Abstract
A Nevanlinna function is a function which is analytic in the open upper half-plane and has a non-negative imaginary paxt there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disk. Applying the transformation p times we find a Nevanlinna function n p which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and n p by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, u-resolvent matrices, and reproducing kernel Hilbert spaces.
D. Alpay acknowledges with thanks the Earl Katz family for endowing the chair which supported this research. The research of A. Dijksma and H. Langer was supported in part by the Center for Advanced Studies in Mathematics (CASM) of the Department of Mathematics, Ben-Gurion University.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
N.I Akhiezer, The classical moment problem and some related topics in analysis. Fizrnatgiz, Moscow, 1961; English transl.: Hafner, New York, 1965.
D, Alpay, R.W. Buursema, A, Dijksnia, and H. Langer, The combined moment and interpolation problem for Nevanlinna, functions. Operator Theory, Structured Matrices, and Dilations, Theta Ser. Adv. Math., 7, Theta, Bucharest, 2007, 1–28.
D. Alpay, A, Dijksma, and H. Langer, Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized, Nevanlinna functions. Linear Algebra Appl. 387 (2004), 313–342
D. Alpay, A. Dijksnia, H. Langer, and Y. Shondin, The Schur transform for generalized Nevanlinna functions: interpolation and self-adjoint operator realizations. Complex Anal. Oper. Theory 1 (2007), 189–120.
D. Alpay, A. Dijksnia, and H. Langer, The transformation of Issai Schur and related topics in an indefinite setting. Oper. Theory Adv. Appl. 176, Birkhäuser, Basel, 2007, 1–98.
D. Alpay and H. Dym, Structure invariant spaces of vector valued functions, Hermitian matrices and a generalization of the Iohvidov laws. Linear Algebra Appl. 137/138 (1990) 137–181.
L. de Branges, Some Hilbert spaces of analytic functions, I. Trans. Amer. Math. Soc. 106 (1963), 445–468.
A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad, Orthogonal rational functions, Cambridge Monographs on Appl. and Comp. Math. 5, Cambridge University Press, Cambridge, 1999.
M. Derevyagin, On the Schur algorithm for indefinite moment problem. Methods Funct, Anal. Topology 9(2) (2003), 133–145.
A. Dijksnia, H. Langer, A. Luger, and Y. Shondin, Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization. Oper. Theory Adv. Appl. 154, Birkhäuser, Basel, 2004, 69–90.
A. Dijksnia, H. Langer, and H.S.V. de Snoo, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions. Math. Nachr. 181 (1993), 107–154.
H. Dym, J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. CBMS Regional Conference Series in Mathematics 71, Amer. Math. Soc., Providence, R.I, 1989.
I.S. Kac and M.G. Krein, R-functions-analytic functions mapping the upper half-plane into itself. Amer. Math. Soc. Transl. 103(2) (1974), 1–18.
M.G. Krein and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren in Raume Πk zusammenhängen. I. Einige Funktionen-klassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236.
M.G. Krem and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren in Raume Πk, zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen, und ganze Operatoren. J. Functional Analysis 30(3) (1978), 390–447.
M.G. Krein and H. Langer, On some continuation problems which are closely related to the theory of Hermitian operators in spaces Πk. IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions, J. Operator Theory 13 (1985), 299–417.
H. Langer and B. Textorius, On generalized resolvents and Q-functions of symmetric linear relations (subspaces). Pacific J. Math. 72 (1977), 135–165.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Mark Krein, on the occasion of his 100th anniversary
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Alpay, D., Dijksma, A., Langer, H. (2009). The Schur Transformation for Nevanlinna Functions: Operator Representations, Resolvent Matrices, and Orthogonal Polynomials. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 190. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9919-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-7643-9919-1_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-9918-4
Online ISBN: 978-3-7643-9919-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)