Direct sums of finite dimensional $S{L}_n^{\infty }$ spaces

Abstract

$S{L}^{\infty }$ denotes the space of functions whose square function is in $L^{\infty }$, and the subspaces $S{L}_n^{\infty }$, $n\in \mathbb{N}$, are the finite dimensional building blocks of $S{L}^{\infty }$. We show that the identity operator $I_{S{L}_n^{\infty }}$ on $S{L}_n^{\infty }$ well factors through operators $T:S{L}_N^{\infty }\to S{L}_N^{\infty }$ having large diagonal with respect to the standard Haar system. Moreover, we prove that $I_{S{L}_n^{\infty }}$ well factors either through any given operator $T:S{L}_N^{\infty }\to S{L}_N^{\infty }$, or through $I_{S{L}_N^{\infty }}-T$. Let $X^{(r)}$ denote the direct sum $({\sum }_{n\in {\mathbb{N}}_0}S{L}_n^{\infty }{)}_r$, where $1\le r\le \infty$. Using Bourgain's localization method, we obtain from the finite dimensional factorization result that for each $1\le r\le \infty$, the identity operator $I_{X^{(r)}}$ on $X^{(r)}$ factors either through any given operator $T:X^{(r)}\to X^{(r)}$, or through $I_{X^{(r)}}-T$. Consequently, the spaces $({\sum }_{n\in {\mathbb{N}}_0}S{L}_n^{\infty }{)}_r$, $1\le r\le \infty$, are all primary.

29 pages, 2 figures

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