HAMMERSTEIN-OPERATORS PRESERVING DISJOINTNESS

This paper deals with the problems of a multiplicative representation and of automatic continuity of linear and nonlinear operators preserving disjointness. The operators satisfying a modified Hammerstein condition are introduced and investigated. In 3 we develop a theory of quasi-linear disjointness-preserving Hammerstein operators. As an application we prove that a bijective disjointness-preserving operator between Banach lattices is a continuous d-isomorphism, thus answering in the affirmative a problem posed by Y. Abramovich in 1992. We also construct an example demonstrating that the completeness of the ''departure'' space cannot be omitted in general.