Commutativity, Majorization, and Reduction in Fan–Theobald–von Neumann Systems

Abstract

A Fan–Theobald–von Neumann system (Gowda in Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ) is a triple $$({{\mathcal {V}}},{{\mathcal {W}}},\lambda )$$ , where $${{\mathcal {V}}}$$ and $${{\mathcal {W}}}$$ are real inner product spaces and $$\lambda :{{\mathcal {V}}}\rightarrow {{\mathcal {W}}}$$ is a norm-preserving map satisfying a Fan–Theobald–von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In Gowda (Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ), we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions. We also introduced the concept of commutativity via the equality in the Fan–Theobald–von Neumann-type inequality. In the present paper, we elaborate on the concept of commutativity and introduce/study automorphisms, majorization, and reduction in Fan–Theobald–von Neumann systems.