A general framework for extending means to higher orders

• Autorzy: Jimmie Lawson , Yongdo Lim
• Języki publikacji: EN

Abstrakty

EN

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)].

Kategorie tematyczne

• 47A64: Operator means, shorted operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2008
• Tom: 113
• Numer: 2
• Strony: 191-221

Daty

wydano

2008

Twórcy

autor

Jimmie Lawson

• Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

autor

Yongdo Lim

• Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea

Identyfikatory

DOI

10.4064/cm113-2-3