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Zero-pole interpolation for meromorphic matrix functions on an algebraic curve and transfer functions of 2D systems. (English) Zbl 0861.47010

Summary: We formulate and solve the problem of constructing a meromorphic bundle map over a compact Riemann surface \(X\) having a prescribed zero-pole structure (including directional information). The output bundle together with the zero-pole data is prespecified while the input bundle and the bundle map are to be determined. The Riemann surface \(X\) is assumed to be (birationally) embedded as an irreducible algebraic curve in \(\mathbb{P}^2\) and both input and output bundles are assumed to be equal to the kernels of determinantal representations for \(X\). In this setting the solution can be found as the joint transfer function of a Livsic-Kravitsky two-operator commutative vessel (2D input-output dynamical system). Also developed is the basic theory of two-operator commutative vessels and the correct analogue of the transfer function for such a system (a meromorphic bundle map between input and output bundles defined over an algebraic curve associated with the vessel) together with a state space realization, a Mittag-Leffler type interpolation theorem and the state space similarity theorem for such bundle mappings. A more abstract version of the zero-pole interpolation problem is also presented.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A57 Linear operator methods in interpolation, moment and extension problems
14H45 Special algebraic curves and curves of low genus
14H60 Vector bundles on curves and their moduli
47A45 Canonical models for contractions and nonselfadjoint linear operators
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
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[1] Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J.: Geometry of Algebraic Curves: Volume I, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[2] Alpay, D. and Vinnikov, V.: Some reproducing kernel Kreîn spaces on Riemann surfaces, Preprint. · Zbl 1067.46034
[3] Ball, J. A. and Clancey, K.: Interpolation with meromorphic matrix functions, Proc. Amer. Math. Soc. 121 (1994), 491-496. · Zbl 0817.47019 · doi:10.1090/S0002-9939-1994-1189741-6
[4] Ball, J. A., Gohberg, I., and Rodman, L.: Interpolation of rational Matrix Functions, Operator Theory Adv. Appl. 45, Birkhäuser-Verlag, Basel, Berlin, Boston, 1990. · Zbl 0708.15011
[5] Ball, J. A., Gohberg, I., and Rodman, L.: Boundary Nevanlinna-Pick interpolation for rational matrix functions, J. Math. Systems, Estimation, Control 1 (1991), 131-164.
[6] Ball, J. A., Groenewald, G., Kaashoek, M. A., and Kim, J.: Column reduced rational matrix functions with given null-pole data in the complex plane, Linear Algebra Appl. 203/204/205 (1994), 67-110. · Zbl 0809.15010 · doi:10.1016/0024-3795(94)90199-6
[7] Ball, J. A. and Ran, A. C. M.: Local inverse spectral problems for rational matrix functions, Integral Equations Operator Theory 10 (1987), 309-348. · Zbl 0621.47020 · doi:10.1007/BF01195034
[8] Ball, J. A. and Ran, A. C. M.: Global inverse spectral problems for rational matrix functions, Linear Algebra Appl. 86 (1987), 237-282. · Zbl 0617.15017 · doi:10.1016/0024-3795(87)90297-7
[9] Bart, H., Gohberg, I., and Kaashoek, M. A.: Minimal Factorization of Matrix and Operator Functions, Operator Theory Adv. Appl. 1, Birkhäuser-Verlag, Basel, Berlin, Boston, 1979. · Zbl 0424.47001
[10] Drezet, J.-M. and Narasimhan, M. S.: Groupe de Picard des varétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53-94. · Zbl 0689.14012 · doi:10.1007/BF01850655
[11] Fay, J.: Theta Functions on Riemann Surfaces, Lecture Notes in Math. 352, Springer-Verlag, New York, 1973. · Zbl 0281.30013
[12] Fay, J.: Kernel Functions, Analytic Torsion, and Moduli Spaces, Mem. Amer. Math. Soc. 464, Amer. Math. Soc., Providence, 1992. · Zbl 0777.32011
[13] Farkas, H. M. and Kra, I.: Riemann Surfaces, Springer-Verlag, New York, 1980, 1992. · Zbl 0475.30001
[14] Forster, O.: Lectures on Riemann Surfaces, Springer-Verlag, New York, 1981. · Zbl 0475.30002
