dorsal/arxiv
View SchemaNorm attaining dual truncated Toeplitz operators
| Authors | Sudip Ranjan Bhuia, Puspendu Nag |
|---|---|
| Categories | |
| ArXiv ID | 2601.09375vv1 |
| URL | https://arxiv.org/abs/2601.09375 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
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"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.",
"arxiv_id": "2601.09375",
"authors": [
"Sudip Ranjan Bhuia",
"Puspendu Nag"
],
"categories": [
"math.FA"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Norm attaining dual truncated Toeplitz operators",
"url": "https://arxiv.org/abs/2601.09375",
"version": "v1"
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