On the Norm of a Generalized Derivation

Abstract

Let H be an infinite dimensional complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For two bounded operators A, B ∈ B(H), the map δAB : B(H) → B(H) is a generalized inner derivation operator induced by A and B defined by δAB(X) = AX − XB (1) In this paper we show that the norm of a generalized inner derivation operator is given by k(δAB/B(B(H)))k = kAk+kBk for all A, B ∈ B(H). Mathematics Subject Classification: Primary 47A30, Secondary 47L25