论文标题
弱$^*$的融合序列几乎到处都是连续操作员
Almost everywhere convergent sequences of weak$^*$-to-norm continuous operators
论文作者
论文摘要
令$ x $和$ y $为Banach Space，而$ t：x^*\ to y $是运营商。我们证明，如果$ x $是asplund，并且$ y $具有近似属性，那么对于$（b_ {x^*}}，w^*）的每个radon概率$μ$，$ w^*$ - to-w^*$ - to-to-to-to-to-to-to-norm连续运算符$ t_n：x^*\ to y $ to y $ to y $ y $ y $ y $ y $ y $ y $ y $ y $ y $ y $ to y $ \ \ | t_n（x^x^^^*） \ to 0 $ for $μ$ -a.e。 $ x^*\ in B_ {X^*} $;如果$ y $具有一些$λ\ geq 1 $的$λ$结合近似属性，则可以以$ \ | t_n \ | \ | \ leqleqλ\ | t \ | $的所有$ n \ in \ Mathbb in \ Mathbb {n} $的方式选择序列。如果$ x $不包含子空间同构为$ \ ell_1 $，$ y $具有近似属性（分别为$λ$ bunded近似属性），并且$ t $具有可分离的范围，则相同的结论得出。这扩展到Mercourakis和Stamati的结果不可分割的设置。
Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $μ$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $μ$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $λ$-bounded approximation property for some $λ\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq λ\|T\|$ 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., $λ$-bounded approximation property) and $T$ has separable range. This extends to the non-separable setting a result by Mercourakis and Stamati.
