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Compact Groups of Operators on Subproportional Quotients of l 1 m

Published online by Cambridge University Press:  20 November 2018

Piotr Mankiewicz
Piotr Mankiewicz*
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw, Poland email: piotr@impan.gov.pl
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Abstract

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It is proved that a “typical” $n$ -dimensional quotient ${{X}_{n}}$ of $l_{1}^{m}$ with $n={{m}^{\sigma }},0<\sigma <1$ , has the property

$$\text{Average}\int_{G}{||Tx|{{|}_{{{X}_{n}}}}d{{h}_{G}}(T)\ge \frac{c}{\sqrt{n{{\log }^{3}}n}}\left( n-\int_{G}{|trT|d{{h}_{G}}(T)} \right)},$$

for every compact group $G$ of operators acting on ${{X}_{n}}$ , where ${{d}_{G}}(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of ${{X}_{n}}$ . Several consequences of this estimate are presented.

Keywords

46B2046B09

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 5 , 01 October 2000 , pp. 999 - 1017
Copyright
Copyright © Canadian Mathematical Society 2000

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