Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators

Let B ( H ) be the algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space H . For every T ∈ B ( H ) , let m ( T ) and q ( T ) denote the minimum modulus and surjectivity modulus of T respectively. Let ϕ : B ( H ) → B ( H ) be a surjective linear map. In this paper, we prove that the following assertions are equivalent:(i) ) = m ( ϕ ( T ) ) for all T ∈ B ( H ) , ) = q ( ϕ ( T ) ) for all T ∈ B ( H ) , exist two unitary operators U , V ∈ B ( H ) such that ϕ ( T ) = U T V for all T ∈ B ( H ) . generalizes the result of Mbekhta [7, Theorem 3.1] to the non-unital case.