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Evolution semigroups, translation algebras, and exponential dichotomy of cocycles. (English) Zbl 0962.34035
The authors study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space $\Theta$ acting on a Banach space $X$. Their main tool is the associated evolution semigroup on $C_0 (\Theta;X)$. They prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, they establish a spectral mapping/annular hull theorem for the evolution semigroup. In addition, the dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on $c_0 (\Bbb Z;X)$. Here, they develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that the dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, the exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, they extend the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous, mild solutions to the inhomogeneous equation.

MSC:
34D09Dichotomy, trichotomy
34G20Nonlinear ODE in abstract spaces
47H20Semigroups of nonlinear operators
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References:
[1] Antonevich, A.: Linear functional equations. Operator approach. Oper. theory adv. Appl. 83 (1996)
[2] Antonevich, A.: Two methods for investigating the invertibility of operators from C*-algebras generated by dynamical systems. Math. USSR-sb 52, 1-20 (1985) · Zbl 0603.47034
[3] Arendt, W.; Greiner, G.: The spectral mapping theorem for one-parameter groups of positive operators on $C0(X)$. Semigroup forum 30, 297-330 (1984) · Zbl 0536.47032
[4] Baskakov, A. G.: Semigroups of difference operators in spectral analysis of linear differential operators. Funct. anal. Appl. 30, 149-157 (1996) · Zbl 0911.47040
[5] Ben-Artzi, A.; Gohberg, I.; Kaashoek, M. A.: Invertibility and dichotomy of differential operators on a half line. J. dynamics differential equations 5, 1-36 (1993) · Zbl 0771.34011
[6] Bhatia, N. P.; Szegö, G. P.: Stability theory of dynamical systems. (1970) · Zbl 0213.10904
[7] Chicone, C.; Swanson, R. C.: The spectrum of the adjoint representation and the hyperbolicity of dynamical systems. J. differential equations 36, 28-39 (1980) · Zbl 0443.58019
[8] Chicone, C.; Swanson, R. C.: A generalized Poincaré stability criterion. Proc. amer. Math. soc. 81, 495-500 (1981) · Zbl 0472.58018
[9] Chicone, C.; Swanson, R.: Spectral theory for linearization of dynamical systems. J. differential equations 40, 155-167 (1981) · Zbl 0432.58016
[10] Chow, S. -N.; Leiva, H.: Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces. J. differential equations 120, 429-477 (1995) · Zbl 0831.34067
[11] Chow, S. -N.; Leiva, H.: Two definitions of the exponential dichotomy for skew-product semiflow in Banach spaces. Proc. amer. Math. soc. 124, 1071-1081 (1996) · Zbl 0845.34064
[12] Chow, S. -N.; Leiva, H.: Unbounded perturbations of the exponential dichotomy for evolution equations. J. differential equations 129, 509-531 (1996) · Zbl 0857.34062
[13] Coppel, W. A.: Dichotomies in stability theory. Lecture notes in math. 629 (1978) · Zbl 0376.34001
[14] Daleckij, J.; Krein, M.: Stability of differential equations in Banach space. (1974)
[15] Engelking, R.: General topology. (1977) · Zbl 0373.54002
[16] Hadwin, D.; Hoover, T.: Representations of weighted translation algebras. Houston J. Math. 18, 295-318 (1992) · Zbl 0833.47027
[17] Hale, J.: Asymptotic behavior of dissipative systems. Math. surveys monographs 25 (1988) · Zbl 0642.58013
[18] Henry, D.: Geometric theory of nonlinear parabolic equations. Lecture notes in math. 840 (1981)
[19] Johnson, R.: Analyticity of spectral subbundles. J. differential equations 35, 366-387 (1980) · Zbl 0458.34017
[20] Latushkin, Y.; Montgomery-Smith, S.: Evolutionary semigroups and Lyapunov theorems in Banach spaces. J. funct. Anal. 127, 173-197 (1995) · Zbl 0878.47024
[21] Latushkin, Y.; Montgomery-Smith, S.; Randolph, T.: Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibers. J. differential equations 125, 73-116 (1996) · Zbl 0881.47020
[22] Latushkin, Y.; Randolph, T.: Dichotomy of differential equations on Banach spaces and an algebra of weighted composition operators. Integral equations operator theory 23, 472-500 (1995) · Zbl 0839.47026
[23] Latushkin, Y.; Randolph, T.; Schnaubelt, R.: Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces. J. dynam. Differential equations 10, 489-510 (1998) · Zbl 0908.34045
[24] Y. Latushkin, and, R. Schnaubelt, The spectral mapping theorem for evolution semigroups on Lp associated with strongly continuous cocycles, Semigroup Forum, to appear. · Zbl 0937.47047
[25] Latushkin, Y.; Stepin, A. M.: Weighted composition operators and linear extensions of dynamical systems. Russian math. Surveys 46, 95-165 (1992)
[26] Y. Latushkin, and, A. M. Stepin, On the perturbation theorem for the dynamical spectrum, preprint.
