论文标题
美元
$\tilde{o}$rder-norm continuous operators and $\tilde{o}$rder weakly compact operators
论文作者
论文摘要
令$ e $为矢量晶格$ f $的议法。从矢量晶格$ e $ $ e $的连续操作员$ t $ $ x $被称为$ \ tilde {o} $ rder-rder-norm连续连续连续连续，每当$x_α\ xrightArrow {fo} 0 $ in $ in $ $tx_α\ xrightArlow \ xrightArrow {\ vert。我们的平均值来自融合$x_α\ stackrel {fo} {\ longrightArow} x $，在$ f $中存在另一个net $ \ weft（y_α\ right）$，具有相同的索引集满足$y__α\y_α\ downarrow 0 $ in $ f $ in $ f $ and $ \ vertx__α$ y_ $ y_ y y y y y $ vert。在本文中，我们将研究这种新的运营商的某些属性及其与某些已知的运营商分类的关系。我们还定义了名为$ \ tilde {o} $ rder弱紧凑的操作员的新的运营商。连续的操作员$ t：e \ rightArrow x $据说为$ \ tilde {o} $ rder弱紧凑，如果$ t（a）$ in $ x $中的$ x $是一个相对较弱的紧凑型设置，对于每个$ fo $ $ bounded $ a $ a $ a $ a \ subseteq e $。在此手稿中，我们研究了此类运营商的一些属性及其与$ \ tilde {o} $ rder-norm连续运算符的关系。
Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_α\xrightarrow{Fo}0$ implies $Tx_α\xrightarrow{\Vert.\Vert}0$ for each $(x_α)_α\subseteq E$. Our mean from the convergence $ x_α\stackrel{Fo} {\longrightarrow} x $ is that there exists another net $ \left(y_α\right) $ in $F $ with the same index set satisfying $ y_α\downarrow 0 $ in $F$ and $ \vert x_α- x \vert \leq y_α$ for all indexes $ α$. In this paper, we will study some properties of this new class of operators and its relationships with some known classifications of operators. We also define the new class of operators that named $\tilde{o}$rder weakly compact operators. A continuous operator $T: E \rightarrow X $ is said to be $\tilde{o}$rder weakly compact, if $ T(A) $ in $X$ is a relatively weakly compact set for each $Fo$-bounded $A\subseteq E$. In this manuscript, we study some properties of this class of operators and its relationships with $\tilde{o}$rder-norm continuous operators.
