The asymptotic Fubini-Study operator over general non-Archimedean fields

Given an ample line bundle $L$ over a projective $\mathbb{K}$-variety $X$, with $\mathbb{K}$ a non-Archimedean field, we study limits of non-Archimedean metrics on $L$ associated to submultiplicative sequences of norms on the graded pieces of the section ring $R(X,L)$. We show that in a rather general case, the corresponding asymptotic Fubini-Study operator yields a one-to-one correspondence between equivalence classes of bounded graded norms and bounded plurisubharmonic metrics that are regularizable from below. This generalizes results of Boucksom-Jonsson where this problem has been studied in the trivially valued case.