On the range of the map N AB

Abstract EN

Let H be an infinite dimensional separable complex Hilbert space and B (H) be the Banach algebra of all bounded linear operators on H.

In this paper we introduce a mapping A^: B (H) -> B (H) .

By ^B (T) =AT- T*B Te B (H) We study some’ properties of it, and we study subjectivity of this mapping when A is pseudo normal operator whose spectrum satisfies certain properties if the analytic function f(A) that belongs to the (RangeNAA ) then f(A) is the zero function .Also we generalize some results for the Jordan * derivation JA and the derivation D when A is normal operator and prove it when A is pseudo normal operator.