Characteristic matrices and spectral properties of evolutionary systems

Kaashoek, M. A.; Verduyn Lunel, S. M.

doi:10.1090/S0002-9947-1992-1155350-0

Characteristic matrices and spectral properties of evolutionary systems
HTML articles powered by AMS MathViewer

by M. A. Kaashoek and S. M. Verduyn Lunel PDF

Trans. Amer. Math. Soc. 334 (1992), 479-517 Request permission

Abstract:

In this paper we introduce the notion of a characteristic matrix for a large class of unbounded operators and study the precise connection between characteristic matrices and spectral properties of evolutionary systems. In particular, we study so-called multiplicity theorems. Several examples will illustrate our results.

References

Harm Bart, Israel Gohberg, and Marinus A. Kaashoek, Minimal factorization of matrix and operator functions, Operator Theory: Advances and Applications, vol. 1, Birkhäuser Verlag, Basel-Boston, Mass., 1979. MR 560504
John A. Burns, Terry L. Herdman, and Harlan W. Stech, Linear functional-differential equations as semigroups on product spaces, SIAM J. Math. Anal. 14 (1983), no. 1, 98–116. MR 686237, DOI 10.1137/0514007
Ph. Clément, O. Diekmann, M. Gyllenberg, H. J. A. M. Heijmans, and H. R. Thieme, Perturbation theory for dual semigroups. I. The sun-reflexive case, Math. Ann. 277 (1987), no. 4, 709–725. MR 901713, DOI 10.1007/BF01457866
M. C. Delfour, The largest class of hereditary systems defining a $C_{0}$ semigroup on the product space, Canadian J. Math. 32 (1980), no. 4, 969–978. MR 590659, DOI 10.4153/CJM-1980-074-8
O. Diekmann and S. M. Verduyn Lunel, A new short proof of an old folk theorem in functional-differential equations, Semigroup theory and evolution equations (Delft, 1989) Lecture Notes in Pure and Appl. Math., vol. 135, Dekker, New York, 1991, pp. 101–106. MR 1164644
W. Desch and W. Schappacher, Spectral properties of finite-dimensional perturbed linear semigroups, J. Differential Equations 59 (1985), no. 1, 80–102. MR 803088, DOI 10.1016/0022-0396(85)90139-1
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162

Matrizenrechnung

I. C. Gohberg, M. A. Kaashoek, and D. C. Lay, Equivalence, linearization, and decomposition of holomorphic operator functions, J. Functional Analysis 28 (1978), no. 1, 102–144. MR 0482317, DOI 10.1016/0022-1236(78)90081-2
Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, vol. 49, Birkhäuser Verlag, Basel, 1990. MR 1130394, DOI 10.1007/978-3-0348-7509-7
I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and Rouché’s theorem, Mat. Sb. (N.S.) 84(126) (1971), 607–629 (Russian). MR 0313856
Jack K. Hale, Functional differential equations with infinite delays, J. Math. Anal. Appl. 48 (1974), 276–283. MR 364813, DOI 10.1016/0022-247X(74)90233-9

Theory of functional differential equations

Jack K. Hale and Junji Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), no. 1, 11–41. MR 492721
Daniel Henry, Linear autonomous neutral functional differential equations, J. Differential Equations 15 (1974), 106–128. MR 338520, DOI 10.1016/0022-0396(74)90089-8
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Nathan Jacobson, Lectures in abstract algebra, Graduate Texts in Mathematics, No. 31, Springer-Verlag, New York-Berlin, 1975. Volume II: Linear algebra; Reprint of the 1953 edition [Van Nostrand, Toronto, Ont.]. MR 0369381
M. A. Kaashoek, Analytic equivalence of the boundary eigenvalue operator function and its characteristic matrix function, Integral Equations Operator Theory 9 (1986), no. 2, 275–285. MR 837519, DOI 10.1007/BF01195011
Franz Kappel and Harald K. Wimmer, An elementary divisor theory for autonomous linear functional differential equations, J. Differential Equations 21 (1976), no. 1, 134–147. MR 407412, DOI 10.1016/0022-0396(76)90021-8
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Heinz Langer and Manfred Möller, Linearization of boundary eigenvalue problems, Integral Equations Operator Theory 14 (1991), no. 1, 105–119. MR 1079819, DOI 10.1007/BF01194929
B. W. Levinger, A folk theorem in functional differential equations, J. Differential Equations 4 (1968), 612–619. MR 231009, DOI 10.1016/0022-0396(68)90011-9
Aloisio Freiria Neves, Hermano de Souza Ribeiro, and Orlando Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal. 67 (1986), no. 3, 320–344. MR 845461, DOI 10.1016/0022-1236(86)90029-7
Reinhard Mennicken and Manfred Möller, Root functions of boundary eigenvalue operator functions, Integral Equations Operator Theory 9 (1986), no. 2, 237–265. MR 837517, DOI 10.1007/BF01195009
Toshiki Naito, On autonomous linear functional differential equations with infinite retardations, J. Differential Equations 21 (1976), no. 2, 297–315. MR 412564, DOI 10.1016/0022-0396(76)90124-8
Aloisio Freiria Neves and Xiao-Biao Lin, A multiplicity theorem for hyperbolic systems, J. Differential Equations 76 (1988), no. 2, 339–352. MR 969429, DOI 10.1016/0022-0396(88)90079-4
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Patrick J. Rabier, Generalized Jordan chains and two bifurcation theorems of Krasnosel′skiĭ, Nonlinear Anal. 13 (1989), no. 8, 903–934. MR 1009078, DOI 10.1016/0362-546X(89)90021-7
Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 564653
S. M. Verduyn Lunel, Exponential type calculus for linear delay equations, CWI Tract, vol. 57, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 990243
G. F. Webb, Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1985. MR 772205