When A ∈ B(H) and B ∈ B(K) are given, we denote by MC an operator acting on the
Hilbert space H ⊕ K of the form \documentclass[12pt]{minimal}
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M_{C} = {\left( {\begin{array}{*{20}c}
{A} & {C} \\
{0} & {B} \\
\end{array} } \right)}
.$$\end{document}In this paper, first we give the necessary and sufficient
condition for MC to be an upper semi-Fredholm (lower semi–Fredholm, or Fredholm) operator for
some C ∈ B(K,H). In addition, let \documentclass[12pt]{minimal}
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\sigma _{{SF_{ + } }}
$$\end{document} (A) ={λ ∈ ℂ : A − λI is not an upper semi-Fredholm
operator} be the upper semi–Fredholm spectrum of A ∈ B(H) and let \documentclass[12pt]{minimal}
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$$
\sigma _{{SF_{ - } }}
$$\end{document}
(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 \documentclass[12pt]{minimal}
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\sigma _{{SF_{ ±} }} {\left( A \right)} \cap \sigma _{{SF_{ ±} }} {\left( B \right)}\;{\text{to}}\;\sigma_{{SF_{±} }} {\left( {M_{C} } \right)}
$$\end{document} is accomplished by removing certain open subsets of
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\sigma _{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)}
$$\end{document} from the former, that is, there is an equality
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\sigma_{{SF_{± } }} {\left( A \right)} \cup \sigma _{{SF_{ ±} }} {\left( B \right)} = \sigma_{{SF_{±} }} {\left( {M_{C} } \right)} \cup {\fancyscript G},$$\end{document} where \documentclass[12pt]{minimal}
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\begin{document}$${\fancyscript G}$$\end{document}is the union of certain of the holes in \documentclass[12pt]{minimal}
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\sigma_{{SF_{± } }} {\left( {M_{C} } \right)}
$$\end{document} which happen to be subsets of \documentclass[12pt]{minimal}
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$$
\sigma_{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)}
.$$\end{document}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.
