Norm attaining dual truncated Toeplitz operators

This paper develops a complete framework for understanding when a dual truncated Toeplitz operator (DTTO) attains its norm. Given a nonconstant inner function $u$, the DTTO associated with a symbol $φ\in L^{\infty}(\mathbb{T})$ acts on the orthogonal complement ${\mathcal{K}_u}^{\perp} = uH^{2} \oplus H^{2}_{-}$ of the model space $\mathcal{K}_u = H^{2}\ominus uH^{2}$. Assuming $\|φ\|_{\infty}=1$, we give a characterization of the norm attaining property of $D_φ$ and describe all extremal vectors. A sharp analytic and coanalytic dichotomy emerges $D_φ$ attains its norm precisely when the symbol admits either $φ=\overline{u}\overlineψ_{+}χ_{+}$ or $φ=uψ_{-}\overlineχ_{-},$ where $ψ_{\pm},χ_{\pm}$ are inner functions. The first condition corresponds to norm attainment on the analytic component $uH^{2}$, while the second corresponds to norm attainment on the coanalytic component $H^{2}_{-}$ via the natural conjugation $C_{u}$. A key feature of the theory is that the dual compressed shift $D_{u}$ (the case $φ(z)=z$) always attains its norm. We also obtain a coupled Toeplitz, Hankel system governing analytic and coanalytic components of extremal vectors, and provide several concrete examples including nonanalytic unimodular symbols illustrating how the factorization criteria govern norm attainment.

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