Norm attaining dual truncated Toeplitz operators
In this paper, we investigate norm attainment for dual truncated Toeplitz operators $D_\vp$ acting on $\clk_u^\perp=uH^2\oplus H^2_{-}$, where $u$ is a nonconstant inner function and $\vp\in L^\infty(\T)$. Our main focus is the structure of extremal vectors and the distinction between global and componentwise norm attainment. For arbitrary $\vp\in L^\infty(\T)$, we establish an exact norm-defect identity and characterize the extremal space in terms of the essential maximum set $E_\vp=\{ζ\in\T:|\vp(ζ)|=\|\vp\|_\infty\}$. As a consequence, when $u$ is a finite Blaschke product, \[ D_\vp\in\mathcal{NA}\quad\Longleftrightarrow\quad m(E_\vp)>0. \] In this case, whenever $D_\vp$ is norm attaining, its extremal space is infinite-dimensional. We further show that the sets of symbols generating norm attaining and non-norm attaining DTTOs are both norm dense in $L^\infty(\T)$. Consequently, both the norm attaining and the non-norm attaining DTTOs are operator-norm dense in the class of all DTTOs associated with $u$. For unimodular symbols, we characterize extremality by the condition $M_\vp f\in\clk_u^\perp$ and equivalently by a truncated Hankel kernel condition. For mixed extremal vectors $f=x\oplus y$, we derive the identity \[ \|D_\vp x\|^2-\|x\|^2=\|D_\vp y\|^2-\|y\|^2=-\langle D_\vp x,D_\vp y\rangle, \] which yields a phase-rotation criterion and coupled Toeplitz--Hankel relations. We also show that global norm attainment may occur even when neither the analytic nor the coanalytic component contains a nonzero extremal vector. Under additional Hardy-space hypotheses, we obtain factorization criteria for componentwise extremals, construct explicit extremal families for quotient-inner symbols, and relate norm attainment of Toeplitz operators to that of dual truncated Toeplitz operators.