Abstract
Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the “nonstationary setting”, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as “transfer functions” of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point α of the open unit disk) for Schur functions were introduced and studied in [3] and [4].
Similar content being viewed by others
References
N.I. Akhiezer and I.M. Glazman.Theory of linear operators. (Vol. I). Pitman Advanced Publishing Program, 1981.
D. Alpay, J. Ball, and Y. Peretz. System theory, operator models and scattering: the time-varying case. Preprint. 1999.
D. Alpay, V. Bolotnikov, A. Dijksma, and H.S.V. de Snoo.On some operator colligations and associated reproducing kernel Hilbert spaces, volume 61 ofOperator theory: Advances and Applications, pages 89–159. Birkhäuser Verlag, Basel, 1993.
D. Alpay, P. Bolttnikov, A. Dijksma, and H.S.V. de Snoo. On some operator colligations and associated reproducing kernel Pontryagin spaces.J. Funcs. Anal., 136:39–80, 1996.
D. Alpay and P. Dewilde. Time-varying signal approximation and estimation. In M. Kaashoek, J.H. van Schuppen, and A.C.M. Ran, editors,Signal processing, scattering and operator theory, and numerical methods (Amsterdam, 1989), volume 5 ofProgress in systems and control theory, pages 1–22. Birkhäuser Boston, boston, MA, 1990.
D. Alpay, P. Dewilde, and H. Dym.Lossless inverse scattering and reproducing kernels for upper triangular operators, volume 47 ofOperator Theory: Advances and Applications, pages 61–133. Birkhäuser Verlag, Basel, 1990.
D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo.Schur junctions, operator colligations and reproducing kernel Pontryagin spaces, volume 96 ofOperator theory: Advances and Applications. Birkhäuser Verlag, Basel, 1997.
D. Alpay and H. Dym.On applications of reproducing kernel spaces to the Schur algorithm and rational J-unitary factorization, volume 18 ofOperator Theory: Advances and Applications, pages 89–159. Birkhäuser Verlag, Basel, 1986.
D. Alpay and H. Dym.On reproducing kernel spaces, the Schur algorithm and interpolation in a general class of domains, volume 59 ofOperator Theory: Advances and Applications, pages 30–77. Birkhäuser Verlag, Basel, 1992.
D. Alpay and H. Dym. On a new class of reproducing kernel Hilbert spaces and a new generalization of the Iohvidov's laws.Linear Algebra Appl., 178:109–183, 1993.
D. Alpay and H. Dym. On a new class of structured reproducing kernel Hilbert spaces.J. Funct. Anal., 111:1–28, 1993.
D. Alpay and Y. Peretz. Realizations for Schur upper triangular operators. In A. Dijksma, I. Gohberg, M. Kaashoek, and R. Mennicken, editors,Contributions to operator theory in spaces with an indefinite metric, volume 106 ofOperator Theory: Advances and Applications, pages 37–90. Birkhäuser Verlag, Basel, 1998.
N. Aronsjan. Theory of reproducing kernels.Trans. Amer. Math. Soc., 68:337–404 (1950).
D. Arov, M. Kaashoek, and D. Pik. Minimal and optimal linear discrete time-varying dissipative scattering systems. Preprint, Vrije Universiteit Amsterdam, 1997.
Gr. Arsene, Z. Ceauşescu, and T. Constantinescu. Schur analysis of some completion problems.Linear Algebra Appl., 109:1–35, 1988.
J. Ball and T. Trent. Unitary colligations, reproducing kernel Hilbert spaces and Nevanlinna-Pick interpolation in several variables.J. Funct. Anal., 157:1–61, 1998.
H. Bart, I. Gohberg, and M. Kaashoek. Convolution equations and linear systems.Integral Equations Operator Theory, 5:283–340, 1982.
L. de Branges. Complementation in Kreîn spaces.Trans. Amer. Math. Soc., 305:277–291, 1988.
L. de Branges and J. Rovnyak. Canonical models in quantum scattering theory. In C. Wilcox. editor,Perturbation theory and its applications in quantum mechanics, pages 295–392. Wiley, New York, 1966.
L. de Branges and J. Rovnyak.Square summable power series. Holt, Rinehart and Winston, New York, 1966.
T. Constantinescu.Schur parameters, factorization and dilation problems, volume 82 ofOperator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1996.
P. Dewilde and H. Dym. Interpolation for upper triangular operators. In I. Gohberg, editor,Time-variant systems and interpolation, volume 56 ofOperator Theory: Advances and Applications, pages 153–260. Birkhäuser Verlag, Basel, 1992.
H. Dym and B. Freydin. Bitangential interpolation for upper triangular operators. In H. Dym, B. Fritzsche, V. Katsnelson, and B. Kirstein, editors,Topics in interpolation theory, volume 95 ofOperator Theory: Advances and Applications, pages 105–142. Birkhäuser Verlag, Basel, 1997.
J. Kos. Higher order time-varying Nevanlinna-Pick interpolation. InChallenges of a generalized system theory (Amsterdam, 1992), pages 59–71. North-Holland, Amsterdam, 1993.
J. Kos,Time-dependent problems in linear operator theory. PhD thesis, Vrije Universiteit, Amsterdam, 1995.
S. Saitoh.Theory of reproducing kernels and its applications, volume 189. Longman scientific and technical, 1988.
D. Sarason.Sub-Hardy Hilbert spaces in the unit disk, volume 10 ofUniversity of Arkansas lecture notes in the mathematical sciences. Wiley, New York, 1994.
A. Sayed, T. Constantinescu, and T. Kailath. Time-variant displacement structure and interpolation problems.IEEE Trans. Automat. Control, 39(5):960–976, 1994.