Let B(H) be the C^∗ -algebra of all bounded linear operators acting on a complex separable Hilbert space H . We shall show that:

1. The class of all selfadjoint operators in B(H) multiplied by scalars is characterized by ∀X ∈ B(H), S²X +XS² =>2||SXS||, (S ∈ B(H)).

2. The class of all normal operators in B(H) is characterized by each of the three following properties (where DS = S^∗S−SS^∗ , for S ∈ B(H)),

(i) ∀X ∈ B(H), S²X + XS² =>2||SXS||,(S ∈ B(H)),

(ii) S∗DSS = 0 = SDSS∗,(S ∈ B(H)),

(iii) S^∗D_SS=> 0 =>SD_SS^∗,(S ∈ B(H)).