[15] Fulton, W.: Algebraic Curves: An Introduction to Algebraic Geometry, Benjamin, New York, 1969. · Zbl 0181.23901
[16] Gantmacher, F. R.: The Theory of Matrices, Chelsea, New York, 1959. · Zbl 0085.01001
[17] Gohberg, I., Kaashoek, M. A., Lerer, L., and Rodman, L.: Minimal divisors of rational matrix functions with prescribed zero and pole structure, in: H. Dym and I. Gohberg (eds), Topics in Operator Theory, Systems and Networks, Operator Theory Adv. Appl. 12, Birkhäuser-Verlag, Basel, 1984, pp. 241-275. · Zbl 0541.47012
[18] Gohberg, I. and Rodman, L.: Interpolation and local data for meromorphic matrix and operator functions, Integral Equations Operator Theory 9 (1986), 60-94. · Zbl 0589.47015 · doi:10.1007/BF01257062
[19] Griffiths, P. A.: Introduction to Algebraic Curves, Transl. Math. Monographs 76, Amer. Math. Soc., Providence, 1989. · Zbl 0696.14012
[20] Griffiths, P. A. and Harris, J.: Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. · Zbl 0408.14001
[21] Gunning, R. C.: Lectures on Vector Bundles over Riemann Surfaces, Princeton University Press, Princeton, N.J., 1967. · Zbl 0163.31903
[22] Hartshorne, R.: Algebraic Geometry, Springer-Verlag, New York, 1977. · Zbl 0367.14001
[23] Kailath, T.: Linear Systems, Prentice-Hall, Englewood Cliffs, N.J., 1980. · Zbl 0454.93001
[24] Kravitsky, N.: Regular colligations for several commuting operators in Banach space, Integral Equations Operator Theory 6 (1983), 224-249. · Zbl 0535.47007 · doi:10.1007/BF01691897
[25] Liv?ic, M. S.: A method for constructing triangular canonical models of commuting operators based on connections with algebraic curves, Integral Equations Operator Theory 3/4 (1980), 489-507. · Zbl 0472.47007
[26] Liv?ic, M. S.: Cayley-Hamilton theorem, vector bundles aand divisors of commuting operators, Integral Equations Operator Theory 6 (1983), 250-273. · Zbl 0515.47020 · doi:10.1007/BF01691898
[27] Liv?ic, M. S.: Commuting nonselfadjoint operators and mappings of vector bundles on algebraic curves, in: H. Bart, I. Gohberg and M. A. Kaashoek (eds), Operator Theory and Systems, Operator Theory Adv. Appl. 19, Birkhäuser-Verlag, Basel, 1986, pp. 255-279. · Zbl 0627.47019
[28] Liv?ic, M. S., Kravitsky, N., Markus, A. S., and Vinnikov, V.: Theory of Commuting Nonselfadjoint Operators, Kluwer, Acad. Publ., Dordrecht, 1995.
[29] Liv?ic, M. S. and Waksman, L. L.: Commuting Nonselfadjoint Operators in Hilbert Space, Lecture Notes in Math. 1272, Springer-Verlag, New York, 1987. · Zbl 0629.47007
[30] Liv?ic, M. S. and Jancevich, A. A.: Theory of Operator Colligations in Hilbert Space, Wiley, New York, 1979.
[31] Mumford, D.: Tata Lectures on Theta, Progr. Math. 28 (Vol. I); 43 (Vol. II), Birkhäuser-Verlag, Basel, 1983 (Vol. I), 1984 (Vol. II). · Zbl 0509.14049
[32] Seshadri, C. S.: Fibrés vectoriels sur les courbes algébriques, astérisque 96, Société Mathématique de France, Paris, 1982.
[33] Vinnikov, V.: Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl. 125 (1989), 103-140. · Zbl 0704.14041 · doi:10.1016/0024-3795(89)90035-9
[34] Vinnikov, V.: Elementary transformations of determinantal representations of algebraic curves, Linear Algebra Appl. 135 (1990), 1-18. · Zbl 0732.14011 · doi:10.1016/0024-3795(90)90113-Q
[35] Vinnikov, V.: Commuting nonselfadjoint operators and algebraic curves, in: T. Ando and I. Gohberg (eds), Operator Theory and Complex Analysis, Operator Theory Adv. Appl. 59, Birkhäuser-Verlag, Basel, 1992, pp. 348-371. · Zbl 0794.47005
[36] Vinnikov, V.: Self-adjoint determinantal representions of real plane curves, Math. Ann. 296 (1993), 453-479. · Zbl 0789.14029 · doi:10.1007/BF01445115
[37] Vinnikov, V.: 2D systems and realization of bundle mappings on compact Riemann surfaces, in U. Helmke, R. Mennicken and J. Saurer (eds), Systems and Networks: Mathematical Theory and Applications, Volume II, Math. Res. 79, Akademie Verlag, Berlin, 1994, pp. 909-912. · Zbl 0925.93440
[38] Vinnikov, V.: in preparation.
[39] Weil, A.: Généralisation des fonctions abéliennes, J. Math. Pures Appl. 17 (1938), 47-87. · JFM 64.0361.02
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