Direct sums of finite dimensional $SL^\infty_n$ spaces

TL;DR

This paper investigates the structure of finite dimensional $SL^ fty_n$ spaces and their direct sums, showing they are primary by demonstrating how the identity operator factors through various operators using Bourgain's localization method.

Contribution

It establishes finite dimensional factorization properties of $SL^ fty_n$ spaces and proves the primarity of their direct sum spaces for all $1 \\leq r \\leq \\infty$.

Findings

The identity operator on $SL^ fty_n$ well factors through operators with large diagonals.

The identity operator on the direct sum spaces factors through any given operator or its complement.

The spaces $(igoplus_{n} SL^ fty_n)_r$ are all primary.

Abstract

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