Compact and AM-compact orthogonally additive operators

We show that compactness of orthogonally additive operators between Banach lattices is not preserved under taking the modulus by providing a nonlinear analogue of Krengel’s example. In contrast, for operators acting from a Banach lattice into an AM-space, AM-compactness is stable under lattice operations. Moreover, the space of absolutely norm bounded AM-compactorthogonally additive operators from a Banach lattice into an AM-space is a Banach lattice, and on such AM-spaces with the weak Fatou property, norm boundedness coincides with absolute norm boundedness.