[27] Levitan, B. M.; Zhikov, V. V.: Almost periodic functions and differential equations. (1982) · Zbl 0499.43005
[28] Magalhães, L. T.: Persistence and smoothness of hyperbolic invariant manifold for functional differential equations. SIAM J. Math. anal. 18, 670-693 (1987) · Zbl 0621.34058
[29] Mañe, R.: Quasi-Anosov diffeomorphisms and hyperbolic manifolds. Trans. amer. Math. soc. 229, 351-370 (1977) · Zbl 0356.58009
[30] Massera, J.; Schaffer, J.: Linear differential equations and function spaces. (1966) · Zbl 0178.50503
[31] Mather, J.: Characterization of Anosov diffeomorphisms. Indag. math. (N.S.) 30, 479-483 (1968) · Zbl 0165.57001
[32] Van Minh, Nguyen; Räbiger, F.; Schnaubelt, R.: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integral equations operator theory 32, 332-353 (1998) · Zbl 0977.34056
[33] Nagel, R.: One parameter semigroups of positive operators. Lecture notes in math. 1184 (1984)
[34] Neerven, J. M. A.M.V.: The asymptotic behavior of semigroups of linear operators. Operator theory adv. Appl. 88 (1996)
[35] Palmer, K.: Exponential dichotomy and Fredholm operators. Proc. amer. Math. soc. 104, 149-156 (1988) · Zbl 0675.34006
[36] Räbiger, F.; Schnaubelt, R.: A spectral characterization of exponentially dichotomic and hyperbolic evolution families. Tübinger berichte zur funktionalanalysis 3, 204-221 (1994)
[37] Räbiger, F.; Schnaubelt, R.: The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions. Semigroup forum 52, 225-239 (1996) · Zbl 0897.47037
[38] Räbiger, F.; Rhandi, A.; Schnaubelt, R.: Perturbation and abstract characterization of evolution semigroups. J. math. Anal. appl. 198, 516-533 (1996) · Zbl 0884.47017
[39] Rau, R.: Hyperbolic evolution semigroups on vector valued function spaces. Semigroup forum 48, 107-118 (1994) · Zbl 0802.47043
[40] Rau, R.: Hyperbolic linear skew-product semiflows. Z. anal. Anwendungen 15, 865-880 (1996) · Zbl 0879.47021
[41] Renardy, M.: On the linear stability of hyperbolic pdes and viscoelastic flows. Z. angew. Math. phys. 45, 854-865 (1994) · Zbl 0820.76008
[42] Sacker, R.; Sell, G.: Existence of dichotomies and invariant splitting for linear differential systems I, II, III. J. differential equations 15, 22, 429-458 (1974, 1976) · Zbl 0294.58008
[43] Sacker, R.; Sell, G.: A spectral theory for linear differential systems. J. differential equations 27, 320-358 (1978) · Zbl 0372.34027
[44] Sacker, R.; Sell, G.: Dichotomies for linear evolutionary equations in Banach spaces. J. differential equations 113, 17-67 (1994) · Zbl 0815.34049
[45] R. Schnaubelt, Sufficient conditions for exponential stability and dichotomy of evolution equations, submitted. · Zbl 0936.34038
[46] Shen, W.; Yi, Y.: On minimal sets of scalar parabolic equations with skew product structures. Trans. amer. Math. soc. 347, 4413-4431 (1995) · Zbl 0849.35005
[47] Shen, W.; Yi, Y.: Almost automorphic and almost periodic dynamics in skew product semiflows. Mem. amer. Math. soc. 136, 93 (1998) · Zbl 0913.58051