  The paper is concerned with compact bilinear operators on asymmetric normed
spaces. The study of multilinear operators on asymmetric normed spaces was
initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this
direction and prove a Schauder type theorem on the compactness of the adjoint
of a compact bilinear operator and study the ideal properties of spaces of
compact bilinear operators. These extend some results of Ramanujan and Schock,
Linear and Multilinear Algebra (1985), and Ruch, ibid. (1989), on compact
bilinear operators on Banach spaces. On the space of bilinear forms one
introduces the analog of the weak$^*$-topology, called the $w^2$-topology, and
one proves an Alaoglu-Bourbaki type theorem -- the $w^2$-compactness of the
closed unit ball.
