Compact-Like Operators in Lattice-Normed Spaces

TL;DR

This paper introduces and studies a new class of operators called p-compact operators in lattice-normed spaces, generalizing many existing operator classes and exploring their properties.

Contribution

It defines p-compact, p-M-weakly, and p-L-weakly compact operators, extending the theory of operator classes in lattice-normed spaces with new properties and relationships.

Findings

p-compact operators generalize known classes like compact and weakly compact operators.

Properties of p-M-weakly and p-L-weakly compact operators are established.

Analysis of up-continuous and up-compact operators in lattice-normed vector lattices.

Abstract

A linear operator $T$ between two lattice-normed spaces is said to be $p$-compact if, for any $p$-bounded net $x_\alpha$, the net $Tx_\alpha$ has a $p$-convergent subnet. $p$-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, $AM$-compact operators, etc. Similar to $M$-weakly and $L$-weakly compact operators, we define $p$-$M$-weakly and $p$-$L$-weakly compact operators and study some of their properties. We also study $up$-continuous and $up$-compact operators between lattice-normed vector lattices.