Compact bilinear operators on asymmetric normed spaces

Abstract

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.