Compact Groups of Operators on Subproportional Quotients of $l^m_1$

摘要

It is proved that a ``typical'' $n$-dimensional quotient $X_n$ of$l^m_1$ with $n = m^{sigma}$, $0 < sigma < 1$, has the property$$Average int_G |Tx|_{X_n} ,dh_G(T) geqfrac{c}{sqrt{nlog^3 n}} iggl( n - int_G | r T| ,dh_G (T)iggr),$$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 averageis taken over all extreme points of the unit ball of $X_n$. Severalconsequences of this estimate are presented.