Norm attaining dual truncated Toeplitz operators

TL;DR

This work resolves the norm attainment problem for dual truncated Toeplitz operators $D_\varphi$ on the orthogonal complement $\mathcal{K}_u^{\perp}$ of model spaces. A sharp analytic/coanalytic dichotomy shows that norm attainment occurs precisely when the symbol factors as $\varphi=\overline{u}\,\overline{\psi}_{+}\,\chi_{+}$ or $\varphi=u\,\psi_{-}\overline{\chi}_{-}$ with inner factors, corresponding respectively to attainment on $uH^2$ and on $H^2_{-}$ via the conjugation $C_u$. The paper provides explicit extremal-vector structures (Beurling-type subspaces) and a coupled Toeplitz–Hankel system governing the analytic and coanalytic components, along with concrete examples including nonanalytic unimodular symbols. A key result is that the dual of the compressed shift $D_u$ always attains its norm. The work also connects DTTO norm attainment to classical Toeplitz norm attainment via symbol factorization, enriching the interplay between inner-outer factorization, model spaces, and operator-norm extremals in this dual setting.

Abstract

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 $\varphi=\overline{u}\overlineψ_{+}χ_{+}$ or $\varphi=uψ_{-}\overlineχ_{-},$ 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 $\varphi(z)=z$) 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.

Theorems & Definitions (49)

• Proposition 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 2.4
• Remark 2.5
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3