On the minimum modulus of dual truncated Toeplitz operators

TL;DR

This paper systematically investigates the minimum modulus of dual truncated Toeplitz operators, providing explicit formulas, spectral bounds, and characterizations, with applications to normal and unimodular symbols, supported by concrete examples.

Contribution

It introduces explicit formulas and bounds for the minimum modulus of DTTOs, extending understanding of their spectral properties and conditions for extremal behavior.

Findings

Explicit formula for the minimum modulus of the compressed shift and its dual.

Sharp spectral bounds for normal DTTOs based on the symbol's essential range.

Exact formulas and estimates for unimodular symbols using Toeplitz and Hankel operators.

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.