Weyl’s Theorem for Operator Matrices

Weyl’s theorem for an operator says that the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. H. Weyl ([22]) discovered that this property holds for hermitian operators and it has been extended from hermitian operators to hyponormal operators and to Toeplitz operators by L. Coburn ([5]), and to several classes of operators including seminormal operators by S. Berberian ([1],[2]). Recently Weyl’s theorem under the “small” perturbations has been considered in [14] and [15]. But Weyl’s theorem is liable to fail for 2× 2 (even diagonal) operator matrices even though Weyl’s theorem holds for the entries in the operator matrices. In this paper we consider Weyl’s theorem and the less restrictive “Browder’s theorem” for 2× 2 operator matrices. Let H and K be Hilbert spaces, let L(H,K) denote the set of bounded linear operators from H to K, and abbreviate L(H,H) to L(H). If A ∈ L(H) is a Fredholm operator, that is, if A has finite dimensional null space and its range of finite co-dimension, then the index of A, denoted ind A, is given by