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A theorem of the Dore-Venni type for noncommuting operators

Authors: Sylvie Monniaux and Jan Prüss
Journal: Trans. Amer. Math. Soc. 349 (1997), 4787-4814
MSC (1991): Primary 47A60, 47B47, 47G20, 47D06; Secondary 45A05, 45D05, 45K05
MathSciNet review: 1433125
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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

Jan Prüss
Affiliation: Fachbereich Mathematik und Informatik, Martin-Luther- Universität Halle-Wittenberg, Theodor-Lieser-Str. 5, D-06120 Halle, Germany

Keywords: Sum of linear operators, bounded imaginary powers of linear operators, commutator conditions, parabolic evolution equations, parabolic evolutionary integral equations, completely positive kernels, fractional derivatives, creep functions, viscoelasticity
Received by editor(s): May 22, 1995
Article copyright: © Copyright 1997 American Mathematical Society