Operator valued fourier multipliers and stability of strongly continuous semigroups

Integral Equations and Operator Theory

Volumn 51, Issue 3, 2005, Pages 375-394

  Latushkin, Yuri ^a   Räbiger, Frank ^b  

^a UNIVERSITY OF MISSOURI   (United States)

^b UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Asymptotics for strongly continuous semigroups; Input output stability; Operator valued Fourier multipliers; Transfer function

Indexed keywords

EID: 16244373947     PISSN: 0378620X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00020-004-1349-x     Document Type: Article

References (26)

1. H. Amann, Operator-valued Fourier multipliers, vector-valued. Besov spaces, and applications, Math. Nachr. 186 (1997), 5-56.
2. W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z. 240 (2002), 311-343.
3. C. Chicone and Y. Latushkin, "Evolution semigroups in dynamical systems and differential equations", Math. Surv. and Mon., 70, AMS, 1999.
4. S. Clark, Y. Latushkin, S. Montgomery-Smith, and T. Randolph, Stability radius and internal versus external stability in Banach spaces: an evolution semigroups approach, SIAM J. Contr. and Optim. 38 (2000), 1757-1793.
5. Ph. Clément, B. de Pagter, F. Sukochev, and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math. 138 (2000), 135-163.
6. R. Curtain, Equivalence of input-output stability and exponential stability for infinite-dimensional systems, Math. Systems Theory 21 (1988), 19-48.
7. R. Curtain and H. J. Zwart, "An introduction to infinite-dimensional linear control systems theory," Springer-Verlag, New York, 1995.
8. R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. 3 (1972), 428-445.
9. K. Engel and R. Nagel, "One-parameter semigroups for linear evolution equations", Springer-Verlag, New York, 2000.
10. 2, J. Oper. Theory 7 (1982), 61-78.
11. M. Hieber, Operator valued Fourier multipliers, in "Progr. Nonlin. Diff. Eqns. Appl.", 35, 363-380, Birkhäuser-Verlag, 1999.
12. M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers. In: Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 121-124, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001.
13. M.A. Kaashoek and S.M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Diff. Eqns. 112 (1994), 374-406.
14. Y. Latushkin and S. Montgomery-Smith, Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal. 127 (1995), 173-197.
15. Y. Latushkin, T. Randolph and R. Schnaubelt, Regularization and frequency-domain stability of well-posed systems, Math. Control Signals Syst., 2004, to appear.
16. Y. Latushkin and R. Shvydkoy, Hyperbolicity of semigroups and Fourier multipliers. In: Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 341-363, Oper. Theory Adv. Appl., 129, Birkhäuser, Basel, 2001.
17. A. Pazy, "Semigroups of linear operators and applications to partial differential equations", Springer-Verlag, New York, 1983.
18. 0-semigroups, Trans. Amer. Math. Soc. 284(2) (1984), 847-857.
19. R. Rebarber, Conditions for the equivalence of internal and external stability for distributed parameter systems, IEEE Trans. on Automat. Control 31(6) (1993), 994-998.
20. J.M. A.M. van Neerven, "The asymptotic behavior of semigroups of linear operators", Oper. Theory Adv. Appl., 88, Birkhäuser-Verlag, 1996.
21. L. Weis, Stability theorems for semigroups via multiplier theorems, in "Differential equations, asymptotic analysis, and mathematical physics" (Potsdam, 1996), Math. Res., 100 (1997), 407-411.
22. p-regularity, Math. Ann. 319 (2001), 735-758.
23. 0-semigroups in Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 3663-3671.
24. G. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces, J. Math. Anal. Appl. 145 (1990), 154-171.
25. ∞, Math. Control Signals Syst. 4 (1991), 193-203.
26. 0-semigroups, Math. Ann. 285 (1989), 201-219.