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 ‖(δAB/B(B(H)))‖ = ‖A‖+‖B‖ for all A,B ∈ B(H). Mathematics Subject Classification: Primary 47A30, Secondary 47L25