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Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras

Department of Mathematics, National University of Singapore, Singapore 119077
Department of Decision Sciences, National University of Singapore, Singapore 119077
matsundf{at}nus.edu.sg
bizsunj{at}nus.edu.sg

We study analyticity, differentiability, and semismoothness of Löwner's operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan algebra is shown to be strongly semismooth. The research also raises several open questions, whose answers would be of strong interest for optimization research.

Key Words: Euclidean Jordan algebras; Löwner's operator; spectral functions; semismoothness
History: Received: December 28, 2004; revision received: July 5, 2007;