论文标题
在操作员的矢量晶格上的收敛结构和本地固体拓扑结构
Convergence structures and locally solid topologies on vector lattices of operators
论文作者
论文摘要
对于矢量晶格$ e $和$ f $，其中$ f $是dedekind的完整并提供了本地固体拓扑，我们将相应的本地绝对坚固的操作员拓扑介绍在订单有限的运算符$ \ mathcal $ \ mathcal l _ {\ mathrm {ob mathrm {ob}}}（e，f）$ f的$ f $中。使用此情况，可以得出$ \ MATHCAL L _ {\ MATHRM {ob}}（E，F）$在$ f $时允许Hausdorff Uo-Lebesgue拓扑。对于每种订单收敛，无界的订单收敛，以及在Hausdorff UO-Lebesgue拓扑中的适用连接时，在$ \ Mathcal l _ {\ Mathrm {ob Mathrm {ob}}}}}（e，f）上都有均匀且强的收敛结构。在这三对中的六个可能的包含物中，只有一对通常是有效的。但是，关于Dedekind完整矢量晶格的正编态性，五个通常是有效的，而第六个对订单有限的网络有效。在我们确定的正常形态的统一界有界原理的结果，在Banach晶格上的正态序列的情况下，后一种条件是多余的。我们还表明，与一般有限的操作员相比，正态不仅保留了网络的收敛，而且保留了无界顺序融合，并且（如果适用）也可以在Hausdorff UO-Lebesgue拓扑中收敛。
For vector lattices $E$ and $F$, where $F$ is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators $\mathcal L_{\mathrm{ob}}(E,F)$ from $E$ into $F$. Using this, it follows that $\mathcal L_{\mathrm{ob}}(E,F)$ admits a Hausdorff uo-Lebesgue topology whenever $F$ does. For each of order convergence, unbounded order convergence, and-when applicable-convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on $\mathcal L_{\mathrm {ob}}(E,F)$. Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms on a Banach lattice, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We also show that, in contrast to general order bounded operators, the orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and -- when applicable -- convergence in the Hausdorff uo-Lebesgue topology as well.
