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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

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A theorem of the Dore-Venni type for noncommuting operators
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by Sylvie Monniaux and Jan Prüss PDF
Trans. Amer. Math. Soc. 349 (1997), 4787-4814 Request permission

Abstract:

A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations $\dot {u}(t)+L(t)u(t)=f(t)$ and evolutionary integral equations $u(t)+\int _0^ta(t-s)L(s)u(s)ds = g(t)$ which are nonautonomous. The domains of the involved operators $L(t)$ may depend on $t$, but $L(t)^{-1}$ is required to satisfy a certain smoothness property. The results are then applied to parabolic partial differential and integro-differential equations.
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Additional Information
  • Sylvie Monniaux
  • Affiliation: Mathematik V, Universität Ulm, D-89069 Ulm, Germany
  • Email: monniaux@mathematik.uni-ulm.de
  • Jan Prüss
  • Affiliation: Fachbereich Mathematik und Informatik, Martin-Luther- Universität Halle-Wittenberg, Theodor-Lieser-Str. 5, D-06120 Halle, Germany
  • Email: anokd@volterra.mathematik.uni-halle.de
  • Received by editor(s): May 22, 1995
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 4787-4814
  • MSC (1991): Primary 47A60, 47B47, 47G20, 47D06; Secondary 45A05, 45D05, 45K05
  • DOI: https://doi.org/10.1090/S0002-9947-97-01997-1
  • MathSciNet review: 1433125