Joint spreading models and uniform approximation of bounded operators

We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C>0$ with the following property: for any $A\in\mathcal{L}(X)$ and convex compact subset $W$ of $\mathcal{L}(X)$ for which there exists $\varepsilon>0$ such that for every $x\in X$ there exists $B\in W$ with $\|A(x)-B(x)\|\le\varepsilon\|x\|$, there exists a subspace $Y$ of $X$ of finite codimension and a $B\in W$ with $\|(A-B)|_Y\|_{\mathcal{L}(Y,X)}\leq C\varepsilon$. We prove that a class of separable Banach spaces including $\ell_p$, for $1\le p< \infty$, and $C(K)$, for $K$ countable and compact, satisfy the UALS. On the other hand every $L_p[0,1]$, for $1\le p\le \infty$ and $p\neq2$, fails the property and the same holds for $C(K)$, where $K$ is an uncountable metrizable compact space. Our sufficient conditions for UALS are based on joint spreading models, a multidimensional extension of the classical concept of spreading model, introduced and studied in the present paper.