Essential approximate point spectra and Weyl's theorem for operator matrices

Abstract

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, it is shown that a 2 × 2 operator matrix M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ B ( K , H ) if and only if A is upper semi-Fredholm and { n ( B ) < ∞ and n ( A ) + n ( B ) ⩽ d ( A ) + d ( B ) or n ( B ) = d ( A ) = ∞ , if R ( B ) is closed , d ( A ) = ∞ , if R ( B ) is not closed . We also give the necessary and sufficient conditions for which M C is Weyl or M C is lower semi-Fredholm with nonnegative index for some C ∈ B ( K , H ) . In addition, we 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.