Subsymmetric bases have the factorization property

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 $Tcolon Xto X$ 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 $1leq pleq infty$. 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 $Tcolon X_nto X_n$ with $delta$-large diagonal can be inverted when restricted to $X_sigma = [e_j : jinsigma]$ for a large set $sigmasubset {1,ldots,n}$ (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 $|sigma|geq c n^{1/4}$.