Random Clarkson inequalities and LP versions of Grothendieck's inequality

✍ Scribed by Andrew Tonge

• Publisher: John Wiley and Sons
• Year: 1987
• Tongue: English
• Weight: 426 KB
• Volume: 131
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19871310127

✦ Synopsis

In a recent paper KATO [3] uscd the LITTLEWOOD matrices to generalise CLARK-SON'S inequalities. Our first aim is to indicate how KATO'S result can be deduced from a neglected version of the HAUSDORFP-YOUXG inequnlity which was proved by WELLS m c t WILLIAXS [12]. \Ve next establish "random CLARKSON inequalities". These show that the expected behaviour of matrices whose coefficients are random j l ' s is, i ~s one might expect, the same as the bchaviour that KATO observed in the LITTLEWOOD matrices. Finally we show how sharp L, versions of GROTHEXDIECK'S inequality can be obtained by combining a I<aro-lilre result with LL theorem of BENXETT [l] on SCHUR multipliers. 3otational conventions. For convenience we confine our attention to contplex BAXACH spaces. All but the group theoretical results have true analogues in the real case. These analogues are simple corollaries of our theorems. We intend to use the standard notations of BANACH space theory. These can be found in the books of LNDEN-STRAUSS and TZAFRIRI [6]. Our basic reference for topological group theory is RUDIN'S book [lo].