Bounded symbols and one-component geometry
Abstract
We prove that every bounded truncated Toeplitz operator on a Hardy model space admits a bounded symbol if and only if its defining inner function is one-component, meaning that it has a connected sublevel set at some level in (0,1). This resolves the bounded symbol conjecture of Baranov, Chalendar, Fricain, Mashreghi and Timotin and, through known equivalences, the embedding and weak factorization conjecture of Baranov, Bessonov and Kapustin. A uniform bound R for the ratio of least bounded symbol norm to operator norm on the rank-one family associated with point evaluation in the disk already forces this geometry. The corresponding bound for all bounded truncated Toeplitz operators is at most CRlog(e+R), where C is absolute, and the logarithmic factor is optimal. In a nonzero de Branges space, the same geometric condition characterizes when every sufficiently small equivalent change of norm preserving the de Branges structure is represented exactly by reweighting the original integral with a positive density bounded above and away from zero. This characterization remains valid when the norm changes are restricted to discrete sampling norms. Outside the one-component class, every model space admits self-adjoint trace class truncated Toeplitz operators without bounded symbols, with arbitrarily small norm and singular values satisfying any prescribed positive upper bounds.
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BibTeX
@misc{looi2026bounded,
title = {{Bounded symbols and one-component geometry}},
author = {Sam Looi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31382},
eprint = {2610.31382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31382},
}
RIS
TY - UNPB
TI - Bounded symbols and one-component geometry
AU - Sam Looi
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31382
UR - https://arxiv.org/abs/2610.31382
AB - We prove that every bounded truncated Toeplitz operator on a Hardy model space admits a bounded symbol if and only if its defining inner function is one-component, meaning that it has a connected sublevel set at some level in (0,1). This resolves the bounded symbol conjecture of Baranov, Chalendar, Fricain, Mashreghi and Timotin and, through known equivalences, the embedding and weak factorization conjecture of Baranov, Bessonov and Kapustin. A uniform bound R for the ratio of least bounded symbol norm to operator norm on the rank-one family associated with point evaluation in the disk already forces this geometry. The corresponding bound for all bounded truncated Toeplitz operators is at most CRlog(e+R), where C is absolute, and the logarithmic factor is optimal. In a nonzero de Branges space, the same geometric condition characterizes when every sufficiently small equivalent change of norm preserving the de Branges structure is represented exactly by reweighting the original integral with a positive density bounded above and away from zero. This characterization remains valid when the norm changes are restricted to discrete sampling norms. Outside the one-component class, every model space admits self-adjoint trace class truncated Toeplitz operators without bounded symbols, with arbitrarily small norm and singular values satisfying any prescribed positive upper bounds.
ER -