Norm of a Derivation and Hyponormal Operators

Let H be a complex Hilbert space and let L(H) be the algebra of all bounded linear operators on H. Following [3], a (symmetric) norm ideal (J, ‖.‖J) in L(H) consists of a proper two-sided ideal J together with a norm ‖.‖J satisfying the conditions: (i) (J, ‖.‖J) is a Banach space; (ii) ‖AXB‖J ≤ ‖A‖‖X‖J‖B ‖ for all X ∈ J and all operators A and B in L(H); (iii) ‖X‖J = ‖X ‖ for X a rank one operator. For a complete account of the theory of norm ideals, we refer to [3], [7] and [8]. In the sequel we will be particularly interested in operators belonging to the Hilbert-Schmidt class C2(H) which represents a Hilbert space when equipped with the inner product < X,Y> = tr(XY ∗), (X,Y ∈ C2(H)) where tr stands for the usual trace functional and Y ∗ denotes the adjoint of Y. For A ∈ L(H), the inner derivation induced by A is the operator δA defined on L(H) by δA(X) = AX −XA, X ∈ L(H)