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Gohberg, I.; Kaashoek, M. A.; Kos, J.

Classification of linear time-varying difference equations under kinematic similarity. (English) Zbl 0860.39008

Integral Equations Oper. Theory 25, No. 4, 445-480 (1996).
This paper concerns the problem of classifying linear time-varying finite dimensional systems of difference equations under kinematic similarity, i.e., under a uniformly bounded time-varying change of variables of which the inverse is also uniformly bounded. Also the problem of reducing difference equations by using such similarity transformations is studied. Both problems are solved for a number of subclasses, including equations with scalar coefficients, time-invariant equations, finitely supported equations, and equations with one jump. For the general case an open problem is formulated.
Reviewer: A.D.Mednykh (Novosibirsk)

Cited in 12 Documents

MSC:

39A10 Additive difference equations
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Keywords:

linear time-varying finite dimensional systems of difference equations; kinematic similarity
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References:

[1] H. Bart, I. Gohberg, and M.A. Kaashoek,Minimal Factorization of Matrix and Operator Functions, OT 1, Birkhäuser Verlag, Basel, 1979. · Zbl 0424.47001
[2] A. Ben-Artzi and I. Gohberg, Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts,Integral Equations Operator Theory 14 (1991), 613-677. · Zbl 0751.47014 · doi:10.1007/BF01200554
[3] C.V. Coffman and J.J. Schäffer, Dichotomies for linear difference equations,Math. Ann. 172 (1967), 139-166. · Zbl 0189.40303 · doi:10.1007/BF01350095
[4] W.A. Coppel,Dichotomies in Stability Theory, Lecture Notes in Math, 629, Springer-Verlag, Berlin, 1978. · Zbl 0376.34001
[5] Ju. L. Daleckñ and M.G. Kre?n,Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs 43, Amer. Math. Soc., Providence, Rhode Island, 1974.
[6] L. Elsner, On some algebraic problems in connection with general eigenvalue algorithms,Linear Algebra Appl. 26 (1979), 123-138. · Zbl 0412.15019 · doi:10.1016/0024-3795(79)90175-7
[7] L.A. Fialkow, A note on quasisimilarity of operators,Acta Sci. Math. (Szeged) 39 (1977), 67-85. · Zbl 0364.47020
[8] P. Gabriel, Unzerlegbare Darstellungen I,Manuscripta Math. 6 (1972), 71-103. · Zbl 0232.08001 · doi:10.1007/BF01298413
[9] I. Gohberg and S. Goldberg, Finite dimensional Wiener-Hopf equations and factorizations of matrices,Linear Algebra Appl. 48 (1982), 219-236. · Zbl 0504.15006 · doi:10.1016/0024-3795(82)90109-4
[10] I. Gohberg, S. Goldberg, and M.A. Kaashoek,Classes of Linear Operators I, OT 49, Birkhäuser Verlag, Basel, 1990. · Zbl 0745.47002
[11] I. Gohberg, S. Goldberg, and M.A. Kaashoek,Classes of Linear Operators II, OT 63, Birkhäuser Verlag, Basel, 1993. · Zbl 0789.47001
[12] I. Gohberg, M.A. Kaashoek, and J. Kos, The asymptotic behaviour of the singular values of matrix powers and applications, accepted for publication inLinear Algebra Appl..
[13] I. Gohberg, M.A. Kaashoek, and J. Kos, Classification of linear periodic difference equations under periodic or kinematic similarity. To appear. · Zbl 0997.39001
[14] R.L. Kelley,Weighted Shifts on Hilbert space, PhD thesis, University of Michigan, 1966.
[15] M.G. Kre?n, Introduction to the geometry of indefiniteJ-spaces and to the theory of operators in these spaces,Transl. Amer. Math. Soc. Ser. 2, 93 (1970), 103-176, 1970.
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