On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities

Let B ( H ) be the C * -algebra of all bounded linear operators acting on a complex Hilbert space H. In this note, we shall show that if S is an invertible normal operator in B ( H ) the following estimation holds ‖ S ⊗ S −1 + S −1 ⊗ S ‖ λ ⩽ ‖ S ‖ ‖ S −1 ‖ + 1 ‖ S ‖ ‖ S −1 ‖ where ‖ . ‖ λ is the injective norm on the tensor product B ( H ) ⊗ B ( H ) . This last inequality becomes an equality when S is invertible self-adjoint. On the other hand, we shall characterize the set of all invertible normal operators S in B ( H ) satisfying the relation ‖ S ⊗ S −1 + S −1 ⊗ S ‖ λ = ‖ S ‖ ‖ S −1 ‖ + 1 ‖ S ‖ ‖ S −1 ‖ and also we shall give some characterizations of some subclasses of normal operators in B ( H ) by inequalities or equalities.