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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Compressions of resolvents and maximal radius of regularity
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by C. Badea and M. Mbekhta PDF
Trans. Amer. Math. Soc. 351 (1999), 2949-2960 Request permission

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
  • MR Author ID: 121980
  • Email: mbekhta@gat.univ-lille1.fr
  • Received by editor(s): February 17, 1997
  • Published electronically: March 8, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 2949-2960
  • MSC (1991): Primary 47A10, 47A20
  • DOI: https://doi.org/10.1090/S0002-9947-99-02365-X
  • MathSciNet review: 1621709