Abstract
Given a regular rational matrix function A(z), we describe the set of vector-valued functions g(z) of the form g(z)=A(z)f(z) for some vector valued function f(z) which is analytic at the prescribed point λ. The description is in terms of a “canonical set of spectral data” for A at λ. We solve the inverse problem of characterizing when a given set of data is a canonical set of spectral data for some rational matrix function at each point in some subset of the complex plane. Our approach is to understand the connection between the rational nxn matrix function A and the shift invariant subspaces AH 2n and AH 2⊥n in L 2n . As an application we show how the linear fractional map which parametrizes the set of all solutions of a matricial Nevanlinna-Pick interpolation problem can be found as the solution of a local inverse spectral problem.
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References
[BH1] J.A. Ball and J.W. Helton, A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation, J. Operator Theory 9 (1983), 107–142.
[BH2] J.A. Ball, and J.W. Helton, Beurling-Lax representations using classical Lie groups with many applications II: GL(n,C) and Wiener-Hopf factorization, Integral Equations and Operator Theory 7 (1984), 291–309.
[BH3] J.A. Ball and J.W. Helton, Beurling-Lax representations using classical Lie groups III: groups preserving forms, Amer. J. Math., 108 (1986), 95–176.
[BH4] J.A. Ball and J.W. Helton, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations and Operator Theory, 9 (1986), 155–203.
[BR] J.A. Ball and A.C.M. Ran, Global inverse spectral problems for rational matrix functions, Linear Algebra and Applications, to appear.
[BGK] H. Bart, I. Gohberg and M.A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, OT1 Birkhäuser (Basel), 1979.
[C] N. Cohen, Spectral analysis of regular matrix polynomials, Integral Equations and Operator Theory 6 (1983), 161–183.
[GKLR] I. Gohberg, M.A. Kaashoek, L. Lerer and L. Rodman, Minimal divisors of rational matrix functions with prescribed zero and pole structure, in Topics in Operator Theory Systems and Networks (ed. H. Dym and I. Gohberg), OT 12 Birkhäuser (Basel) (1983), 241–275.
[GKvS] I. Gohberg, M.A. Kaashoek and F. van Schagen, Rational matrix and operator functions with prescribed singularities, Integral Equations and Operator Theory 5 (1982), 673–717.
[GLR] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press (New York), 1982.
[GR1] I. Gohberg and L. Rodman, Analytic matrix polynomials with prescribed local data, J. d' Analyse Mathematique 40 (1981), 90–128.
[GR2] I. Gohberg and L. Rodman, Interpolation and local data for meromorphic matrix and operator functions, Integral Equations and Operator Theory 9 (1986), 60–94.
[G] L.B. Golinsky, On a generalization of the matrix Nevanlinna-Pick problem, Proc. of the Armenian Academy of Science 18(1983), 187–205 [in Russian].
[Hel] H. Helson, Lectures on Invariant Subspaces, Academic Press (New York), 1964.
[K] V.E. Katsnelson, Methods of J-theory in continuous interpolation problems of analysis, Part I, Private translation by T.Ando (1982), Sapporo, Japan.
[KP] I.V. Kovalishina and V.P. Potapov, Integral representations of Hermitian positive functions, Private Translation by T. Ando (1982), Sapporo, Japan.
[RR] M. Rosenblum and J. Rovnyak, An operator-theoretic approach to theorems of the Pick-Nevanlinna and Loewner types I, Integral Equations and Operator Theory 3 (1980), 408–436.
[S1] D. Sarason, Generalized interpolation in H∞, Trans. Amer. Math. Soc. 127 (1967), 179–203.
[S2] D. Sarason, Operator-theoretic aspects of the Nevanlinna-Pick interpolation problem, in Operators and Function Theory (ed. S.C. Power), D. Reidel (Dordrecht, Holland) (1985), 279–314.
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The first author was partially supported by a grant from the National Science Foundation.
The second author was supported by a grant for the Niels Stichting at Amsterdam.
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Ball, J.A., Ran, A.C.M. Local inverse spectral problems for rational matrix functions. Integr equ oper theory 10, 349–415 (1987). https://doi.org/10.1007/BF01195035
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DOI: https://doi.org/10.1007/BF01195035