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Triangular Toeplitz contractions and Cowen sets for analytic polynomials

Authors: Muneo Cho, Raúl E. Curto and Woo Young Lee
Journal: Proc. Amer. Math. Soc. 130 (2002), 3597-3604
MSC (2000): Primary 47B35, 15A57, 15A60; Secondary 47B20, 30D50
Published electronically: May 8, 2002
MathSciNet review: 1920039
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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 $\partial _{\mathfrak{L}_{N}} \mathfrak{T}_{N} \subseteq \text{\rm {ext}}\,\mathfrak{T}_{N}$, 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 $G_{f}^{\prime }:=\{g\in H^{\infty }(\mathbb{T}): \text{$g(0)=0$\space and the Toeplitz operator $T_{f+\bar g}$\space is hyponormal}\}$, then $G_{f}^{\prime }$ is strictly convex. This answers a question of C. Cowen for the case of analytic polynomials.

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Additional Information

Muneo Cho
Affiliation: Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan

Raúl E. Curto
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Woo Young Lee
Affiliation: Department of Mathematics, SungKyunKwan University, Suwon 440-746, Korea

Keywords: Triangular Toeplitz contractions, hyponormal Toeplitz operators
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
Article copyright: © Copyright 2002 American Mathematical Society

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