Linear maps preserving the minimum and surjectivity moduli of operators

Abstract

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H, and denote by m(T) and q(T) respectively the minimum modulus and the surjectivity mod- ulus for every T ∈ B(H). In this paper, we prove that if φ is a surjective unital linear map on B(H) ,t hen m(φ(T)) = m(T) for every T ∈ B(H) if and only if q(φ(T)) = q(T) for every T ∈ B(H) if and only if there exists an unitary operator U ∈ B(H) such that φ(T )= UTU ∗ for all T ∈ B(H).