Weyl's theorem for algebraically k-quasiclass A operators

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If T or T is an algebraically k-quasiclass A operator acting on an infinite di- mensional separable Hilbert space and F is an operator commuting with T, and there exists a positive integer n such that F n has a finite rank, then we prove that Weyl's theorem holds for f(T)+F for every f2 H( (T)), where H( (T)) denotes the set of all analytic functions in a neighborhood of (T). Moreover, if T is an algebraically k-quasiclass A operator, then -Weyl's theorem holds for f(T). Also, we prove that if T or T is an algebraically k-quasiclass A operator then both the Weyl spectrum and the approximate point spectrum of T obey the spectral mapping theorem for every f2 H( (T)). We begin with some standard notation on Fredholm theory. Throughout this paper letH be a separable complex Hilbert space with inner producth ;i . Let B(H) and K(H) denote respectively, theC -algebra of all bounded linear operators and the ideal of compact operators acting on H. If T 2 B(H), we shall write kerT and ranT for the null space and the range of T respectively. Also let (T ) =dim kerT, (T ) =dim kerT and let (T ), a(T ) denote the spectrum, approximate point spectrum of T, respectively. Let p = p(T ) be the ascent of T; i.e., the smallest nonnegative integer p such that kerT p = kerT p+1 . If such an integer does not exist, we put p(T ) =1. Analogously, let q = q(T ) be the descent of T; i.e., the smallest nonnegative integer q such that ran T q =ranT q+1 , and if such an integer does not exist, we put q(T ) =1. It is well known that if p(T ) and q(T ) are both finite then p(T ) = q(T ). Moreover, 0 < p( T ) = q( T ) <1 precisely when is a pole of the resolvent of T, see Heuser (20, Proposition 50.2). An operator T 2 B(H) is called Fredholm if ranT is