Browse > Article
http://dx.doi.org/10.4134/JKMS.2015.52.5.1003

COMPLETION OF HANKEL PARTIAL CONTRACTIONS OF NON-EXTREMAL TYPE  

KIM, IN HYOUN (Department of Mathematics Incheon National University)
YOO, SEONGUK (Department of Mathematics Inha University)
YOON, JASANG (School of Mathematical and Statistical Sciences The University of Texas Rio Grande Valley)
Publication Information
Journal of the Korean Mathematical Society / v.52, no.5, 2015 , pp. 1003-1021 More about this Journal
Abstract
A matrix completion problem has been exploited amply because of its abundant applications and the analysis of contractions enables us to have insight into structure and space of operators. In this article, we focus on a specific completion problem related to Hankel partial contractions. We provide concrete necessary and sufficient conditions for the existence of completion of Hankel partial contractions for both extremal and non-extremal types with lower dimensional matrices. Moreover, we give a negative answer for the conjecture presented in [8]. For our results, we use several tools such as the Nested Determinants Test (or Choleski's Algorithm), the Moore-Penrose inverse, the Schur product techniques, and a congruence of two positive semi-definite matrices; all these suggest an algorithmic approach to solve the contractive completion problem for general Hankel matrices of size $n{\times}n$ in both types.
Keywords
Hankel partial contraction; contractive completion; extremal type; non-extremal type;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. G. Crandall, Norm preserving extensions of linear transformations on Hilbert space, Proc. Amer. Math. Soc. 21 (1969), 335-340.
2 G. M. Crippen and T. F. Havel, Distance Geometry and Molecular Conformation, Wiley, New York, 1988.
3 R. Curto, C. Hernandez, and E. de Oteyza, Contractive completions of Hankel partial contractions, J. Math. Anal. Appl. 203 (1996), no. 2, 303-332.   DOI   ScienceOn
4 R. Curto and W. Y. Lee, Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc. 150 (2001) no. 712, 65 pp.
5 R. Curto, S. H. Lee, and J. Yoon, Completion of Hankel partial contractions of extremal type, J. Math. Phys. 53 (2012), 123526.   DOI   ScienceOn
6 C. Davis, An extremal problem for extensions of a sesquilinear form, Linear Algebra Appl. 13 (1976), no. 1-2, 91-102.   DOI   ScienceOn
7 C. Davis, W. M. Kahan, and H. F. Weinberger, Norm-preserving dilations and their application to optimal error bounds, SIAM J. Numer. Anal. 19 (1982), no. 3, 445-469.   DOI   ScienceOn
8 C. Foias and A. E. Frazho, Redheffer products and the lifting of contractions on Hilbert space, J. Operator Theory 11 (1984), no. 1, 193-196.
9 R. M. Gray, On unbounded Toeplitz matrices and nonstationary time series with an application to information theory, Inform. Control 24 (1974), 181-196.   DOI   ScienceOn
10 R. M. Gray and L. D. Davisson, An Introduction to Statistical Signal Processing, Cambridge University Press, London, 2005.
11 L. Hogben, Matrix completion problems for pairs of related classes of matrices, Numer. Linear Algebra Appl. 373 (2003), 13-19.   DOI   ScienceOn
12 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, London, 1985.
13 I. S. Iohvidov, Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkhauser- Verlag, Boston, 1982.
14 C. R. Johnson and L. Rodman, Completion of partial matrices to contractions, J. Funct. Anal. 69 (1986), no. 2, 260-267.   DOI
15 C. R. Johnson and L. Rodman, Completion of Toeplitz partial contractions, SIAM J. Matrix Anal. Appl. 9 (1988), no. 2, 159-167.   DOI
16 M. Laurent, A connection between positive semidefinite and Euclidean distance matrix completion problems, Linear Algebra Appl. 273 (1998), 9-22.   DOI   ScienceOn
17 S. Parrott, On a quotient norm and Sz.-Nagy-Foias lifting theorem, J. Funct. Anal. 30 (1978), no. 3, 311-328.   DOI
18 V. Paulsen, Completely bounded maps and dilations, Pitmam Research Notes in Mathematics Series, vol. 146, Longman Sci. Tech., New York, 1986.
19 S. Power, The distance to upper triangular operators, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 327-329.   DOI
20 J. L. Smul'jan, An operator Hellinger integral, Mat. Sb. (N.S.) 49 (1959), 381-430 (in Russian).
21 Y. L. Shmul'yan and R. N. Yanovskaya, Blocks of a contractive operator matrix, (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 25 (1981), no. 7, 72-75
22 (English translation) Soviet Math. (Iz. VUZ) 25 (1981), 82-86.
23 H. J. Woerdeman, Strictly contractive and positive completions for block matrices, Linear Algebra Appl. 136 (1990), 63-105.   DOI   ScienceOn
24 J. Bowers, J. Evers, L. Hogben, S. Shaner, K Sinder, and A. Wangsness, On completion problems for various classes of P-matrices, Linear Algebra Appl. 413 (2006), no. 2-3, 342-354.   DOI   ScienceOn
25 Wolfram Research, Inc., Mathematica, Version 9, Wolfram Research Inc., Champaign, IL, 2013.
26 G. Arsene and A. Gheondea, Completing matrix contractions, J. Operator Theory 7 (1982), no. 1, 179-189.
27 W. Arveson, Interpolation problems in nest algebra, J. Funct. Anal. 20 (1975), no. 3, 208-233.   DOI