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*DSS=> 0 =>SDSS*,(S ∈ B(H)).  

In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (self-adjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities.