ON THE RANGE OF THE MAP N AB

Abstract

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 : B(H) → B(H) . By (T)=AT-T*B , T B(H). ABNABN
We study some properties of it , and we study surjectivity of this mapping when A is pseudonormal operator whose spectrum satisfies certain properties if the analytic function f(A) that belongs to the *AARangeN then f(A) is the zero function .Also we generalize some results for the Jordan * derivation and the derivation when A is normal operator and prove it when A is pseudonormal operator.