Norm attaining dual truncated Toeplitz operators

TL;DR

This paper characterizes when dual truncated Toeplitz operators attain their norm, providing explicit conditions on symbols and describing extremal vectors, with applications to specific examples and a focus on analytic and coanalytic components.

Contribution

It introduces a complete characterization of norm attainment for dual truncated Toeplitz operators, including explicit symbol conditions and extremal vector descriptions, advancing understanding of their functional structure.

Findings

Dual DTTOs attain their norm under specific symbol factorizations.

The dual compressed shift always attains its norm.

A coupled Toeplitz-Hankel system governs extremal vectors.

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 $\varphi=\overline{u}\overline{\psi}_{+}\chi_{+}$ 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.