ON THE RANGE OF THE MAP N AB

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 :N AB→ B(H) → B(H) . By N AB(T)=AT-T*B , Tϵ 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 (Range NAA)* then f(A) is the zero function .Also we generalize some results for the Jordan * derivation J A and the derivatio D A when A is normal operator and prove it when A is pseudonormal operator.