On the minimum modulus of dual truncated Toeplitz operators

TL;DR

The paper develops a systematic framework to quantify the minimum modulus of dual truncated Toeplitz operators on the orthogonal complement of model spaces. It first establishes explicit minimum-modulus formulas for the compressed shift and its dual in terms of $|u(0)|$, and proves that these minima are attained; it then derives sharp spectral bounds for normal DTTOs via the symbol’s essential range, yielding exact values when convexity of the essential range holds. For unimodular symbols, the authors provide exact formulas and two-sided estimates by analyzing restricted Toeplitz and Hankel norms, and they connect DTTOs to standard TTOs through operator identities that facilitate computation of $m(D_\varphi)$. The results are complemented by concrete examples illustrating sharpness and by detailed treatment of special symbol classes, including inner and analytic cases, with explicit formulas for the minimum modulus in those regimes. This work advances the quantitative understanding of extremal properties of DTTOs and their spectral behavior, with potential implications for complex symmetry and operator theory on model spaces.

Abstract

This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) $D_{\varphi}$ acting on the orthogonal complement of the model space $\mathcal{K}_u^{\perp}$, where $u$ is a nonconstant inner function and $\varphi \in L^\infty(\T)$. We first establish an explicit formula for the minimum modulus of the compressed shift $S_u$ and its dual $D_u$ in terms of $|u(0)|$, and prove that the minimum is always attained. For normal DTTOs, we derive sharp spectral bounds utilizing the essential range of the symbol and characterize the conditions under which $m(D_{\varphi})$ coincides with the essential infimum of $|\varphi|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_{\varphi})$ by analyzing the norms of associated Toeplitz and Hankel operators restricted to the model space. Finally, we provide several concrete examples to illustrate our results.

Theorems & Definitions (55)

• Proposition 2.1
• Corollary 2.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Proposition 3.4
• Theorem 3.5
• proof