The Kadec-Pełczyński theorem in $L^p$, $1\le p<2$

Abstract:

By a classical result of Kadec and Pełczyński (1962), every normalized weakly null sequence in $L^p$, $p>2$, contains a subsequence equivalent to the unit vector basis of $\ell ^2$ or to the unit vector basis of $\ell ^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczyński theorem is that the limit random measure $\mu$ of the sequence satisfies $\int _{\mathbb {R}} x^2 d\mu (x)\in L^{p/2}$.