A general framework for extending means to higher orders

Jimmie Lawson, and Yongdo Lim. "A general framework for extending means to higher orders." Colloquium Mathematicae 113.2 (2008): 191-221. <http://eudml.org/doc/284197>.

@article{JimmieLawson2008,
abstract = {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)].},
author = {Jimmie Lawson, Yongdo Lim},
journal = {Colloquium Mathematicae},
keywords = {mean; geometric mean; iterated mean; convex mean},
language = {eng},
number = {2},
pages = {191-221},
title = {A general framework for extending means to higher orders},
url = {http://eudml.org/doc/284197},
volume = {113},
year = {2008},
}

TY - JOUR
AU - Jimmie Lawson
AU - Yongdo Lim
TI - A general framework for extending means to higher orders
JO - Colloquium Mathematicae
PY - 2008
VL - 113
IS - 2
SP - 191
EP - 221
AB - 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)].
LA - eng
KW - mean; geometric mean; iterated mean; convex mean
UR - http://eudml.org/doc/284197
ER -