# The Schur Algorithm in Terms of System Realizations

• Bernd Fritzsche
• Victor Katsnelson
• Bernd Kirstein
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 197)

## Abstract

The main goal of this paper is to demonstrate the usefulness of certain ideas from System Theory in the study of problems from complex analysis. With this paper, we also aim to encourage analysts, who might not be familiar with System Theory, colligations or operator models to take a closer look at these topics. For this reason, we present a short introduction to the necessary background. The method of system realizations of analytic functions often provides new insights into and interpretations of results relating to the objects under consideration. In this paper we will use a well-studied topic from classical analysis as an example. More precisely, we will look at the classical Schur algorithm from the perspective of System Theory. We will confine our considerations to rational inner functions. This will allow us to avoid questions involving limits and will enable us to concentrate on the algebraic aspects of the problem at hand. Given a non-negative integer n, we describe all system realizations of a given rational inner function of degree n in terms of an appropriately constructed equivalence relation in the set of all unitary (n+1)×(n+1)-matrices. The concept of Redheffer coupling of colligations gives us the possibility to choose a particular representative from each equivalence class. The Schur algorithm for a rational inner function is, consequently, described in terms of the state space representation.

## Mathematics Subject Classification (2000)

Primary 30D50 47A48 47A57 Secondary 93B28

## Keywords

Schur algorithm rational inner functions state space method characteristic functions of unitary colligations Redheffer coupling of colligations Hessenberg matrices

