Polaroid operators satisfying Weyl’s theorem. (English) Zbl 1096.47039
Let \(T\) be a continuous linear operator on a given Banach space \(X\). \(T\) is said to be polaroid if every isolated point in the spectrum \(\sigma(T)\) of \(T\) is a pole of the resolvent set of \(T\). \(T\) satisfies Weyl’s theorem if \(\sigma(T)\setminus\sigma_w(T)\) consists of the isolated points of \(\lambda\in\sigma(T)\) with \(0<\dim(T-\lambda)^{-1}(0)<\infty\), where \(\sigma_w(T)\) stands for the Weyl spectrum of \(T\).
The author gives characterizations of when the operator \(T\) is polaroid and satisfies Weyl’s theorem. As a matter of fact, he proves that \(T\) has the above-mentioned properties if and only if \(T\) has the single-valued extension property at every point not in \(\sigma_w(T)\) and is of Kato type at every isolated point of \(\sigma(T)\).
The author gives characterizations of when the operator \(T\) is polaroid and satisfies Weyl’s theorem. As a matter of fact, he proves that \(T\) has the above-mentioned properties if and only if \(T\) has the single-valued extension property at every point not in \(\sigma_w(T)\) and is of Kato type at every isolated point of \(\sigma(T)\).
Reviewer: Armando R. Villena (Granada)
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 47A10 | Spectrum, resolvent |
| 47A11 | Local spectral properties of linear operators |
Keywords:
polaroid operator; Weyl’s theorem; single-valued extension property; Kato type; hereditarily normaloid operatorsReferences:
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