Abstract
Recently, Ben-Artzi and Gohberg [2] used the concept ofC 0-semigroups in order to characterize the existence of dichotomies for nonautonomous differential equations on ℂn. A similar task was performed by Latushkin and Stepin [11] for dichotomies of linear skew-product flows. In this paper we will useC o-semigroups to characterize existence of dichotomies for strongly continuous evolution families (U(t,s)) t.s∃ℝ on Hilbert and Banach spaces. Under an exponential growth condition we show that the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.
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References
Arendt, W., and Greiner, G. (1984). A spectral mapping theorem for one-parameter groups of positive operators onC 0(X).Semigroup Forum 30, 297–330.
Ben-Artzi, A., and Gohberg, I. (1993). Dichotomy of a systems and invertibility of linear ordinary differential operators (in press).
Daleckii, J. L., and Krein, M. G. (1974).Stability of Solutions of Differential Equations in Banach space. Am. Math. Soc., Providence, RI.
Diestel, J., and Uhl, J. J. (1977). Vector measures. Mathematical Surveys 15, Am. Math. Soc., Providence, RI.
Evans, D. E. (1976). Time dependent perturbations and scattering of strongly continuous groups on Banach spaces.Math. Ann. 221, 275–290.
Goldstein, J. A. (1985).Semigroups of Linear Operators and Applications, Oxford University Press, New York.
Greiner, G., and Schwarz, M. (1991). Weak spectral mapping theorems for functional differential equations.J. Diff. Eq. 94, 205–216.
Howland, J. S. (1974). Stationary scattering theory for time-dependent hamiltonians.Math. Ann. 207, 315–335.
Kato, T. (1970). Linear evolution equations of “hyperbolic” type.J. Fac. Tokyo 17, 241–258.
Lumer, G. (1989). Equations de diffusion dans le domaines (x, t) non-cylindriques et semigroupes “espace-temps,” Séminaire de Théorie du Potential, Lecture Notes in Mathematics 1393, Springer-Verlag, Berlin-Heidelberg, pp. 161–179.
Latushkin, Y. D., and Stepin, A. M. (1991). Weighted translation operators and linear extensions of dynamical systems.Russ. Math. Surv. 46(2), 95–165.
Massera, J. L., and Schaffer, J. J. (1966).Linear Differential Equations and Function Spaces, Academic Press, New York.
Nagel, R. (ed.) (1984).One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics 1184, Springer-Verlag, Berlin-Heidelberg.
Pazy, A. (1983).Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg.
Sacker, R. J., and Sell, G. R. (1978). A spectral theory for linear differential systems.J. Diff. Eq. 27, 330–358.
Sacker, R. J., and Sell, G. R. (1993). Dichotomies for linear evolutionary equations in Banach spaces.J. Dyn. Diff. Eq. (in press).
Tanabe, H. (1979).Equations of Evolution, Pitman, London.
Walters, P. (1982).An Introduction to Ergodic Theory, Springer-Verlag, New York.
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Rau, R.T. Hyperbolic evolution groups and dichotomic evolution families. J Dyn Diff Equat 6, 335–350 (1994). https://doi.org/10.1007/BF02218534
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DOI: https://doi.org/10.1007/BF02218534