A general framework for extending means to higher orders

Jimmie Lawson¹ ; Yongdo Lim²

¹ Department of Mathematics Louisiana State University Baton Rouge, LA 70803, U.S.A.
² Department of Mathematics Kyungpook National University Taegu 702-701, Korea

Colloquium Mathematicum, Tome 113 (2008) no. 2, pp. 191-221

Résumé

Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It has been an open problem to define and prove the existence of these higher order means in a general setting. In this paper we lay the foundations for such a theory by showing how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. We consider uniqueness properties and preservation properties of these extensions, properties which provide validation to our approach. As our targeted application, we consider the positive operators on a Hilbert space under the Thompson metric and apply our methods to derive higher order extensions of a variety of standard operator means such as the geometric mean, the Gauss mean, and the logarithmic mean. That the operator logarithmic mean admits extensions of all higher orders provides a positive solution to a problem of Petz and Temesi [SIAM J. Matrix Anal. Appl. 27 (2005)].

Détail

DOI : 10.4064/cm113-2-3

Keywords: although there extensive literature various means positive operators their applications these means typically readily extend means three operators has problem define prove existence these higher order means general setting paper lay foundations theory showing higher order means inductively defined established general metric spaces particular convex metric spaces consider uniqueness properties preservation properties these extensions properties which provide validation approach targeted application consider positive operators hilbert space under thompson metric apply methods derive higher order extensions variety standard operator means geometric mean gauss mean logarithmic mean operator logarithmic mean admits extensions higher orders provides positive solution problem petz temesi siam matrix anal appl