Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. P. Acquistapace. Evolution operators and strong solutions of abstract linear parabolic equations. Diff. Int. Equations, 1:433-457, 1988. MR 90b:34094
  • 2. P. Acquistapace and B. Terreni. A unified approach to abstract linear non-autonomous parabolic equations. Rend. Sem. Mat. Univ. Padova, 78:47-107, 1987. MR 89e:34099
  • 3. J.B. Baillon and Ph. Clément. Examples of unbounded imaginary powers of operators. J. Funct. Anal., 100:419-434, 1991. MR 92j:47036
  • 4. J. Bourgain. Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat., 22:163-168, 1983. MR 85a:46011
  • 5. J. Bourgain. Vector-valued singular integrals and the $H^1$-$BMO$ duality. In D. Burkholder, editor, Probability Theory and Harmonic Analysis, pages 1-19, New-York, 1986. Marcel Dekker. MR 87j:42049b
  • 6. D.L. Burkholder. Martingales and Fourier analysis in Banach spaces. In G. Letta and M. Pratelli, editors, Probability and Analysis, volume 1206 of Lect. Notes Math., pages 61-108, Berlin, 1986. Springer Verlag. MR 88c:42017
  • 7. Ph. Clément and J. Prüss. Completely positive measures and Feller semigroups. Math. Ann., 287:73-105, 1990. MR 91d:45011
  • 8. G. Da Prato and P. Grisvard. Sommes d'opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl., 54:305-387, 1975. MR 56:1129
  • 9. G. Dore and A. Venni. On the closedness of the sum of two closed operators. Math. Z., 196:189-201, 1987. MR 88m:47072
  • 10. X. T. Duong. $H_{\infty}$- functional calculus for second order elliptic partial differential operators on $L^p$-spaces. In I. Doust, B. Jefferies, C. Li, and A. McIntosh, editors, Operators in Analysis, pages 91-102, Sydney, 1989. Australian National University. MR 91m:35072
  • 11. X. T. Duong and D. W. Robinson. Gaussian bounds, Brownian estimates and $H_{\infty}$-functional calculus. Preprint, 1994.
  • 12. P.L. Duren. Theory of $H^p$ Spaces, volume 38 of Pure Appl. Math. Acad. Press, New York, 1970. MR 42:3552
  • 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. R. Labbas and B. Terreni. Somme d'opérateurs linéaires de type parabolique. Boll. Un. Mat. Ital., (7) 1-B:545-569, 1987. MR 89g:47016
  • 15. A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser-Verlag, Basel, 1995. MR 96e:47039
  • 16. J. Prüss. Evolutionary Integral Equations and Applications. Birkhäuser Verlag, Basel, 1993. MR 94h:45010
  • 17. J. Prüss and H. Sohr. On operators with bounded imaginary powers in Banach spaces. Math. Z., 203:429-452, 1990. MR 91b:47030
  • 18. J. Prüss and H. Sohr. Boundedness of imaginary powers of second-order elliptic differential operators in $L^p$. Hiroshima Math. J., 23:161-192, 1993. MR 94d:47051
  • 19. P.E. Sobolevskii. On equations of parabolic type. Amer. Math. Soc. Transl., 49:1-62, 1965. MR 25:5297
  • 20. H. Tanabe. Equations of Evolution, volume 6 of Monographs and Studies in Mathematics. Pitman, London, 1979. MR 82g:47032
  • 21. H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, 1978. MR 80i:46032
  • 22. F. Zimmermann. On vector-valued Fourier multiplier theorems. Studia Math., 93:201-222, 1989. MR 91b:46031

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47A60, 47B47, 47G20, 47D06, 45A05, 45D05, 45K05

Retrieve articles in all journals with MSC (1991): 47A60, 47B47, 47G20, 47D06, 45A05, 45D05, 45K05

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

American Mathematical Society