Subsymmetric bases have the factorization property
英文摘要：
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 $δ$-large diagonal (that is $\inf_j |\langle Te_j, e_j^*\rangle| \geq δ> 0$, 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 $δ$-large diagonal can be inverted when restricted to $X_σ= [e_j : j\inσ]$ for a "large" set $σ\subset \{1,\ldots,n\}$ (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 $δ$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_σ$ for some $σ$ with $|σ|\geq c n^{1/4}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
