Joint spreading models and uniform approximation of bounded operators
Volume 253 / 2020
Abstract
We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C \gt 0$ with the following property: for any $A\in \mathcal {L}(X)$ and convex compact subset $W$ of $\mathcal {L}(X)$ for which there exists $\varepsilon \gt 0$ 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 \lt \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 $p\neq 2$, 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.
