Complement of forms

Abstract

The notions of the parallel sum, the parallel difference, and the complement of two nonnegative sesquilinear forms were introduced and studied by Hassi, Sebestyé and de Snoo in Hassi et al. (Oper Theory Adv Appl 198:211–227, 2010) and Hassi et al. (J Funct Anal 257(12):3858–3894, 2009). In this paper we continue these investigations. The Galois correspondence induced by the map ${\mathfrak{m} \mapsto \mathfrak{m}_\mathfrak{t}}$ (where ${\mathfrak{m}_\mathfrak{t}}$ denotes the ${\mathfrak{t}}$ -complement of ${\mathfrak{m}}$) is also studied. Inspired by the work of Eriksson and Leutwiler Eriksson and Leutwiler (Math Ann 274:301–317, 1986), we introduce the notion of quasi-unit for nonnegative sesquilinear forms. The quasi-units are characterized by means of the complement and the disjoint part. It is also shown that the ${{\mathfrak{t}}}$ -quasi-units coincide with the extreme points of the convex set ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .