WEYL'S AND BROWDER'S THEOREM FOR AN ELEMENTARY OPERATOR

Let H be a separable infinite dimensional complex Hilbert space and let B(H) denote the algebra of bounded operators on H into itself. The generalized derivation delta(A,B) is defined by delta(A,B)(X) = AX - XB. For pairs C = (A(1), A(2)) and D = (B-1, B-2) of operators, we define the elementary operator Phi(C,) (D) by Phi(C,) (D)(X) = A(1) X B-1 - A(2) X B-2. If A(2) = B-2 = I, we get the elementary operator Delta(A1,) (B1) (X) = A(1) X B-1 - X. Let d(A,) (B) = delta(A,) (B) or Delta(A,) (B). We prove that if A, B* are log-hyponormal, then f (d(A,) (B)) satisfies (generalized) Weyl's Theorem for each analytic function f on a neighborhood of sigma(d(A,B)), we also prove that f(Phi(C,D)) satisfies Browder's Theorem for each analytic function f on a neighborhood of sigma(Phi(C,D)).