A note on the range of generalized derivation

Abstract

Let ℒ(H) denote the algebra of bounded linear operators on a complex separable and infinite dimensional Hilbert space H: For A;B ϵ ℒ(H); the generalized derivation σA;B associated with (A;B); is de¯ned by σᴀ,ʙ; (X) = AX-XB for X ϵ ℒ(H): In this note we give some su±cient conditions for A and B under which the intersection between the closure of the range of σᴀʙ, respect to the given topology and the kernel of σᴀ∗,ʙ∗;