GENERALIZED BROWDER, WEYL SPECTRA AND THE POLAROID PROPERTY UNDER COMPACT PERTURBATIONS
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For a Banach space operator $A\in\b$, let $\sigma(A)$, $\sigma_a(A)$, $\sigma_w(A)$ and $\sigma_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator $A$ is polaroid (resp., left polaroid), if the points $\iso\sigma(A)$ (resp., $\iso\sigma_a(A)$) are poles (resp., left poles) of the resolvent of $A$. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A\in\b$, we prove that a sufficient condition for: (i) $A+K$ to have SVEP on the complement of $\sigma_w(A)$ (resp., $\sigma_{aw}(A)$) for every compact operator $K\in\b$ is that $\sigma_w(A)$ (resp., $\sigma_{aw}(A)$) has no holes; (ii) $A+K$ to be polaroid (resp., left polaroid) for every compact operator $K\in\b$ is that $\iso\sigma_w(A)=\emptyset$ (resp., $\iso\sigma_{aw}(A)=\emptyset$). It is seen that these conditions are also necessary in the case in which the Banach space $\X$ is a Hilbert space.
Abstract
For a Banach space operator $A\in\b$, let $\sigma(A)$, $\sigma_a(A)$, $\sigma_w(A)$ and $\sigma_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator $A$ is polaroid (resp., left polaroid), if the points $\iso\sigma(A)$ (resp., $\iso\sigma_a(A)$) are poles (resp., left poles) of the resolvent of $A$. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A\in\b$, we prove that a sufficient condition for: (i) $A+K$ to have SVEP on the complement of $\sigma_w(A)$ (resp., $\sigma_{aw}(A)$) for every compact operator $K\in\b$ is that $\sigma_w(A)$ (resp., $\sigma_{aw}(A)$) has no holes; (ii) $A+K$ to be polaroid (resp., left polaroid) for every compact operator $K\in\b$ is that $\iso\sigma_w(A)=\emptyset$ (resp., $\iso\sigma_{aw}(A)=\emptyset$). It is seen that these conditions are also necessary in the case in which the Banach space $\X$ is a Hilbert space.
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