# Complement of forms

Complement of forms 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 $${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Complement of forms

, Volume 17 (1) – Jul 29, 2011
15 pages

/lp/springer_journal/complement-of-forms-9w0nSwjnPC
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0138-4
Publisher site
See Article on Publisher Site

### 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 $${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$$ .

### Journal

PositivitySpringer Journals

Published: Jul 29, 2011

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