Abstract
In this paper structured reproducing kernel Hilbert spaces are used to solve matrix versions of a number of classical interpolation problems. Enroute a characterization of a class of such spaces which originates with de Branges is reformulated in terms of matrix equations of the Liapunov and Stein type in the finite dimensional case. Some generalizations to indefinite inner product spaces are also formulated. Finally it is shown that every invertible Hermitean matrix with displacement rank m is the Gram matrix of a “chain” of m x 1 vector valued functions in a suitably defined reproducing kernel Pontryagin space.
The author would like to acknowledge with thanks Renee and Jay Weiss for endowing the chair which supported this research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Alpay and H. Dym, Hilbert spaces of analytic functions, inverse scattering, and operator models I, Integral Equations and Operator Theory 7 (1984), 589–641.
D. Alpay and H. Dym, On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization, in I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications, OT18, Birkhäuser Verlag, Basel, 1986, pp. 89–159.
D. Alpay and I. Gohberg, Unitary rational matrix functions, in Topics in Interpolation Theory of Rational Matrix-valued Functions (I. Gohberg, ed.), Operator Theory: Advances and Applications OT33, Birkhäuser Verlag, Basel, 1988, pp.175–222.
J.A. Ball, Models for non contractions, J. Math. Anal. Appl. 52 (1975), 235–254.
J.A. Ball, Nevanlinna-Pick interpolation: Generalizations and applications, in Surveys of Some Recent Results in Operator Theory (J.B. Conway and B.B. Morrel, eds.), Pitman Research Notes in Mathematics, Longman, in press.
R.R. Bitmead and B.D.O. Anderson, Asymptotically fast solution of Toeplitz and related systems of linear equations, Linear Algebra Appl. 34 (1980), 117–124.
J.A. Ball, I. Gohberg and L. Rodman, Realization and interpolation of rational matrix functions, in Topics in Interpolation Theory of Rational Matrix-valued Functions (I. Gohberg, ed.), Operator Theory: Advances and Applications OT33, Birkhäuser Verlag, Basel, 1988, pp.1–72.
[BGR2]J.A. Ball, I. Gohberg and L. Rodman, Two-sided Nudelman interpolation problem for rational matrix functions, preprint, 1988.
[BGR3]J.A. Ball, I. Gohberg and L. Rodman, Interpolation Problems for Matrix Valued Functions, Part I: Rational Functions, Monograph in preparation.
J.A. Ball and A.C.M. Ran, Local inverse spectral problems for rational matrix functions, Integral Equations and Operator Theory 10 (1987), 349–415.
H. Dym, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, CBMS Lecture Notes, in press.
L. de Branges, Some Hilbert spaces of analytic functions I, Trans. Amer. Math. Soc. 106 (1963), 445–468.
L. de Branges and J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966.
L. de Branges and J. Rovnyak, Canonical Models in Quantum Scattering Theory, in Perturbation Theory and its Applications in Quantum Mechanics (C. Wilcox, ed.), Wiley, New York, 1966, pp. 295–392.
I. Gohberg, M.A. Kaashoek, L. Lerer and L. Rodman, Minimal divisors of rational matrix function with prescribed zero and pole structure, in Topics in Operator Theory Systems and Networks (H. Dym and I. Gohberg, eds.), Operator Theory: Advances and Applications OT12, Birkhäuser Verlag, Basel, 1984, pp.241–275.
J.W. Helton, Operator Theory, Analytic Functions, Matrices and Electrical Engineering, Regional Conference Series in Mathematics, 68, Amer. Math. Soc., Providence, R.I., 1987.
G. Heinig and K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Birkhäuser Verlag, Basel, 1984.
H. Lev-Ari and T. Kailath, Triangular factorization of structured Hermitean matrices, in I. Schur Methods in Operator Theory and Signal Processing (I. Gohberg, ed.), Operator Theory: Advances and Applications OT18, Birkhäuser-Verlag, Basel, 1986, pp.301–324.
T. Kailath, S.-Y. Kung and M. Morf, Displacement ranks of matrices and linear equations, J. Math. Anal, and Appl. 68 (1979), 395–407.
J. Rovnyak, Characterization of spaces K(M), unpublished manuscript.
D.L. Russell, Mathematics of Finite-Dimensional Control Systems, Lecture Notes in Pure and Applied Mathematics, Vol. 43, Marcel Dekker, New York, 1979.
L.A. Sakhnovich, Factorization problems and operator identities, Russian Math. Surveys 41 (1986), 1–64.
A.L. Sakhnovich, On a class of extremal problems, Math. USSR Izvestiya, 30 (1988), 411–418.
Author information
Authors and Affiliations
Editor information
Additional information
To Israel Gohberg: teacher, colleague and valued friend, with admiration and affection, on being sixty years young.
Rights and permissions
Copyright information
© 1989 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Dym, H. (1989). On Reproducing Kernel Spaces, J Unitary Matrix Functions, Interpolation and Displacement Rank. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds) The Gohberg Anniversary Collection. Operator Theory: Advances and Applications, vol 41. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9278-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9278-0_11
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9975-8
Online ISBN: 978-3-0348-9278-0
eBook Packages: Springer Book Archive