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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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  • 1. Paolo Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations 1 (1988), no. 4, 433–457. MR 945820
  • 2. Paolo Acquistapace and Brunello Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987), 47–107. MR 934508
  • 3. J.-B. Baillon and Ph. Clément, Examples of unbounded imaginary powers of operators, J. Funct. Anal. 100 (1991), no. 2, 419–434. MR 1125234, 10.1016/0022-1236(91)90119-P
  • 4. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163–168. MR 727340, 10.1007/BF02384306
  • 5. Jean Bourgain, Vector-valued singular integrals and the 𝐻¹-BMO duality, Probability theory and harmonic analysis (Cleveland, Ohio, 1983) Monogr. Textbooks Pure Appl. Math., vol. 98, Dekker, New York, 1986, pp. 1–19. MR 830227
  • 6. Donald L. Burkholder, Martingales and Fourier analysis in Banach spaces, Probability and analysis (Varenna, 1985) Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, pp. 61–108. MR 864712, 10.1007/BFb0076300
  • 7. Philippe Clément and Jan Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), no. 1, 73–105. MR 1048282, 10.1007/BF01446879
  • 8. G. Da Prato and P. Grisvard, Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. (9) 54 (1975), no. 3, 305–387 (French). MR 0442749
  • 9. Giovanni Dore and Alberto Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), no. 2, 189–201. MR 910825, 10.1007/BF01163654
  • 10. Xuan Thinh Duong, 𝐻_{∞} functional calculus of second order elliptic partial differential operators on 𝐿^{𝑝} spaces, Miniconference on Operators in Analysis (Sydney, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., Canberra, 1990, pp. 91–102. MR 1060114
  • 11. X. T. Duong and D. W. Robinson. Gaussian bounds, Brownian estimates and $H_{\infty}$-functional calculus. Preprint, 1994.
  • 12. Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • 13. M. Fuhrmann. Sums of linear operators of parabolic type : a priori estimates and strong solutions. Ann. Mat. Pura Appl., 164:229-257, 1993.
  • 14. Rabah Labbas and Brunello Terreni, Somme d’opérateurs linéaires de type parabolique. I, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 2, 545–569 (French, with Italian summary). MR 896340
  • 15. Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. MR 1329547
  • 16. Jan Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939
  • 17. Jan Prüss and Hermann Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), no. 3, 429–452. MR 1038710, 10.1007/BF02570748
  • 18. Jan Prüss and Hermann Sohr, Imaginary powers of elliptic second order differential operators in 𝐿^{𝑝}-spaces, Hiroshima Math. J. 23 (1993), no. 1, 161–192. MR 1211773
  • 19. P. E. Sobolevskiĭ, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč. 10 (1961), 297–350 (Russian). MR 0141900
  • 20. Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824
  • 21. H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, 1978. MR 80i:46032
  • 22. Frank Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), no. 3, 201–222. MR 1030488

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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

DOI: https://doi.org/10.1090/S0002-9947-97-01997-1
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