Compact-like operators in lattice-normed spaces

A linear operator T between two lattice-normed spaces is said to be p-compact if, for any p-bounded net xα, the net Txα 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.