Triangular Toeplitz contractions and Cowen sets for analytic polynomials
HTML articles powered by AMS MathViewer
- by Muneo Chō, Raúl E. Curto and Woo Young Lee
- Proc. Amer. Math. Soc. 130 (2002), 3597-3604
- DOI: https://doi.org/10.1090/S0002-9939-02-06628-5
- Published electronically: May 8, 2002
- PDF | Request permission
Abstract:
Let $\mathfrak {L}_{N}$ be the collection of $N\times N$ lower triangular Toeplitz matrices and let $\mathfrak {T}_{N}$ be the collection of $N\times N$ lower triangular Toeplitz contractions. We show that $\mathfrak {T}_{N}$ is compact and strictly convex, in the spectral norm, with respect to $\mathfrak {L}_{N}$; that is, $\mathfrak {T}_{N}$ is compact, convex and $\partial _{\mathfrak {L}_{N}} \mathfrak {T}_{N} \subseteq \operatorname {ext}\mathfrak {T}_{N}$, where $\partial _{\mathfrak {L}_{N}}(\cdot )$ and $\operatorname {ext}(\cdot )$ denote the topological boundary with respect to $\mathfrak {L}_{N}$ and the set of extreme points, respectively. As an application, we show that the reduced Cowen set for an analytic polynomial is strictly convex; more precisely, if $f$ is an analytic polynomial and if $G_f’ := \{\, g\in H^\infty (\mathbb {T}): g(0)=0$ and the Toeplitz operator $T_{f+\bar {sg}}$ is hyponormal$\,\}$, then $G_{f}’$ is strictly convex. This answers a question of C. Cowen for the case of analytic polynomials.References
- John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
- Carl C. Cowen, Hyponormal and subnormal Toeplitz operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 155–167. MR 958573
- Carl C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), no. 3, 809–812. MR 947663, DOI 10.1090/S0002-9939-1988-0947663-4
- Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
- Karel de Leeuw and Walter Rudin, Extreme points and extremum problems in $H_{1}$, Pacific J. Math. 8 (1958), 467–485. MR 98981
- Douglas R. Farenick and Woo Young Lee, Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4153–4174. MR 1363943, DOI 10.1090/S0002-9947-96-01683-2
- 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
- Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Classes of linear operators. Vol. II, Operator Theory: Advances and Applications, vol. 63, Birkhäuser Verlag, Basel, 1993. MR 1246332, DOI 10.1007/978-3-0348-8558-4_{1}
- Charles R. Johnson and Leiba Rodman, Completion of Toeplitz partial contractions, SIAM J. Matrix Anal. Appl. 9 (1988), no. 2, 159–167. MR 938495, DOI 10.1137/0609013
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
- Takahiko Nakazi and Katsutoshi Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), no. 2, 753–767. MR 1162103, DOI 10.1090/S0002-9947-1993-1162103-7
- Geir Nævdal, On the completion of partially given triangular Toeplitz matrices to contractions, SIAM J. Matrix Anal. Appl. 14 (1993), no. 2, 545–552. MR 1211806, DOI 10.1137/0614038
- I. Schur, Über Potenzreihen die im Innern des Einheitskreises beschränkt, J. Reine Angew. Math. 147 (1917), 205–232.
- Sechiko Takahashi, Extension of the theorems of Carathéodory-Toeplitz-Schur and Pick, Pacific J. Math. 138 (1989), no. 2, 391–399. MR 996207
- Ke He Zhu, Hyponormal Toeplitz operators with polynomial symbols, Integral Equations Operator Theory 21 (1995), no. 3, 376–381. MR 1316550, DOI 10.1007/BF01299971
Bibliographic Information
- Muneo Chō
- Affiliation: Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan
- Email: chiyom01@kanagawa-u.ac.jp
- Raúl E. Curto
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- MR Author ID: 53500
- Email: curto@math.uiowa.edu
- Woo Young Lee
- Affiliation: Department of Mathematics, SungKyunKwan University, Suwon 440-746, Korea
- MR Author ID: 263789
- Email: wylee@yurim.skku.ac.kr
- Received by editor(s): September 7, 2000
- Received by editor(s) in revised form: July 2, 2001
- Published electronically: May 8, 2002
- Additional Notes: The second author’s work was partially supported by NSF research grant DMS-9800931
The third author’s work was partially supported by KOSEF research project No. R01-2000-00003 - Communicated by: David R. Larson
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3597-3604
- MSC (2000): Primary 47B35, 15A57, 15A60; Secondary 47B20, 30D50
- DOI: https://doi.org/10.1090/S0002-9939-02-06628-5
- MathSciNet review: 1920039