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Parallel Addition and Fixed Points of Compressions on Symmetric Cones
Author(s) -
Bae Heekyung,
Lim Yongdo
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200212)246:1<20::aid-mana20>3.0.co;2-g
Subject(s) - mathematics , semigroup , automorphism , pure mathematics , combinatorics , cone (formal languages) , fixed point , commutative property , identity (music) , jordan algebra , algebra over a field , mathematical analysis , algebra representation , physics , algorithm , acoustics
Let V be a Euclidean Jordan algebra with identity e , and let Ω be the corresponding symmetric cone. In this paper, we introduce a partially ordered commutative semigroup structure on the closed convex cone $ \bar \Omega $ extending the binary operation a : b = ( a –1 + b –1 ) –1 on Ω and consider compressions of the symmetric cone Ω of the form$$ \Phi _{a, b, kP(w)}(x) = a + kP(w)(x : b), \; a \in \Omega, \; b \in \bar \Omega, \; kP(w) \in {\rm Aut}(V)P(V), $$where P is the quadratic representation of the Jordan algebra V and Aut( V ) is the Jordan automorphism group of V . The aim of this paper is to show that Φ a,b,kP ( w ) has a unique fixed point p ( a, b, kP ( w )) on Ω and the fixed point map$$ p : \Omega \times \bar \Omega \times V \times {\rm Aut}(V) \, \longrightarrow \, \Omega, \; (a, b, w, k) \, \longmapsto \, p(a, b, kP(w)) $$is continuous.