References
A. B. Antonevich,Two methods for investigating the invertibility of operators from C *-algebras generated by dynamical systems, Math. USSR-Sbornik52 (1985), 1–20.
W. B. Arveson and K. B. Josephson,Operator algebras and measure preserving automorphisms. II, J. Funct. Anal.4 (1969), no. 1, 100–334.
S. Bochner and R. S. Phillips,Absolutely convergent Fourier expansions for noncommutative normed rings, Annals of Math.43 (3) (1942), 409–418.
H. Bart, I. Gohberg, and M. A. Kaashoek,Wiener-Hopf factorization, inverse Fourier transform and exponentially dichotomous operators, J. Funct. Anal.68 (1986), 1–42.
A. Ben-Artzi and I. Gohberg,Band matrices and dichotomy, Oper. Theory Adv. Appl.50 (1991), 137–170.
A. Ben-Artzi and I. Gohberg,Dichotomy of systems and invertibility of linear ordinary differential operators, Oper. Theory Adv. Appl.56 (1992), 90–119.
A. Ben-Artzi and I. Gohberg,Dichotomies of perturbed time-varying systems and the power method, Indiana Univ. Math. J.42 (1983), no. 3, 699–720.
A. Ben-Artzi, I. Gohberg, and M. A. Kaashoek,Invertibility and dichotomy of singular difference equations, Oper. Theory Adv. Appl.48 (1990), 157–184.
A. Ben-Artzi, I. Gohberg and M. A. Kaashoek,Invertibility and dichotomy of differential operators on a half line, J. Dynamics and Diff. Eqns.5 (1993), 1–36.
C. Chicone and R. Swanson,Spectral theory of linearizations of dynamical systems, J. Diff. Eqns.40 (1981), 155–167.
J. Daleckij and M. Krein,Stability of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974.
J. Diestel and J. J. Uhl,Vector Measures, Math. Surv. no. 15, AMS, Providence, RI, 1977.
I. Gohberg and J. Leiterer,Factorization of operator functions with respect to a contour. II. Factorization of operator functions close to the identity, Math. Nach.54 (1973), 41–74.
J. Goldstein,Semigroups of Linear Operators and Applications, Oxford Univ. Press, N.Y., 1985.
—,Asymptotics for bounded semigroups on Hilbert space, Aspects of Positivity in Functional Analysis122 (1986), North-Holland, Amsterdam.
D. E. Evans,Time dependent perturbations and scattering of strongly continuous groups on Banach spaces, Math. Ann.221 (1976), 275–290.
D. Hadwin and T. Hoover,Representations of weighted translation algebras, Houston J. Math.18 (1992), 295–318.
J. Hale,Asymptotic Behavior of Dissipative Systems, Math. Surv. and Mongr., vol. 25, Amer. Math. Soc., Providence, RI, 1988.
D. Henry,Geometric Theory of Nonlinear Parabolic Equations, Lect. Notes Math., vol. 840, Springer-Verlag, NY, 1981.
J. S. Howland,Stationary scattering theory for time-dependent hamiltonians, Math. Ann.207 (1974), 315–335.
R. Johnson,Analyticity of spectral subbundles, J. Diff. Eqns.35 (1980), 366–387.
Yu. I. Karlovich,The local-trajectory method of studying invertibility in C *-algebras of operators with discrete groups of shifts, Soviet Math. Dokl.37 (1988), 407–412.
V. G. Kurbatov,Lyneinye differentsial'no-rasnostnye uravneniya (Linear differential-difference equations), Voronez University, Voronez (1990).
Y. Latushkin,Green's function, continual weighted composition operators along trajectories, and hyperbolicity of linear extensions for dynamical systems, J. Dynamics and Diff. Eqns.6 (1994), 1–21.
Y. Latushkin and S. Montgomery-Smith,Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal.127 (1995), 173–197.
—,Lyapunov theorems for Banach spaces, Bull. Amer. Math. Soc.31 (1994), 44–49.
Y. Latushkin, S. Montgomery-Smith and T. Randolph, J. Diff. Eqns. (to appear).
Y. Latushkin and A. Stepin,Weighted translations operators and linear extensions of dynamical systems, Russian Math. Surveys46 (1991), 95–165.
A. V. Lebedev,Some C *-methods used in the study of algebras associated with automorphisms and endomorphisms, VINITI Preprint no. 5351-B87 (1987).
G. Lumer,Equations de diffusion dans le domaines (x, t) non-cylindriques et semigroupes “espacetemps”, Lect. Notes Math.1393 (1989), 161–179.
J. Massera and J. Schaffer,Linear Differential Equations and Function Spaces, Acad. Press, NY, 1966.
M. Megan, R. Latcu,Exponential dichotomy of evolution operators in Banach spaces, Int. Ser. of Numerical Math.107 (1992), 47–52.
Nguyen Van Minh,Semigroups and stability of nonautonomous differential equations in Banach spaces, Trans. Amer. Math. Soc.345 (1994), 223–241.
Y. A. Mitropolskij, A. M. Samojlenko, and V. L. Kulik,Issledovanija Dichtomii Lineinyh Sistem Differential'nyh Uravnenij, Russian (Dichotomy of Systems of Linear Differential Equations), Naukova Dumka, Kiev, 1990.
R. Nagel,Semigroup methods for non-autonomous Cauchy problems, Tübingen Berichte zur Funktionalanalysis, Heft 2, Jahrang 1992/93 (1993), 103–118.
R. Nagel and A. Rhandi,A characterization of Lipschitz continuous evolution families on Banach spaces, J. Int. Eqns. Oper. Th. (to appear).
K. Palmer,Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc.104 (1988), 149–156.
K. Palmer,Two linear systems criteria for exponential dichotomy, Ann. Math. Pura Appl.124 (1980), 199–216.
J. Prüss,On the spectrum of C 0-semigroups, Trans. Amer. Math. Soc.284, no. 2 (1984), 847–857.
R. Rau,Hyperbolic evolutionary semigroups on vector-valued function spaces, Semigroup Forum48 (1994), 107–118.
R. Rau,Hyperbolic evolution groups and exponential dichotomy of evolution families, J. Funct. Anal. (to appear.).
R. Sacker and G. Sell,Existemce of dichotomies and invariant splitting for linear differential systems. I, II, III, J. Diff. Eqns15, 22 (1974, 1976), 429–458, 478–522.
R. Sacker and G. Sell,Dichotomies for Linear Evolutionary Equations in Banach Spaces, IMA Preprint No. 838 (1991).
H. Tanabe,Equations of Evolution, Pitman, London, 1979.
P. P. Zabreiko and Nguyen Van Minh,Group of characteristic operators and its applications in the theory of linear differential equations, Russian Acad. Sci. Dokl. Math.45 (3) (1992), 517–521.
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The first author was supported by the NSF grant DMS 9400518
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Latushkin, Y., Randolph, T. Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators. Integr equ oper theory 23, 472–500 (1995). https://doi.org/10.1007/BF01203919
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DOI: https://doi.org/10.1007/BF01203919