Ando’s theorem for nonnegative forms

Abstract

In this paper, we present a generalization of Ando’s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1–12, 1999). That is, [A]B ≤ [B]A or [B]A ≤ [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form ${\mathfrak{t}}$ into an almost dominated and a singular part with respect to a nonnegative form ${\mathfrak{w}}$ (see Hassi et al. in J. Funct. Anal. 257(12), 3858–3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.