Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices

Abstract

A linear operator $T$ between two vector lattices normed by locally solid Riesz spaces is said to be $p_{\tau }$-continuous if, for any $p_{\tau }$-null net $(x_{\alpha })$, the net $(T{x}_{\alpha })$ is $p_{\tau }$-null, and $T$ is said to be $p_{\tau }$-bounded operator if it sends $p_{\tau }$-bounded subsets to $p_{\tau }$-bounded subsets. Also, $T$ is called $p_{\tau }$-compact if, for any $p_{\tau }$-bounded net $(x_{\alpha })$, the net $(T{x}_{\alpha })$ has a $p_{\tau }$-convergent subnet. They generalize several known classes of operators such as norm continuous, order continuous, $p$-continuous, order bounded, $p$-bounded, compact and AM-compact operators. We study the general properties of these operators.

arXiv comment10. arXiv admin note: text overlap with arXiv:1711.10263