Norm attaining dual truncated Toeplitz operators

Abstract

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.

arXiv comment34 pages. Substantially revised version. Several proofs have been corrected and reorganized, and new results on norm attainment for arbitrary symbols and finite Blaschke products have been added