## References

1. [AADL] Alpay, D., Azizov, T., Dijksma, A., Langer, H. The Schur algorithm for generalized Schur functions I: coisometric realizations. Pp. 1–36 in: Operator Theory: Adv. Appl. 129, Borichev, A., Nikolski, N. — editors. Birkhäuser Verlag, Basel 2001.Google Scholar
2. [AG] Alpay, D., Gohberg, I. Inverse problem for Sturm-Liouville operators with rational reflection-coefficient. Integral Equations Operator Theory, 30:3, (1998), pp. 317–325.
3. [Arl] Arlinskii, Yu.M. Iterates of the Schur class operator-valued functions and their conservative realizations. arXiv:0801.4267 [math.FA] Submitted to: Operators and Matrices.Google Scholar
4. [Ar] APOB, д.з. Лассивные линейные стационарные динамические системы. Сибиск. Мат. журн., 20:2(1979), 211–28. English transl.: Arov, D.Z. Passive linear stationary dynamic systems. Siberian Math. J., 20:2 (1979), 149-162.Google Scholar
5. [Ausg] Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis (German). [Selected papers on the origins of Schur analysis]. (Fritzsche, B. and B. Kirstein-editors) (German). (Series: Teubner-Archiv zur Mathematik, Volume 16). B.G. Teubner Verlagsgesellschaft, Stuttgart-Leipzig 1991, 290pp.Google Scholar
6. [ARC] Automatic and Remote Control. (Proc. of the First Int. Congress of International Federation of Autom. Control (IFAC). Moscow 1960.) Coales, J.F., Ragazzini, J.R., Fuller, A.T. — editors. Botterworth, London 1961.Google Scholar
7. [BC] Baconyi, M., Constantinescu, T. Schur’s algorithm and several applications. Pitman Research Notes in Math., Volume 261, Longman, Harlow 1992.Google Scholar
8. [BGR] Ball, J.A., Gohberg, I., Rodman, L. Interpolation of rational matrixfunctions. Operator Theory: Advances and Applications, OT 45. Birkhäuser Verlag, Basel, 1990. xii+605 pp.Google Scholar
9. [BGK] Bart, H., Gohberg, I., Kaashoek, M.A. Minimal Factorization of Matrix and Operator Functions. Operator Theory: Advances and Applications, OT 1. Birkhäuser, Basel·Boston·Stuttgart, 1979.Google Scholar
10. [BGKR] Bart, H., Gohberg, I., Kaashoek, M.A., Ran, A. Factorization of matrix and operator functions: the state space method. Operator Theory: Advances and Applications, OT 178. Birkhäuser, Basel·Boston·Stuttgart, 2008. xii+409 pp.Google Scholar
11. [BGKV] Bart, H., Gohberg, I., Kaashoek, M.A., Van Dooren, P. Factorizations of Transfer Functions. SIAM J. Control and Optimization. 18:6 (1980), 675–696.
12. [BFK1] Bogner, S., Fritzsche, B., Kirstein, B. The Schur-Potapov algorithm for sequences of complex p×q-matrices. I. Compl. Anal. Oper. Theory 1 (2007), 55–95.
13. [BFK2] Bogner, S., Fritzsche, B., Kirstein, B. The Schur-Potapov algorithm for sequences of complex p×q-matrices. II. Compl. Anal. Oper. Theory 1 (2007), 235–278.
14. [Br] ъродский, М.С. Треуголlьные и жордановы представления линейных операторов. Hayka, Москва 1969, 287 cc. English transl.: Brodski…, M.S. Triangular and Jordan Representation of linear operators. Transl. of Math. Monogr. 32. Amer. Mat. Soc, Providence, RI, 1971. viii + 246 pp.Google Scholar
15. [BrSv1] Ъродский, В. М., (варцман, Я. С. Об инвариантных подпространствах сжатнй. доклады Академии наук CCCP, 201:3 (1957), 519–522. English transl.: Brodski…, V.M., švarcman, Ja.S. On invariant subspaces of contractions. Soviet. Math. Dokl., 12:6 (1971), 1659–1663.Google Scholar
16. [BrSv2] Ъродский, В. М., (варцман, Я. С. Инвариантные подпостранства сжатия и факторизация характеристической. (Russian). Теория рия Функций, Функциональй анализ и и х приложения. [Teor. Funkciî, Funkcional. Anal. i Priložen.] 12:6 (1971), 15–35, 160.Google Scholar
17. [BrLi] Бродский, М. С., Лившиц, М. С. Спектральий анализ несамосопряженных оператороб и промежуточные системы. Успехи Мат. Наук, том /13:1 (1957), 3–85. English transl.: Brodski…, M.S., Livšic, M.S. Spectral analysis of non-selfadjoint operators and intermediate systems. Amer. Math. Soc. Transl. (2), 13 (1958), 265–346.Google Scholar
18. [CWHF] Constructive Methods of Wiener-Hopf Factorization. Gohberg, I., Kaashoek, M.A. — editors. Birkhäuser, Basel·Boston·Stuttgart 1986. 324 pp.Google Scholar
