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Schur Multipliers with Unequal Operator and Completely Bounded Norms on Sp, $1<p\ne 2<\infty $
Abstract
For every 1<p≠2<∞, we exhibit an explicit Schur multiplier on Sp whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such p, we construct a finitely supported Schur symbol mp such that |Mmp|p→ p < |Mmp|cb,p. This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.
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BibTeX
@misc{huang2026schur,
title = {{Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $}},
author = {J. Huang and F. Sukochev and A. Tomskova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20933},
eprint = {2608.20933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20933},
}
RIS
TY - UNPB
TI - Schur Multipliers with Unequal Operator and Completely Bounded Norms on S_p, $1<p\ne 2<\infty $
AU - J. Huang
AU - F. Sukochev
AU - A. Tomskova
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.20933
UR - https://arxiv.org/abs/2608.20933
AB - For every 1<p≠2<∞, we exhibit an explicit Schur multiplier on S_p whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such p, we construct a finitely supported Schur symbol m_p such that |M_m_p|_p→ p < |M_m_p|_cb,p. This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.
ER -