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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Compressions of resolvents
and maximal radius of regularity


Authors: C. Badea and M. Mbekhta
Journal: Trans. Amer. Math. Soc. 351 (1999), 2949-2960
MSC (1991): Primary 47A10, 47A20
Published electronically: March 8, 1999
MathSciNet review: 1621709
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Abstract: Suppose that $\lambda - T$ is left invertible in $L(H)$ 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 $T$ in $\Omega$ if and only if $T$ 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 $U$, with $\overline{U}$ included in the regular domain of $T$. This implies a formula for the maximal radius of regularity of $T$ in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.


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Additional Information

C. Badea
Affiliation: URA 751 au CNRS & UFR de Mathématiques, Université de Lille I, F–59655 Villeneuve d’Ascq, France
Email: badea@gat.univ-lille1.fr

M. Mbekhta
Affiliation: URA 751 au CNRS & UFR de Mathématiques, Université de Lille I, F–59655 Villeneuve d’Ascq, France
Address at time of publication: University of Galatasaray, Çiragan Cad no 102, Ortakoy 80840, Istanbul, Turkey
Email: mbekhta@gat.univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02365-X
PII: S 0002-9947(99)02365-X
Keywords: One-sided resolvents, Hilbert space operators, spectral radius, dilations and compressions
Received by editor(s): February 17, 1997
Published electronically: March 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society