Abstract
We study the class \(\mathcal {M}_p\) of Schur multipliers on the Schatten-von Neumann class \(\mathcal {S}_p\) with \(1 \le p \le \infty \) as well as the class of completely bounded Schur multipliers \(\mathcal {M}^{cb}_p\). We first show that for \(2 \le p < q \le \infty \) there exists \(m \in \mathcal {M}^{cb}_p\) with \(m \not \in \mathcal {M}_q\), so in particular the following inclusions that follow from interpolation are strict: \(\mathcal {M}_q \subsetneq \mathcal {M}_p\) and \(\mathcal {M}^{cb}_q \subsetneq \mathcal {M}^{cb}_p\). In the remainder of the paper we collect computational evidence that for \(p\not = 1,2, \infty \) we have \(\mathcal {M}_p = \mathcal {M}^{cb}_p\), moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.
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The authors thank Cédric Arhancet and the anonymous referee for useful comments on the contents of this paper.
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Caspers, M., Wildschut, G. On the complete bounds of \(L_p\)-Schur multipliers. Arch. Math. 113, 189–200 (2019). https://doi.org/10.1007/s00013-019-01316-7
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DOI: https://doi.org/10.1007/s00013-019-01316-7