Variations on Weyl’s theorem. (English) Zbl 1101.47001
Summary: We study the property \((w)\), a variant of Weyl’s theorem introduced by V. Rakočević [Mat.Vesn.37, 423–426 (1985; Zbl 0596.47001)], by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property \((w)\) holds. We also relate this property with Weyl’s theorem and with another variant of it, \(a\)-Weyl’s theorem. We show that Weyl’s theorem, \(a\)-Weyl’s theorem and property \((w)\) for \(T\)(respectively, \(T^{*}\)) coincide whenever \(T^{*}\) (respectively, \(T\)) satisfies the SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property \((w)\).
MSC:
| 47A10 | Spectrum, resolvent |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 47B20 | Subnormal operators, hyponormal operators, etc. |
Citations:
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