On almost interval preserving linear operators

A lattice homomorphism between quasi-Banach lattices is known to be compact if and only if it is a sum of a series of rank one lattice homomorphisms converging in the operator norm with pairwise disjoint images. We obtain an analogous description for the dual class of $AM$-compact and compact linear operators that almost preserve intervals and act in quasi-Banach lattices. As a corollary, we get a characterization of a pair of quasi-Banach lattices having no nonzero $AM$-compact (compact) operators that almost preserve intervals. Also, we prove some theorems of the Radon–Nikodym type for almost interval preserving $AM$-compact (compact) operators.