Proceedings of the American Mathematical Society

Author(s): S. ter Horst
Journal: Proc. Amer. Math. Soc. 137 (2009), 2697-2707.
MSC (2000): Primary 47A20, 47A56, 47A57
Posted: February 4, 2009
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Abstract: In this paper we present necessary and sufficient conditions for the existence of a unique solution to the relaxed commutant lifting problem. The obtained conditions are more complicated than those for the classical commutant lifting setting, and earlier obtained sufficient conditions turn out not to be necessary conditions. It is also shown that these conditions simplify in certain special cases.

1.: A. Biswas, C. Foias and A. E. Frazho, Weighted commutant lifting, Acta Sci. Math. (Szeged) 65 (1999), 657-686. MR 1737279 (2001b:47027)
2.: R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415. MR 0203464 (34:3315)
3.: C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Operator Theory 44, Birkhäuser-Verlag, Basel, 1990. MR 1120546 (92k:47033)
4.: C. Foias, A. E. Frazho, I. Gohberg and M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems, Operator Theory 100, Birkhäuser-Verlag, Basel, 1998. MR 1635831 (99i:47027)
5.: C. Foias, A.E. Frazho, and M.A. Kaashoek, Relaxation of metric constrained interpolation and a new lifting theorem, Integral Equations Operator Theory 42 (2002), 253-310. MR 1875186 (2002j:47009)
6.: B.A. Francis, A Course in $H_\infty$ Control Theory, Lecture Notes in Control and Information Sciences 88, Springer-Verlag, Berlin, 1987. MR 932459 (89i:93002)
7.: A.E. Frazho, S. ter Horst, and M.A. Kaashoek, Coupling and relaxed commutant lifting, Integral Equations Operator Theory 54 (2006), 33-67. MR 2195229 (2007c:47007)
8.: A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), 299-318. MR 2249492 (2008d:47020)
9.: A.E. Frazho, S. ter Horst, and M.A. Kaashoek, Relaxed commutant lifting: An equivalent version and a new application, Operator Theory 187, pp. 157-168, Birkhäuser-Verlag, Basel, 2008.
10.: K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L\sp {\infty }$ -error bounds, Internat. J. Control 39 (1984), 1115-1193. MR 748558 (86a:93029)
11.: S. ter Horst, Relaxed commutant lifting and Nehari interpolation, Ph.D. Thesis, Vrije Universiteit, Amsterdam, 2007, available online: www.darenet.nl.
12.: S. ter Horst, Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas, Math. Nachr., to appear.
13.: W.S. Li and D. Timotin, The relaxed intertwining lifting in the coupling approach, Integral Equations Operator Theory 54 (2006), 97-111. MR 2195232 (2006i:47021)
14.: B. Sz.-Nagy and C. Foias, Dilation des commutants d'opérateurs, C. R. Acad. Sci. Paris, Série A 266 (1968), 493-495. MR 0236755 (38:5049)
15.: S. Parrott, On a quotient norm and the Sz.-Nagy-Foiaş lifting theorem, J. Funct. Anal. 30 (1978), 311-328. MR 518338 (81h:47006)
16.: D.R. Pik, Block lower triangular operators and optimal contractive systems, Ph.D. Thesis, Vrije Universiteit, Amsterdam, 1999.
17.: D. Sarason, Generalized interpolation in $H^\infty$ , Trans. Amer. Math. Soc. 127 (1967), 179-203. MR 0208383 (34:8193)
18.: S. Treil and A. Volberg, A fixed point approach to Nehari's problem and its applications, in: Toeplitz Operators and Related Topics (Santa Cruz, CA, 1992), The Harold Widom Anniversary Volume, Operator Theory 71, pp. 165-186, Birkhäuser-Verlag, Basel, 1994. MR 1300219 (95i:47026)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A20, 47A56, 47A57

S. ter Horst
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
Email: terhorst@math.vt.edu

DOI: 10.1090/S0002-9939-09-09813-X
PII: S 0002-9939(09)09813-X
Keywords: Commutant lifting, unique solution, linear fractional maps
Received by editor(s): March 3, 2008,
Received by editor(s) in revised form: October 20, 2008
Posted: February 4, 2009
Communicated by: Marius Junge
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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