Triangular Toeplitz contractions and Cowen sets for analytic polynomials

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.