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

Let B(H)B(H) be the C*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)B(H) the following estimation holds‖S⊗S−1+S−1⊗S‖λ⩽‖S‖‖S−1‖+1‖S‖‖S−1‖ where ‖.‖λ‖.‖λ is the injective norm on the tensor product B(H)⊗B(H)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)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)B(H) by inequalities or equalities.