19. [Con1] Constantinescu, T. On the structure of the Naimark dilation. Journ. Operator Theory, 12 (1984), pp. 159–175.
20. [Con2] Constantinescu, T. Schur Parameters, Factorization and Dilations Problems. Operator Theory: Advances and Applications, OT 82. Birkhäuser Verlag, Basel 1996. ix+253pp.Google Scholar
21. [Dew1] Dewilde, P. Cascade scattering matrix synthesis. Tech. Rep. 6560-21, Information Systems Lab., Stanford University, Stanford 1970.Google Scholar
22. [Dew2] Dewilde, P. Input-output description of roomy systems. SIAM Journ. Control and Optim., 14:4 (1976), 712–736.
23. [Dub] Dubovoy, V.K. Shift operators contained in contractions, Schur parameters and pseudocontinuable Schur functions. Pp. 175–250 in: Interpolation, Schur Functions and Moment Problems.Google Scholar
24. Alpay, D. and Gohberg, I. — eds., Operator Theory: Advances and Applications, 165. Birkhäuser Verlag, Basel 2006. xi+302pp.Google Scholar
25. [DFK] Dubovoy, V.K., Fritzsche, B., Kirstein, B. Matricial Version of the Classical Schur Problem, Teubner Texte zur Mathematik, Band 129, B. 6. Teubner Verlagsgesellschaft, Stuttgart-Leipzig, 1992.Google Scholar
26. [DuHa] Duffin, R.J., Hazony D. The degree of a rational matrix-function. Journ. of Soc. for Industr. Appl. Math.11:3 (1963), pp. 645–658.
27. [Ger] Геронимус, Я.Л. О полиномах, ортогональных на круте, о тригонометрической проблеме моментов и об ассоцированных с нею Функциях типа Carathéodory и Schur’a. (In Russian.) Матем. Сборник, 15(57):11 (1944), 99–130.Google Scholar
28. [Gil] Gilbert, E.G. Controllability and observability in multivariate systems. Journ. of SIAM, Ser. A: Control. Vol. 1 (1962–1963), 128–151.Google Scholar
29. [GolV] Golub, G.H., van Loan, C.F. Matrix Computations. 2nd edition. John Hopkins University Press. Baltimore 1989.
30. [Grg] Gragg, W.B. Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle. Journ. of Comput. and Appl. Math., 46 (1993), pp. 183–198.
31. [He1] Helton, J.W. The characteristic function of operator theory and electrical network realization. Indiana Univ. Math. Journ., 22:5, (1972), 403–414.
32. [He2] Helton, J.W. Discrete time systems, operator models, and scattering theory. Indiana Journal of Funct. Anal., 16, (1974), 15–38.
33. [He3] Helton, J.W. Systems with infinite-dimensional state space: the Hilbert space Approac. Proc. IEEE, 64:1, (1976), 145–160.
34. [HeBa] Helton, J.W., Ball, J.A. The cascade decomposition of a given system vs. the linear fractional decompositions of its transfer function. Integral Equations and Operator Theory, 5 (1982), pp. 341–385.
35. [Hou] Householder, A.S. The Theory of Matrices in Numerical Analysis. Blasdell Publishing, New York·Toronto·London 1964. xi+257 pp. Reprint: Dover Publications, Inc., New York, 1974. x+274 pp.
36. [Fuh] Fuhrmann, P.A. Linear Systems and Operators in Hilbert Space. McGraw Hill, 1981, x+325 pp.Google Scholar
37. [Kaa] Kaashoek, M.A. Minimal factorization, linear systems and integral operators. Pp. 41–86 in: Operators and function theory (Lancaster, 1984), (edited by S.C. Power). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 153, Reidel, Dordrecht, 1985.Google Scholar
38. [Kai] Kailath, T. A theorem of I. Schur and its impact on modern signal processing. In [S: Meth], pp. 9–30.Google Scholar
39. [KFA] Kalman, R.E., Falb, P.L., Arbib, M.A. Topics in mathematical system Theory. McGraw-Hill, New York-Toronto-London, 1969. xiv+358.
40. [Kal1] Kalman, R.E. On the general theory of control systems. In: Ragazzini, J.R., Fuller, A.T. — editors. Botterworth, London 1961 [ARC], Vol. 1, pp. 481–492.Google Scholar
41. [Kal2] Kalman, R.E. Canonical structure of linear dynamical system. Proc. Nat. Acad. Sci. USA, Vol. 48, No. 4 (1962), pp. 596–600.
42. [Kal3] Kalman, R.E. Mathematical description of linear dynamical systems. Journ. of Soc. for Industr. Appl. Math., ser. A: Control, Vol. 1, No. 2 (1962–1963), pp. 152–192.
