On the minimum modulus of dual truncated Toeplitz operators

This article provides a systematic investigation of the minimum modulus of dual truncated Toeplitz operators (DTTOs) Dφ acting on the orthogonal complement Ku⊥ of the model space Ku:=H2⊖uH2, where u is a nonconstant inner function and φ∈L∞(T). We first establish an explicit formula for the minimum modulus of the compressed shift Su and its dual Du in terms of |u(0)|, and prove that the minimum modulus 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(Dφ) coincides with the essential infimum of |φ|. In the general setting, for unimodular φ, we obtain exact formulas and two-sided estimates for m(Dφ) by analysing the norms of associated Toeplitz and Hankel operators restricted to the model space. Finally, we provide several concrete examples to illustrate our results.