Completely J-positive (bi-)linear maps on Krein spaces and their Choi J-matrices

Abstract

In this paper, we consider $J$-completely positive linear maps and $(p,q,r)$-$J$-positive bilinear maps for $p,q,r \in \mathbb{N}$ in the Krein space setting. We investigate relations between $J$-completely positive (bi-)linear maps and their Choi $J$-matrices. We prove that the dual cone of the set of $J$-completely positive linear maps, under a bilinear $J$-pairing, coincides with itself. Several characterizations of various $J$-positivity of bilinear maps are proved, including the one in terms of the $J$-positivity of the corresponding Choi $J$-matrices. Finally, we introduce a notion of a partial $J$-positivity of bilinear maps to clarify the relationship between $J$-positivity of bilinear maps and $J$-positivity of their linearization.