Compact disjointness preserving polynomials on quasi-Banach lattices

Abstract

The paper is devoted to a description of compact disjointness preserving homogeneous polynomials on quasi-Banach lattices. De Pagter and Wickstead found independently that a compact linear operator between Banach lattices is a lattice homomorphism if and only if it is representable as the sum of a norm convergent series of one-dimensional lattice homomorphisms with pairwise disjoint ranges. It is shown that this result also holds for a wider class of operators and spaces, namely, for compact disjointness preserving homogeneous polynomials acting from one quasi-Banach lattice into another. As an auxiliary result it is proved that a lattice homomorphism between quasi-Banach lattices dominated by a positive compact operator is compact.