Dichotomy and Fredholm properties of evolution equations

Journal of Operator Theory

Volumn 58, Issue 2, 2007, Pages 387-414

  Latushkin, Yuri ^a   Pogan, Alin ^a   Schnaubelt, Roland ^b  

^a UNIVERSITY OF MISSOURI   (United States)

^b UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Evolution equation; Evolution family; Exponential dichotomy; Fredholm operator and index; Input output method; Node operator; Weighted shift operator

Indexed keywords

EID: 42149148773     PISSN: 03794024     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. A. ABBONDANDOLO, P. MAJER, Ordinary differential operators in Hilbert spaces and Fredholm pairs, Math. Z. 243(2003), 525-562.
2. A.G. BASKAKOV, Semigroups of difference operators in the spectral analysis of linear differential operators, Functional Anal. Appl. 30(1996), 149-157.
3. A.G. BASKAKOV, Spectral analysis of linear differential operators, and semigroups of difference operators. I, Differential Equations 33(1997), 1305-1312.
4. A.G. BASKAKOV, On correct linear differential operators, Math. Sb. 190(1999), 323-348.
5. A.G. BASKAKOV, On invertibility and the Fredholm property of parabolic differential operators, Dokl. Akad. Nauk 383(2002), 583-585.
6. A. BEN-ARTZI, I. GOHBERG, Dichotomy of systems and invertibility of linear ordinary differential operators, in Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl., vol. 56, Birkhauser, Basel 1992, pp. 91-119.
7. C. CHICONE, Y. LATUSHKIN, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monographs, vol. 70, Amer. Math. Soc, Providence, RI 1999.
8. Y.L. DALETSKII, M.G. KREIN, Stability of Solutions of Differential Equations in Banach Spaces, Amer. Math. Soc. Transl., vol. 43, Amer. Math. Soc, Providence, RI 1974.
9. D. DI GIORGIO, A. LUNARDI, On Fredholm properties of Lu = u' - A(t)u for paths of sectorial operators, Proc. Roy. Soc. Edinburgh Sect. A 135(2005), 39-59.
10. p spaces on the real line, Proc. London Math. Soc. 91(2005), 703-737.
11. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., vol. 840, Springer-Verlag, New York 1981.
12. Y. LATUSHKIN, Y. TOMILOV, Fredholm properties of evolution semigroups, Illinois J. Math. 48(2004), 999-1020.
13. Y. LATUSHKIN, Y. TOMILOV, Fredholm differential operators with unbounded coefficients, J. Differential Equations 208(2005), 388-429.
14. B.M. LEVITAN, V.V. ZHIKOV, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge 1982.
15. N. VAN MINH, N.T. HUY, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl. 261(2001), 28-44.
16. N. VAN MINH, F. RÄBIGER, R. SCHNAUBELT, Exponential stability, exponential ex-pansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equations Opererator Theory 32(1998), 332-353.
17. K.J. PALMER, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55(1984), 225-256.
18. K.J. PALMER, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc. 104(1988), 149-156.
19. D. PETERHOF, B. SANDSTEDE, A. SCHEEL, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Differential Equations 140(1997), 266-308.
20. P. RABIER, An isomorphism theorem for linear evolution problems on the line, J. Dynamics Differential Equations 15(2003), 779-806.
21. P. RABIER, The Robbin-Salamon index theorem in Banach spaces with UMD, Dyn. Partial Differ. Equ. 1(2004), 303-337.
22. J. ROBBIN, D. SALAMON, The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1995), 1-33.
23. R. SACKER, The splitting index for linear differential systems, J. Differential Equations 33(1979), 368-405.
24. B. SANDSTEDE, A. SCHEEL, On the structure of spectra of modulated travelling waves, Math. Nachr. 232(2001), 39-93.
25. R. SCHNAUBELT, Well-posedness and asymptotic behavior of non-autonomous linear evolution equations, in Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhauser, Basel 2002, pp. 311-338.