A spectral commutant lifting theorem

Bercovici, Hari; Foias, Ciprian; Tannenbaum, Allen

doi:10.1090/S0002-9947-1991-1000144-9

A spectral commutant lifting theorem
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by Hari Bercovici, Ciprian Foias and Allen Tannenbaum PDF

Trans. Amer. Math. Soc. 325 (1991), 741-763 Request permission

Abstract:

The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.

References

V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S.) 86(128) (1971), 34–75 (Russian). MR 0298453

Infinite Hankel block matrices and related extension problems

111

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
Joseph A. Ball and J. William Helton, A Beurling-Lax theorem for the Lie group $\textrm {U}(m,\,n)$ which contains most classical interpolation theory, J. Operator Theory 9 (1983), no. 1, 107–142. MR 695942
Hari Bercovici, Operator theory and arithmetic in $H^\infty$, Mathematical Surveys and Monographs, vol. 26, American Mathematical Society, Providence, RI, 1988. MR 954383, DOI 10.1090/surv/026
Hari Bercovici, Ciprian Foias, and Allen Tannenbaum, On skew Toeplitz operators. I, Topics in operator theory and interpolation, Oper. Theory Adv. Appl., vol. 29, Birkhäuser, Basel, 1988, pp. 21–43. MR 945002, DOI 10.1007/978-3-0348-9162-2_{1}

Spectral radius interpolation and robust control

Grigore Arsene and Zoia Ceauşescu, On intertwining dilations. IV, Tohoku Math. J. (2) 30 (1978), no. 3, 423–438. MR 509024, DOI 10.2748/tmj/1178229978
Grigore Arsene and Zoia Ceauşescu, On intertwining dilations. IV, Tohoku Math. J. (2) 30 (1978), no. 3, 423–438. MR 509024, DOI 10.2748/tmj/1178229978

Lecture notes in advances in multivariable control

Harry Dym and Israel Gohberg, A maximum entropy principle for contractive interpolants, J. Funct. Anal. 65 (1986), no. 1, 83–125. MR 819176, DOI 10.1016/0022-1236(86)90018-2
I. P. Fedčina, A description of the solutions of the Nevanlinna-Pick tangent problem, Akad. Nauk Armjan. SSR Dokl. 60 (1975), no. 1, 37–42 (Russian, with Armenian summary). MR 385104
Ciprian Foiaş, Contractive intertwining dilations and waves in layered media, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 605–613. MR 562662
Bruce A. Francis, A course in $H_\infty$ control theory, Lecture Notes in Control and Information Sciences, vol. 88, Springer-Verlag, Berlin, 1987. MR 932459, DOI 10.1007/BFb0007371
Bruce A. Francis and Allen Tannenbaum, Generalized interpolation in control theory, Math. Intelligencer 10 (1988), no. 4, 48–52. MR 966131, DOI 10.1007/BF03023747
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971

Optimization in operator theory, analytic function theory and electrical engineering

Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770

Über beschrankte Funftionen, die in gegebenen Punkten vorgeschriebene Werte annehmen

13

Über die Beschränkungen analytischer Funktionen, welche durch vorgegebenen Funktionswerte bewiskt sind

77

Gian-Carlo Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469–472. MR 112040, DOI 10.1002/cpa.3160130309
Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
Béla Sz.-Nagy, Sur les contractions de l’espace de Hilbert, Acta Sci. Math. (Szeged) 15 (1953), 87–92 (French). MR 58128
Béla Sz.-Nagy and Ciprian Foiaş, Dilatation des commutants d’opérateurs, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A493–A495 (French). MR 236755

Harmonic analysis of operators on Hilbert space

Allen Tannenbaum, Invariance and system theory: algebraic and geometric aspects, Lecture Notes in Mathematics, vol. 845, Springer-Verlag, Berlin-New York, 1981. MR 611155