Abstract
Writing the form associated with a positive matrix as a sum of positive squares, there appears a sequence of complex numbers determining the given matrix. In a certain sense, these calculations are equivalent to the classical algorithm of I.Schur [24] which associates a similar sequence of parameters to an analytic contractive function on the unit disc (such sort of parameters are known as Schur-Szegö parameters). The formalism using these Schur-Szegö parameters (we call it Schur analysis) can be extended to a general framework (operators on Hilbert spaces — see [5], where the operatorial version of the Schur-Szegö parameters is called choice sequence) and can be used to solve some extension problems (as Carathéodory-Fejér problem, Nehari problem and so on — see [1], [5]) and to describe the Kolmogorov decomposition of positive-definite kernels on the set of integers — see [13].
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
Adamjan, V.M.; Arov, D.Z.; Krein, M.G.: Infinite Hankel matrices and generalized Caratheodory-Fejer and Schur problems (Russian), Funck. Analiz. Prilozen. 2: 4 (1968), 1–17.
Adamjan,V.M.; Arov,D.Z.; Krein, M.G.: Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem (Russian), Math. Sb. 86 (128) (1971), 34–75.
Adamjan, V.M.; Arov, D.Z.; Krein, M.G.: Infinite Hankel block matrices and related extension problems (Russian), Izv. Akad. Nauk. Arm jan. SSR. Ser. Mat. 6 (1971), 87–112.
Arov, D.Z.; Krein, M.G.: Problem of search of the minimum of entropy in indeterminate extension problems (Russian), Funck. Analiz. Prilozen. 15 (1981), 61–64.
Arsene, Gr.; Ceausescu, Z.; Foias, C.: On intertwining dilations. VIII, J. Operator Theory 4 (1980), 55–91.
Arsene, Gr.; Constantinescu, T.: The structure of the Naimark dilation and Gaussian stationary processes, Integral Equations Operator Theory, 8 (1985), 181–204.
Arsene, Gr.; Gheondea, A.: Completing matrix contractions, J. Operator Theory, 7 (1982), 179–189.
Ball, J.A.; Helton, J.W.: A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory, J. Operator Theory 9 (1983), 107–142.
Burg, J. P.: Maximal entropy spectral analysis, Ph.D. dissertation, Geophysics Dept., Stanford University, Stanford, CA, May 1975.
Chamfy, C.: Functions méromorphes dans le cercle-unité et leurs series de Taylor, Ann. Inst. Fourier 8 (1958), 211–251.
Constantinescu, T.: On the structure of positive Toeplitz forms, in Dilation theory, Toeplitz operators and other topics,Birkhäuser Verlag, 1983, pp.127–149.
Constantinescu, T.: On the structure of the Naimark dilation, J. Operator Theory, 12 (1984), 159–175.
Constantinescu, T.: The structure of nx n positive operator matrices, INCREST preprint, no.14/1984, 54/1984.
Geronimus, Ja. L.: Orthogonal polynomials, New York, Consultants Bureau, 1961.
Grenander, U.; Szegö, G.: Toeplitz forms and their applications, University of California Press, 1950.
Iohvidov, I. S.; On the theory of indefinite Toeplitz forms (Russian), Dok1. Akad. Nauk. SSSR 101: 2 (1955), 213–216.
Iohvidov, I. S.; Krein, M. G.: Spectral theory of operators in spaces with an indefinite metric. II (Russian), Trudy Moskov. Mat. Obsc. 8 (1959), 413–496.
Iohvidov, I. S.; Krein, M.G.; Langer, H.: Introduction to the spectral theory of operators in spaces with an indefinite metric, Akademie-Verlag, Berlin, 1982.
Kailath, T.; Porat, B. State-space generators for orthogonal polynomials, in Prediction theory and harmonic analysis, North-Holland, 1983.
Krein, M.G.: On the location of the roots of polynomials which are orthogonal on the unit circle with respect to an indefinite weight (Russian), Teor. Funkcii, Funck. Analiz. Prilozen 2 (1966), 131–137.
Krein, M. G.; Langer, H.: Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Tr zusammenhangen. Teil I., Math. Nachr. 77 (1977), 187–236;
Krein, M. G.; Langer, H.: Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Tr zusammenhangen. Teil II, J. Function5I Analysis 30 (1978), 390–447;
Krein, M. G.; Langer, H.: Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Tr zusammenhangen. III Part(1), Beiträge zur Analysis 14 (1981), 25–40;
Krein, M. G.; Langer, H.: Uber einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Tr zusammenhangen. III Part(2), Beiträge zur Analysis 15 (1981), 27–45.
Potapov, V.P.: The multiplicative structure of J-contractive (Russian), Trudy Moskov. Mat. Obsc. 4 (1955), 125–236.
Redheffer, R. M.: Inequalities for a matrix Ricatti equation, 8 (1959), 349–377.
Schur, I.: Über Potenzreihen, die im Innern des Einheitskreises J.Reine Angew. Math. 148 (1918), 122–145.
Sz.-Nagy, B.; Foias, C.: Harmonic analysis of operators on Amsterdam-Budapest, 1970.
Takagi, T.: On an algebraic problem related to an analytic theorem Carathéodory and Fejér, Japan. J. Math. 1 (1924), 83–93.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer Basel AG
About this chapter
Cite this chapter
Constantinescu, T. (1986). Schur Analysis for Matrices with a Finite Number of Negative Squares. In: Douglas, R.G., Pearcy, C.M., Sz.-Nagy, B., Vasilescu, FH., Voiculescu, D., Arsene, G. (eds) Advances in Invariant Subspaces and Other Results of Operator Theory. Operator Theory: Advances and Applications, vol 17. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7698-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7698-8_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7700-8
Online ISBN: 978-3-0348-7698-8
eBook Packages: Springer Book Archive