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Stability of the local spectrum. (English) Zbl 0903.47001

The paper gives a summary of the results that appeared in [Proc. Am. Math. Soc. 125, No. 2, 417-425 (1997; Zbl 0861.47012)]. A local functional calculus for an operator \(T\in {\mathcal B}(X)\) is defined by the integral \[ f[T]x={1\over 2\pi i}\int_{\Gamma}f(\lambda)\widehat{x}_T(\lambda) d\lambda, \] where \(f\) is a holomorphic function defined on an open set containing the spectrum of the operator \(T\), \(\Gamma\) is a suitably chosen contour surrounding the spectrum, and \(\widehat{x}_T\) is the local resolvent of \(T\). This calculus was studied by P. McGuire [Integral Equations Oper. Theory 9, 218-236 (1986; Zbl 0589.47018)] for the case of a Hilbert space operator with empty point spectrum. The present authors use this formula to define a local functional calculus for Banach space operators with the single-valued extension property. The main result of the paper is the following:
Theorem. Assume that an operator \(T\in {\mathcal B}(X)\) satisfies the single-valued extension property. Let \(x\in X\) and let a function \(f\) be holomorphic in a neighbourhood of \(\sigma(x,T)\). Let \(\alpha_1, \ldots, \alpha_p\) be the zeros of \(f\) in \(\sigma(x,T)\) with multiplicities \(n_1, \ldots, n_p\), respectively. Then \(\sigma(f[T]x,T)=\sigma(x,T)\) if and only if there is no \(i\in\{1,\ldots,p\}\) such that \(\alpha_i\) is a pole of \(\widehat{x}_T\) of order not greater than \(n_i\).

MSC:

47A11 Local spectral properties of linear operators
47A60 Functional calculus for linear operators
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