mapping.…”