Abstract

An Eaton triple is an algebraic system related to a decomposition statement for vectors of an inner product space VV and to some special inner product inequality connected with this decomposition. The Spectral Decomposition for the space of Hermitian matrices associated with Fan-Theobald's trace inequality is a typical example of such a situation. In this paper, for a given Eaton triple (V,G,D)(V,G,D) and for a function F ⁣:V→RF\colon V \to \mathbb{R}, invariant with respect to the group GG acting on VV, we study the problem of extending convexity of FF from the convex cone D⊂VD \subset V to the space VV. In our approach we reduce the problem from E-system (V,G,D)(V,G,D) to its subsystem (W,H,E)(W,H,E). Thus we obtain some results related to theorems due to J. von Neumann, C. Davis, A. S. Lewis and T.-Y. Tam et al. Analogous problems are discussed for ψ\psi-uniform convex functions and cc-strongly convex functions. Finally, applications are given for matrix spaces endowed with the structure of Eaton triple.

Author information

Contact details are reproduced from the original publication and may be historical.

Marek Niezgoda

Dept. of Applied Mathematics and Computer Science, University of Life Sciences, 20-950 Lublin, Poland

marek.niezgoda@up.lublin.pl

Keywords

  • Convex function
  • eigenvalues
  • singular value
  • G-invariant function
  • G-majorization
  • Eaton triple
  • normal decomposition system
  • normal map
  • psi-uniformly convex function

Mathematics Subject Classification

15A30, 15A21; 26B25, 06F20

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Suggested citation

M. Niezgoda. “On Convexity and ψ-Uniform Convexity of G-Invariant Functions on an Eaton Triple.” Journal of Convex Analysis 26 (2019), No. 3, 1001–1019. https://doi.org/10.68381/jca26054

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.