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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.
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참고문헌 (22)
참고문헌 신청
A. Amouch / 2006 / On the equivalence of Browder’s and generalized Browder’s theorem / Glasgow Math. J 48 (1) : 179 ~ 185
A. E. Taylor / 1980 / Introduction to Functional Analysis / John Wiley & Sons
B. P. Duggal / / Topicas de Theoria de la Approximacion III / Textos Cientificos BUAP Puebla : 107 ~ 146
B. P. Duggal / 2005 / Hereditarily normaloid operators / Extracta Math 20 (2) : 203 ~ 217
B. P. Duggal / 2007 / Polaroid operators, SVEP and perturbed Browder, Weyl theorems / Rend. Circ. Mat. Palermo (2) 56 (3) : 317 ~ 330
A. E. Taylor / 1980 / Introduction to Functional Analysis / John Wiley & Sons
B. P. Duggal / / Topicas de Theoria de la Approximacion III / Textos Cientificos BUAP Puebla : 107 ~ 146
B. P. Duggal / 2005 / Hereditarily normaloid operators / Extracta Math 20 (2) : 203 ~ 217
B. P. Duggal / 2007 / Polaroid operators, SVEP and perturbed Browder, Weyl theorems / Rend. Circ. Mat. Palermo (2) 56 (3) : 317 ~ 330
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