Almost everywhere convergent sequences of weak${}^{\ast }$-to-norm continuous operators

Abstract

Let $X$ and $Y$ be Banach spaces, and $T:X^{\ast }\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $\mu$ on $(B_{X^{\ast }},w^{\ast })$ there is a sequence of $w^{\ast }$-to-norm continuous operators $T_n:X^{\ast }\to Y$ such that $\Vert T_n(x^{\ast })-T(x^{\ast })\Vert \to 0$ for $\mu$-a.e. $x^{\ast }\in B_{X^{\ast }}$; if $Y$ has the $\lambda$-bounded approximation property for some $\lambda \ge 1$, then the sequence can be chosen in such a way that $\Vert T_n\Vert \le \lambda \Vert T\Vert$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to ${\ell }_1$, $Y$ has the approximation property (resp., $\lambda$-bounded approximation property) and $T$ has separable range. This extends to the non-separable setting a result by Mercourakis and Stamati.

References in corpus (1)

• A class of summing operators acting in spaces of operators