Subsymmetric bases have the factorization property

TL;DR

This paper proves that subsymmetric bases in Banach spaces have the factorization property, allowing certain operators with large diagonals to be factored through the identity, with applications to invertibility on large subspaces.

Contribution

It establishes the factorization property for subsymmetric bases, including in non-separable dual spaces, and provides conditions for restricted invertibility of operators with large diagonals.

Findings

Subsymmetric bases have the factorization property.

Operators with large diagonals can be inverted on large subspaces.

Application to restricted invertibility in Banach spaces.

Abstract

We show that every subsymmetric Schauder basis $(e_j)$ of a Banach space $X$ has the factorization property, i.e. $I_X$ factors through every bounded operator $T\colon X\to X$ with a $\delta$-large diagonal (that is $\inf_j |\langle Te_j, e_j^*\rangle| \geq \delta > 0$, where the $(e_j^*)$ are the biorthogonal functionals to $(e_j)$). Even if $X$ is a non-separable dual space with a subsymmetric weak$^*$ Schauder basis $(e_j)$, we prove that if $(e_j)$ is non-$\ell^1$-splicing (there is no disjointly supported $\ell^1$-sequence in $X$), then $(e_j)$ has the factorization property. The same is true for $\ell^p$-direct sums of such Banach spaces for all $1\leq p\leq \infty$. Moreover, we find a condition for an unconditional basis $(e_j)_{j=1}^n$ of a Banach space $X_n$ in terms of the quantities $\|e_1+\ldots+e_n\|$ and $\|e_1^*+\ldots+e_n^*\|$ under which an operator $T\colon X_n\to X_n$ with $\delta$-large diagonal can be inverted when restricted to $X_\sigma = [e_j : j\in\sigma]$ for a "large" set $\sigma\subset \{1,\ldots,n\}$ (restricted invertibility of $T$; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989). We then apply this result to subsymmetric bases to obtain that operators $T$ with a $\delta$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_\sigma$ for some $\sigma$ with $|\sigma|\geq c n^{1/4}$.