43. [Kal4] Kalman, R.E. Irreducible realizations and the degree of a rational matrix. Journ. of Soc. for Industr. Appl. Math., 13:2 (1965), pp. 520–544.
44. [Ka] Katsnelson, V.E. Right and left joint system representation of a rational matrix-function in general position (system representation theory for dummies). Pp. 337–400 in: Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997). Oper. Theory Adv. Appl., OT 123, Birkhäuser, Basel, 2001.Google Scholar
45. [KaVo1] Katsnelson, V., Volok, D. Rational solutions of the Schlesinger system and isoprincipal deformations of rational matrix-functions. II. pp. 165–203 in: Operator theory, systems theory and scattering theory: multidimensional generalizations. Oper. Theory Adv. Appl., OT 157, Birkhäuser, Basel, 2005.
46. [KaVo2] Katsnelson, V., Volok, D. Deformations of Fuchsian systems of linear differential equations and the Schlesinger system. Math. Phys. Anal. Geom. 9 (2006), no. 2, 135–186.
47. [KiNe] Killip, R. and Nenciu, I. Matrix models for circular ensembles. Intern.Math. Research Notes, 50 (2004), 2665–2701.
48. [LaPhi] Lax, P. and R. Phillips. Scattering Theory. Academic Press, New York London 1967.
49. [Liv1] Лившиц, М. С. Об одном классе линейных операторов в гильбертовом пространстве, Математический Сборник, том (19) (61):2 (1946), 239–260. English transl.: Livšic, M.S. On a class of linear operators in Hilbert space. Amer. Math. Soc. Transl. (2), 13 (1960), 61–82.Google Scholar
50. [Liv2] Ливщиц, М. С. Лзометрические операторы с равными дефектными числами, Математический Сборник, том 26 (68):1 (1950), 247–264. English transl.: Livšic, M.S. Isometric operators with equal deficiency indices, quasiunitary Operators. Amer. Math. Soc. Transl. (2), 13 (1960), 85–102.Google Scholar
51. [Liv3] Ливщиц, М. С. О спектральном разложении линейных несамосопряженных операторов. Математический Сборник, Том 34:1 (1954), 145–199. English transl.: Livšic, M.S. On the spectral resolution of linear non-selfadjoint operator. Amer. Math. Soc. Transl. (2), 5 (1957), 67–114.Google Scholar
52. [Liv4] Ливщиц, М. С. О применении теории несамосопряженных операторов в теории расаеяния. ЖЭТФ, том 31:1 (1956,), 121–131. English transl.: Livshitz, M.S. The application of non-self-adjoint operators to scattering Theory. Soviet Physics JETP 4:1 (1957), 91–98.Google Scholar
53. [Liv5] Ливщиц, М. С. Метод несамосопряженных операторов в теории рассеяния. Успехи Мат. Наук, том 12:1 (1957), 212–218. English transl.: Livšic, M.S. [itThe method of non-selfadjoint operators in dispersion theory. Amer. Math. Soc. Transl. (2), 16 (1960), 427–434.Google Scholar
54. [Liv6] Ливщиц, М. С. Метод несамосопряженных операторов в теории волноводов. Радиотехника и Электроника, том 7 (1962), 281–297. English transl.: Livšic, M.S. The method of non-selfadjoint operators in the theory of waveguides. Radio Engineering and Electronic Physics, 7 (1962), 260–276.Google Scholar
55. [Liv7] Ливщиц, М. С. О линейных Физических системах, соединенных с внешним миром каналами связи. Известия AH CCCP, сер. математическая, 27 (1963), 993–1030.Google Scholar
56. [Liv8] Ливщиц, М. С. Открытые системы как линейные автоматы. Известия AH CCCP, qsер. математическая 27 (1963), 1215–1228.Google Scholar
57. [Liv9] Ливщиц, М. С. Операторы, колебания, волны. Открытые Системы. Наука, Москва 1966. English transl.: Livshitz, M.S. Operators, Oscillations, Waves. Open Systems. (Transl. of Math. Monogr., 34.) Amer. Math. Soc., Providence, RI, 1973. vi+274 pp.Google Scholar
58. [LiFl] Лившиц, М. С. Флексер, М. (. Разложение реактивного четыренполQuсника в цепочку простейших четырехполюсников. доклады Акад. Наук CCCP (кибернетика и теория регулирования), 135:3 (1960, 542–544. English transl.: Livshitz, M.S., Flekser, M.S. Expansion of a reactive four-terminal network into a chain of simplest four-terminal networks. Soviet Physics-Doklady (cybernetics and control theory), 135:3 (1960), 1150-1152.Google Scholar
