The Kadec-Pe{\l} czynski theorem in $L^p$, $1\le p<2$

TL;DR

This paper extends the Kadec-Pełczynski theorem to the range 1 ≤ p < 2 in L^p spaces, characterizing when sequences contain subsequences equivalent to ℓ^2 or ℓ^p based on the limit measure.

Contribution

It provides a necessary and sufficient condition involving the limit random measure for the subsequence structure in L^p spaces when 1 ≤ p < 2.

Findings

Characterization of subsequences equivalent to ℓ^2 in L^p for 1 ≤ p < 2

Condition involving the limit measure's second moment in L^{p/2}

Extension of classical theorem to the case p<2

Abstract

By a classical result of Kadec and Pe\l czynski (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-Pe\l czynski theorem is that the limit random measure $\mu$ of the sequence satisfies $\int_{\mathbb{R}} x^2 d\mu (x)\in L^{p/2}$.