Abstract. An asymmetric Fuglede-Putnam's Theorem for w -hyponormal operators and dominant operators is proved, as a consequence of this result, we obtain that the range of the generalized derivation induced by the above classes of operators is orthogonal to its kernel. Mathematics subject classification (2010): 47B47, 47A30, 47B20.

Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.2000 Mathematics Subject Classification. 47B47, 47B20, 47B10.

We give necessary and sufficient conditions under which the norm of basic elementary operators attains its optimal value in terms of the numerical range.

Given Banach spaces X and Y and Banach space operators A ∈ L(X ) andThis paper is concerned with the problem of the transferring the left polaroid property, from operators A and B * to the generalized derivation δ A,B . As a consequence, we give necessary and sufficient conditions for δ A,B to satisfy generalized a-Browder's theorem and generalized a-Weyl's theorem. As application, we extend some recent results concerning Weyl type theorems.2000 Mathematics Subject Classification. 47A10, 47A53, 47B47.

No abstract