On the complete bounds of Lp -Schur multipliers

Martijn Caspers; Guillermo  Wildschut

doi:10.1007/s00013-019-01316-7

On the complete bounds of L_p -Schur multipliers

Martijn Caspers^*, Guillermo Wildschut

^*Corresponding author for this work

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Abstract

We study the class M_p of Schur multipliers on the Schatten-von Neumann class S_p with 1 ≤ p≤ ∞ as well as the class of completely bounded Schur multipliers Mpcb. We first show that for 2 ≤ p< q≤ ∞ there exists m∈Mpcb with m∉ M_q, so in particular the following inclusions that follow from interpolation are strict: M_q⊊ M_p and Mqcb⊊Mpcb. In the remainder of the paper we collect computational evidence that for p≠ 1 , 2 , ∞ we have Mp=Mpcb, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.

Original language	English
Pages (from-to)	189-200
Number of pages	12
Journal	Archiv der Mathematik
Volume	113
Issue number	2
DOIs	https://doi.org/10.1007/s00013-019-01316-7
Publication status	Published - 2019

Keywords

Non-commutative L-spaces
Operator spaces
Schur multipliers

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10.1007/s00013-019-01316-7

Caspers-Wildschut2019_Article_OnTheCompleteBoundsOfLPLp-SchuFinal published version, 361 KBLicence: CC BY

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AU - Caspers, Martijn

AU - Wildschut, Guillermo

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N2 - We study the class Mp of Schur multipliers on the Schatten-von Neumann class Sp with 1 ≤ p≤ ∞ as well as the class of completely bounded Schur multipliers Mpcb. We first show that for 2 ≤ p< q≤ ∞ there exists m∈Mpcb with m∉ Mq, so in particular the following inclusions that follow from interpolation are strict: Mq⊊ Mp and Mqcb⊊Mpcb. In the remainder of the paper we collect computational evidence that for p≠ 1 , 2 , ∞ we have Mp=Mpcb, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.

AB - We study the class Mp of Schur multipliers on the Schatten-von Neumann class Sp with 1 ≤ p≤ ∞ as well as the class of completely bounded Schur multipliers Mpcb. We first show that for 2 ≤ p< q≤ ∞ there exists m∈Mpcb with m∉ Mq, so in particular the following inclusions that follow from interpolation are strict: Mq⊊ Mp and Mqcb⊊Mpcb. In the remainder of the paper we collect computational evidence that for p≠ 1 , 2 , ∞ we have Mp=Mpcb, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false.

KW - Non-commutative L-spaces

KW - Operator spaces

KW - Schur multipliers

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DO - 10.1007/s00013-019-01316-7

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SP - 189

EP - 200

JO - Archiv der Mathematik

JF - Archiv der Mathematik

IS - 2

ER -

On the complete bounds of L_p -Schur multipliers

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