Lyapunov theorems for Banach spaces
Authors:
Yu. Latushkin and S. Montgomery-Smith
Journal:
Bull. Amer. Math. Soc. 31 (1994), 44-49
MSC:
Primary 47D06; Secondary 34D05, 34D20, 34G10, 47N20
DOI:
https://doi.org/10.1090/S0273-0979-1994-00495-1
MathSciNet review:
1249356
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We present a spectral mapping theorem for semigroups on any Banach space E. From this, we obtain a characterization of exponential dichotomy for nonautonomous differential equations for E-valued functions. This characterization is given in terms of the spectrum of the generator of the semigroup of evolutionary operators.
- [BG] A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators, Oper. Theory Adv. Appl., vol. 56, Birkhaüser, Basel, 1992, pp. 90-119. MR 1173919
- [CS] C. Chicone and R. Swanson, Spectral theory of linearizations of dynamical systems, J. Differential Equations 40 (1981), 155-167. MR 619131 (82h:58039)
- [DK] J. Daleckij and M. Krein, Stability of differential equations in Banach space, Transl. Math. Mono., vol. 43, Amer. Math. Soc., Providence, RI, 1974.
- [H] J. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monographs, vol. 25, Amer. Math. Soc., Providence, RI, 1988. MR 941371 (89g:58059)
- [Ho] J. S. Howland, Stationary scattering theory for time-dependent hamiltonians, Math. Annal. 207 (1974), 315-335. MR 0346559 (49:11284)
- [J] R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980), 366-387. MR 563387 (81c:58056)
- [LM] Y. Latushkin and S. Montogomery-Smith, Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal. (to appear). MR 1308621 (96k:47072)
- [LR] Y. Latushkin and T. Randolph, Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators, Trans. Amer. Math. Soc., submitted.
- [LS] Y. Latushkin and A. Stepin, Weighted translations operators and linear extensions of dynamical systems, Russian Math. Surveys 46 (1991), 95-165. MR 1125273 (92k:47062)
- [N] R. Nagel (ed.), One parameters semigroups of positive operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin, 1984.
- [M] J. Mather, Characterization of Anosov diffeomorphisms, Indag. Math. 30 (1968), 479-483. MR 0248879 (40:2129)
- [P] K. Palmer, Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), 149-156. MR 958058 (89k:34052)
- [R] R. Rau, Hyperbolic evolution groups and exponentially dichotomic evolution families, J. Funct. Anal. (to appear).
- [R1] -, Hyperbolic evolutionary semigroups on vector-valued function spaces, Semigroup Forum 48 (1994), 107-118. MR 1245910 (95c:47045)
- [SS] R. Sacker and G. Sell, Dichotomies for linear evolutionary equations in Banach spaces, IMA preprint no. 838, 1991. MR 1296160 (96k:34136)
Retrieve articles in Bulletin of the American Mathematical Society with MSC: 47D06, 34D05, 34D20, 34G10, 47N20
Retrieve articles in all journals with MSC: 47D06, 34D05, 34D20, 34G10, 47N20
Additional Information
DOI:
https://doi.org/10.1090/S0273-0979-1994-00495-1
Keywords:
Hyperbolicity,
evolution family,
exponential dichotomy,
weighted composition operators,
spectral mapping theorem
Article copyright:
© Copyright 1994
American Mathematical Society