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Compressions of resolvents and maximal radius of regularity

Authors:	C Badea M Mbekhta

Institution:	URA 751 au CNRS & UFR de Mathématiques, Université de Lille I, F--59655 Villeneuve d'Ascq, France ; URA 751 au CNRS & UFR de Mathématiques, Université de Lille I, F--59655 Villeneuve d'Ascq, France

Abstract:	Suppose that $\lambda - T$ is left invertible in for all $\lambda \in \Omega$ , where $\Omega$ is an open subset of the complex plane. Then an operator-valued function $L(\lambda)$ is a left resolvent of in $\Omega$ if and only if has an extension $\tilde{T}$ , the resolvent of which is a dilation of $L(\lambda)$ of a particular form. Generalized resolvents exist on every open set , with $\overline{U}$ included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obtained.

Keywords:	One-sided resolvents Hilbert space operators spectral radius dilations and compressions

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