On the minimum modulus of dual truncated Toeplitz operators

Abstract

This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) $D_{\ddot{I}}†$ acting on the orthogonal complement of the model space ${\mathcal{K}}_u^{\perp }$, where $u$ is a nonconstant inner function and $φ\in L^\infty(\T)$. 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 $m(D_{\ddot{I}}†)$ coincides with the essential infimum of $|\ddot{I}†|$. In the general setting, for unimodular $\vp$, we obtain exact formulas and two sided estimates for $m(D_{\ddot{I}}†)$ 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.

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