Journal of the
Korean Mathematical Society JKMS

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.

Keywords: Banach space operator, compact perturbation, SVEP, polaroid, left polaroid, B-Fredholm spectrum, Browder's theorem, Weyl's theorem, abstract shift condition

MSC numbers: 47A10, 47A55, 47A53, 47B40