Weyl spectra of operator matrices

Lee, Woo

doi:10.1090/S0002-9939-00-05846-9

Weyl spectra of operator matrices
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Proc. Amer. Math. Soc. 129 (2001), 131-138 Request permission

Abstract:

In this paper it is shown that if $M_{C}=\left (\begin {smallmatrix}A&C 0&B\end {smallmatrix} \right )$ is a $2\times 2$ upper triangular operator matrix acting on the Hilbert space $\mathcal {H}\oplus \mathcal {K}$ and if $\omega (\cdot )$ denotes the “Weyl spectrum", then the passage from $\omega (A)\cup \omega (B)$ to $\omega (M_{C})$ is accomplished by removing certain open subsets of $\omega (A) \cap \omega (B)$ from the former, that is, there is equality \begin{equation*}\omega (A)\cup \omega (B)=\omega (M_{C}) \cup \mathfrak {S}, \end{equation*} where $\mathfrak {S}$ is the union of certain of the holes in $\omega (M_{C})$ which happen to be subsets of $\omega (A)\cap \omega (B)$.

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