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On isolated points of the spectrum of a bounded linear operator. (English) Zbl 0780.47019

M. Mbekhta [Glasg. Math. J. 29, 159-175 (1987; Zbl 0657.47038)] introduced, for a closed linear operator \(A\) in a Banach space (actually, in a Hilbert space) \(E\), two subspaces of \(E\): \[ \begin{split} K(A) = \{x\in E: \text{ there exist } c>0 \text{ and a sequence } (x_ n)_{n\geq 1} \subset E \text{ such that }\\ Ax_ 1=x,\;Ax_{n+1}=x_ n \text{ for all } n\in\mathbb{N}, \text{ and } \| x_ n\|\leq c^ n\| x\| \text{ for all } n\in\mathbb{N}\},\end{split} \]
\[ H_ 0(A) = \{x\in E:\;\lim_{n\to\infty} \| A^ n x\|^{1/n}=0\}, \] and proved that a point \(\lambda_ 0\) in the spectrum \(\sigma(A)\) of \(A\) is isolated in \(\sigma(A)\) if and only if \(E\) is decomposed into a topological direct sum \[ E=K(\lambda_ 0 I-A)\oplus H_ 0(\lambda_ 0 I-A) \tag{*} \] with \(H_ 0(\lambda_ 0 I-A)\neq\{0\}\). The paper under review shows for a bounded linear operator \(A\) on a Banach space \(E\) that \(\lambda_ 0\in\sigma(A)\) is isolated in \(\sigma(A)\) if and only if \(K(\lambda_ 0 I-A)\) is closed and \(E\) is decomposed into an algebraic direct sum (*) with \(H_ 0(\lambda_ 0 I-A)\neq\{0\}\). This result is used to characterize the Riesz points of \(A\) and the poles of the resolvent of \(A\).

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A10 Spectrum, resolvent
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

Citations:

Zbl 0657.47038
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References:

[1] James K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), no. 1, 61 – 69. · Zbl 0315.47002
[2] Harro Heuser, Funktionalanalysis, 2nd ed., Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1986 (German). Theorie und Anwendung. [Theory and application]. · Zbl 0653.46002
[3] Mostafa Mbekhta, Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Math. J. 29 (1987), no. 2, 159 – 175 (French). · Zbl 0657.47038 · doi:10.1017/S0017089500006807
[4] Mostafa Mbekhta, Sur la théorie spectrale locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), no. 3, 621 – 631 (French). · Zbl 0721.47006
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