Triangular Toeplitz contractions and Cowen sets for analytic polynomials

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.