Semi-fredholm spectrum and Weyl's theorem for operator matrices

When A is an element of B( H) and B is an element of B( K) are given, we denote by M-C an operator acting on the Hilbert space H circle plus K of the form M-C = (A)(0) (C)(B). In this paper, first we give the necessary and sufficient condition for M-C to be an upper semi-Fredholm ( lower semi-Fredholm, or Fredholm) operator for some C is an element of B( K, H). In addition, let sigma(SF+)(A) = {lambda is an element of C : A - lambda I is not an upper semi-Fredholm operator} be the upper semi-Fredholm spectrum of A is an element of B( H) and let sigma(SF-)(A) = {lambda is an element of C : A - lambda I is not a lower semi-Fredholm operator} be the lower semi-Fredholm spectrum of A. We show that the passage from sigma(SF +/-)(A) boolean OR sigma(SF +/-)(B) to sigma(SF +/-)(M-C) is accomplished by removing certain open subsets of sigma(SF-)(A) boolean AND sigma(SF+)(B) from the former, that is, there is an equality sigma(SF +/-)(A) boolean OR sigma(SF +/-)(B) = sigma(SF +/-)(M-C) boolean OR G, where G is the union of certain of the holes in sigma(SF +/-)(M-C) which happen to be subsets of sigma(SF-)(A) boolean AND sigma(SF+)(B). Weyl's theorem and Browder's theorem are liable to fail for 2 x 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for 2 x 2 upper triangular operator matrices on the Hilbert space.