Minimal representations of unitary operators and orthogonal polynomials on the unit circle

Abstract In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parameterization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein’s Theorem; the second one deals with the movement of mass points of the orthogonality measure under mono-parametric perturbations of the Schur parameters.

[1]  On the Hellmann-Feynman theorem and the variation of zeros of certain special functions , 1988 .

[2]  M. Ismail.  Polynomials Orthogonal on the Unit Circle , 2005 .

[3]  Concavity and convexity of eigenvalues , 1991 .

[4]  William B. Gragg,et al.  The QR algorithm for unitary Hessenberg matrices , 1986 .

[5]  W. J. Thron,et al.  Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle , 1989 .

[6]  M. G. Kreĭn,et al.  Some questions in the theory of moments , 1962 .

[7]  W. Gragg.  Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle , 1993 .

[8]  M. Stone.  Linear transformations in Hilbert space and their applications to analysis , 1932 .

[9]  L. Reichel,et al.  A divide and conquer method for unitary and orthogonal eigenproblems , 1990 .

[10]  R. Feynman.  Forces in Molecules , 1939 .

[11]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[12]  J. Geronimus.  On the Trigonometric Moment Problem , 1946 .

[13]  Tosio Kato.  Perturbation of Continuous Spectra by Trace Class Operators , 1957 .

[14]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .

[15]  Leandro Moral,et al.  Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle , 2002 .

[16]  Y. Genin,et al.  On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal Processing Applications , 1990 .

[17]  W. J. Thron.  L-Polynomials Orthogonal on the Unit Circle , 1988 .

[18]  Walter Van Assche,et al.  Perturbation of Orthogonal Polynomials on an Arc of the Unit Circle, II , 1995 .

[19]  Orthogonal polynomials on the unit circle: distribution of zeros , 1991 .

[20]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[21]  H. Weyl,et al.  Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen , 1910 .

[22]  Barry Simon,et al.  Orthogonal Polynomials on the Unit Circle , 2004, Encyclopedia of Special Functions: The Askey-Bateman Project.

[23]  L. Moral,et al.  Measures and para orthogonal polynomials on the unit circle , 2002 .

[24]  L. Golinskii.  Singular measures on the unit circle and their reflection coefficients , 2000 .

[25]  L. Golinskii.  Geronimus polynomials and weak convergence on a circular arc , 1999 .

[26]  P. Nevai,et al.  Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle , 2001 .

[27]  M. Rosenblum.  Perturbation of the continuous spectrum and unitary equivalence. , 1957 .

[28]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[29]  Alain Joye,et al.  Spectral Analysis of Unitary Band Matrices , 2003 .