59. [LiYa] Лившиц, М. С., Янцевич, А. А. Теория Операторных Узлов в Гильбертовом пространстве. Изд-во Харьковского Чнив-та, Харьков 1971. English transl.: Livshitz, M.S., Yantsevich, A.A. Operator colligations in Hilbert Spaces. Winston & Sons, Washington, D.C., 1979, x+212.Google Scholar
60. [McM] McMillan, B. Introduction to formal realizability theory. Bell Syst. Techn. Journ., textbf31 (1952), Part I — pp. 217–279. Part II — pp. 541–600.Google Scholar
61. [Nik] Nikol’skiî, N.K. (= Nikolski, N.K. Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence, RI, 2002. xiv+461 pp. Vol. 2. Model operators and systems. Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI, 2002. xiv+439 pp.Google Scholar
62. [Red1] Redheffer, R. Remarks on the basis of network theory. J. Math. and Phys. 28 (1949), 237–258.Google Scholar
63. [Red2] Redheffer, R. Inequalities for a matrix Riccati equation. J. Math. Mech. 8 (1959), pp. 349–367.
64. [Red3] Redheffer, R. On a certain linear fractional transformation. J. Math. and Phys. 39 (1960), 269–286.
65. [Red4] Redheffer, R. Difference equations and functional equations in transmissionline Theory. Pp. 282–337 in: Modern mathematics for the engineer: Second Series, Beckenbach, E.F. — ed., McGraw-Hill, New York, 1961.Google Scholar
66. [Red5] Redheffer, R. On the relation of transmission-line theory to scattering and Transfer. J. Math. and Phys. 41 (1962), 1–41.
67. [Sakh1] Сахнович, Л. А. О факторизации передаточной оператор-фчнкции. доклады Акад. Наук CCCP, 226:4 (1976), 781–784. English transl.: Sahnovic, L.A.(=Sakhnovich, L.A.) On the factorization of an operatorvalued transfer function. Soviet Math. — Doklady, 17:1 (1976), 203–207.Google Scholar
68. [Sakh2] Сахнович, Л. А. адачи факторизации и операторие тожества., Успехи Матем. Наук, 41:1 (1986), 3–55. English transl.: Sahnovič, L.A.(=Sakhnovich, L.A.) Factorization problems and operator Identities. Russian Math. Surveys, 41:1 (1986), pp. 1–64.Google Scholar
69. [Sakh3] Sahnovič, L.A. (=Sakhnovich, L.A.) Spectral Theory of Canonical Differential Systems: Method of Operator Identities. Birkhäuser, Basel 1999, vi+202 pp.Google Scholar
70. [Sim] Simon, B. Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications, 54: 1. American Mathematical Society, Providence, RI, 2005. xxvi+466 pp. Part 2. Spectral theory. American Mathematical Society Colloquium Publications, 54:2. American Mathematical Society, Providence, RI, 2005. pp. i-xxii and 467-1044.Google Scholar
71. [Sch] Schur, I. Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I. (in German) J. reine und angewandte Math. 147(1917), 205–232. Reprinted in: [Sch: Ges], Vol. II, pp. 137–164. Reprinted also in: [Ausg], pp. 22–49. English translation: On power series which are bounded in the interior of the unit circle. I., In: [S:Meth], pp. 31–59.
72. [S:Meth] I. Schur Methods in Operator Theory and Signal Processing. Operator Theory: Advances and Applications. Vol. 18. I. Gohberg — editor. Birkhäuser, Basel·Boston·Stuttgart 1986.Google Scholar
73. [Sch: Ges] Schur, I.: Gesammelte Abhandlungen [Collected Works]. Vol. II. Springer-Verlag, Berlin· Heidelberg·New York, 1973.Google Scholar
74. [Str] Strang, G. Linear Algebra and its Applications. Academic Press, 1976.Google Scholar
75. [SzNFo] Sz.-Nagy, B. and C. Foias. Analyse Harmonique des Opérateurs de l’espace de Hilbert (French). Masson and Académiae Kiado, 1967. English transl.: Harmonic Analysis of Operators in Hilbert Space. North Holland, Amsterdam 1970.Google Scholar
76. [Tep] Тепляев, А. Б. Чимо мочечнью. спекмр случайньх ормодоналвных на окружносм п мнодочленоб. доклады Акад. Наук CCCP, 320:1 (1991), 49–53. English Transl.: Teplyaev, A.V. The pure point spectrum of random polynomials orthogonal on the circle. Sov. Math., Dokl. 44:2 (1992), 407–411.Google Scholar
77. [Wil] Wilkinson, J.H. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965.Google Scholar

## Authors and Affiliations

• Bernd Fritzsche
• 1
• Victor Katsnelson
• 2
• Bernd Kirstein
• 1
1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany
2. 2.Department of Mathematicsthe Weizmann InstituteRehovotIsrael