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

Abstract

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 $M_{C} = {\left( {\begin{array}{*{20}c} {A} & {C} \\ {0} & {B} \\ \end{array} } \right)} .$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 $\sigma _{{SF_{ + } }}$ (A) ={λ ∈ ℂ : A − λI is not an upper semi-Fredholm operator} be the upper semi–Fredholm spectrum of A ∈ B(H) and let $\sigma _{{SF_{ - } }}$ (A) = {λ ∈ ℂ : A − λI is not a lower semi–Fredholm operator} be the lower semi–Fredholm spectrum of A. We show that the passage from $\sigma _{{SF_{ ±} }} {\left( A \right)} \cap \sigma _{{SF_{ ±} }} {\left( B \right)}\;{\text{to}}\;\sigma_{{SF_{±} }} {\left( {M_{C} } \right)}$ is accomplished by removing certain open subsets of $\sigma _{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)}$ from the former, that is, there is an equality

$$ \sigma_{{SF_{± } }} {\left( A \right)} \cup \sigma _{{SF_{ ±} }} {\left( B \right)} = \sigma_{{SF_{±} }} {\left( {M_{C} } \right)} \cup {\fancyscript G},$$

where ${\fancyscript G}$is the union of certain of the holes in $\sigma_{{SF_{± } }} {\left( {M_{C} } \right)}$ which happen to be subsets of $\sigma_{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} .$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.