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The intersection of essential approximate point spectra of operator matrices. (English) Zbl 1108.47013

Suppose that $𝒦$ and $ℋ$ are separable Hilbert spaces. Let $B\left(𝒦,ℋ\right)$, ${B}_{l}\left(𝒦,ℋ\right)$, and $\text{Inv}\left(𝒦,ℋ\right)$ be the sets of all bounded linear operators, left invertible bounded linear operators, and invertible bounded linear operators from $𝒦$ to $ℋ$, respectively. Let ${{\Phi }}_{+}$ be the set of all upper semi-Fredholm operators. Consider the sets ${{\Phi }}_{+}^{-}=\left\{T\in {{\Phi }}_{+}:\text{ind}\left(T\right)\le 0\right\}$ and ${\sigma }_{ea}\left(T\right)=\left\{\lambda \in ℂ:T-\lambda \notin {{\Phi }}_{+}^{-}\right\}$. Suppose that $A\in B\left(ℋ\right)$ and $B\in B\left(𝒦\right)$ are fixed. For $C\in B\left(𝒦,ℋ\right)$, ${M}_{C}=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)\in B\left(ℋ\oplus 𝒦\right)$. In the present paper, the sets ${\bigcap }_{C\in {B}_{l}\left(𝒦,ℋ\right)}{\sigma }_{ea}\left({M}_{C}\right)$, ${\bigcap }_{C\in \text{Inv}\left(𝒦,ℋ\right)}{\sigma }_{ea}\left({M}_{C}\right)$, and ${\bigcup }_{C\in \text{Inv}\left(𝒦,ℋ\right)}{\sigma }_{ea}\left({M}_{C}\right)$ are characterized. In particular,

$\bigcap _{C\in \text{Inv}\left(𝒦,ℋ\right)}{\sigma }_{ea}\left({M}_{C}\right)=\bigcap _{C\in B\left(𝒦,ℋ\right)}{\sigma }_{ea}\left({M}_{C}\right)\cup \left\{\lambda \in ℂ:B-\lambda \phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{compact}\right\}·$

##### MSC:
 47A53 (Semi-)Fredholm operators; index theories 47A10 Spectrum and resolvent of linear operators 47A55 Perturbation theory of linear operators