Subsymmetric bases have the factorization property
Volume 170 / 2022
Abstract
Let $(e_j)_{j=1}^\infty $ denote a Schauder basis for a Banach space $X$, and let $(e_j^*)_{j=1}^\infty $ denote the biorthogonal functionals. We say that $(e_j^*)_{j=1}^\infty $ has the factorization property if the identity operator $I_{X^*}$ on $X^*$ factors through every bounded operator $T\colon X^*\to X^*$ with a $\delta $-large diagonal, i.e., $\inf _j |\langle T e_j^*, e_j\rangle |\geq \delta \gt 0$. We show that if $(e_j^*)_{j=1}^\infty $ is subsymmetric and non-$\ell ^1$-splicing (there is no disjointly supported $\ell ^1$-sequence in $X^*$), then $(e_j^*)_{j=1}^\infty $ has the factorization property, even when $X^*$ is non-separable. This property is stable under $\ell ^p$-direct sums of such Banach spaces for all $1\leq p\leq \infty $, i.e., the standard basis $(e_{n,j}^*)_{n,j=1}^\infty $ of $\ell ^p(X^*)$ also has the factorization property.
Moreover, we find a condition ($\star $) for unconditional bases $(e_j)_{j=1}^n$ of finite-dimensional Banach spaces $X_n$, which is expressed in terms of the quantities $\|e_1+\cdots +e_n\|$ and $\|e_1^*+\cdots +e_n^*\|$, under which any operator $T\colon X_n\to X_n$ with 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). In their seminal works [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989], J. Bourgain and L. Tzafriri proved restricted invertibility results for unconditional bases satisfying a lower $r$-estimate and for subsymmetric bases which satisfy certain conditions in terms of Boyd indices. Using condition $(\star )$, we are able to prove restricted invertibility results with reasonably large sets $\sigma $ which apply to a wider range of bases.
