In this paper, we establish some new characterizations of operator monotone functions using matrix mean inequalities.
In this paper, we establish some new characterizations of operator monotone functions using matrix mean inequalities.
Recently the Karcher mean has been extended to the case of probability measures of positive operators on infinite-dimensional Hilbert spaces as the unique solution of a nonlinear operator equation on the convex Banach-Finsler manifold of positive operators. Let ((Omega ,mu )) be a probability space, and let (tau :Omega rightarrow Omega ) be a totally ergodic map. The main result of this paper is a new ergodic theorem for functions ( Fin L^1(Omega ,mathbb {P})), where (mathbb {P}) is the open cone of the strictly positive operators acting on a (separable) Hilbert space. In our result, we use inductive means to average the elements of the orbit, and we prove that almost surely these averages converge to the Karcher mean of the push-forward measure (F_*(mu )). From our result, we recover the strong law of large numbers and the “no dice” results proved by the third and fourth authors in the article Strong law of large numbers for the (L^1)-Karcher mean, Journal of Func. Anal. 279 (2020). From our main result, we also deduce an ergodic theorem for Markov chains with state space included in (mathbb {P}).
Algorithms for the computation of the (weighted) geometric mean G of two positive definite matrices are described and discussed. For large and sparse matrices the problem of computing the product (y=Gb), and of solving the linear system (Gx=b), without forming G, is addressed. An analysis of the conditioning is provided. Substantial numerical experimentation is carried out to test and compare the performances of these algorithms in terms of CPU time, numerical stability, and number of iterative steps.
We study operator means more general than the Kubo-Ando means. They are given as minima of geodesically convex functions and allow uniquely defined extensions to any number of variables. Multivariate hyper-means is a class of means of this type bounded from below by the arithmetic mean. We extend the definition of a hyper-man in the bivariate case and discover bivariate means that are not restrictions of the multivariate means studied earlier. We introduce the notion of reduced relative quantum entropy and prove that it is convex. The result is used to give a simplified proof of a theorem of Lieb and Seiringer.
The seminal work of Kubo and Ando (Math Ann 246:205–224, 1979/80) provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavour. On the other hand, it is highly natural to take the geomeric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.
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