Semi-Fredholm Spectrum and Weyl's Theorem for Operator Matrices

When A . B(H) and B . B(K) are given, we denote by M C an operator acting on the Hilbert space H . K of the form MC=(A0CB).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 . B(K,H). In addition, let .SF+ (A) ={. . . : A . .I is not an upper semi-Fredholm operator} be the upper semi.Fredholm spectrum of A . B(H) and let .SF. (A) = {. . . : A . .I is not a lower semi.Fredholm operator} be the lower semi.Fredholm spectrum of A. We show that the passage from .SF±(A)..SF±(B)to.SF±(MC) is accomplished by removing certain open subsets of .SF.(A)..SF+(B) from the former, that is, there is an equality .SF±(A)..SF±(B)=.SF±(MC).G, where Gis the union of certain of the holes in .SF±(MC) which happen to be subsets of .SF.(A)..SF+(B).Weyl's theorem and Browder's theorem are liable to fail for 2 . 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 . 2 upper triangular operator matrices on the Hilbert space.