The resolvent average on symmetric cones

Recently Bauschke et al. introduced a new notion of proximal average in the context of convex analysis, and studied this subject systemically in Bauschke et al. (2004, 2007, 2008, 2009) [3–7] from various viewpoints. In addition, this new concept was applied to positive semidefinite matrices under the name of resolvent average, and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis [8] (Bauschke et al., 2010). Inspired by their works and the well-known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous cone), we study the resolvent average on symmetric cones, and derive corresponding results in a different manner based on a purely Jordan algebraic technique.