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 $\varphi \in L^{ \infty}(\mathbb{ T})$ 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 $\|\varphi\|_{ \infty}=1$, we give a characterization of the norm attaining property of $D_{ \varphi}$ and describe all extremal vectors. A sharp analytic and coanalytic dichotomy emerges $D_{ \varphi}$ attains its norm precisely when the symbol admits either $\varphi=\overline{ u}\overline{ \psi}_{ +}\chi_{ +}$ or $\varphi=u\psi_{ -}\overline{ \chi}_{ -},$ where $\psi_{ \pm},\chi_{ \pm}$ 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 $\varphi(z)=z$) 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.
Categories: math.FA