Related papers: Operator means, barycenters, and fixed point equat…
In this paper we present some characterizations for quasi-arithmetic operator means (among them the arithmetic and harmonic means) on the positive definite cone of the full algebra of Hilbert space operators, and also for the Kubo-Ando…
In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a…
An axiomatic theory of operator connections and operator means was investigated by Kubo and Ando in 1980. A connection is a binary operation for positive operators satisfying the monotonicity, the transformer inequality and the…
In this paper, we establish a bijection between Kubo-Ando operator means defined on the cones $\mathscr{P}_{n}(\mathbb{H})$, $\mathscr{P}_{2n}(\mathbb{C})$, and $\mathscr{P}_{4n}(\mathbb{R})$. This correspondence is induced by the canonical…
In this article we consider means of positive operators on a Hilbert space. We extend the theory of matrix power means to arbitrary operator means in the sense of Kubo-Ando. The basis of the extension is relying on ideas coming from…
The main goal of this paper is to discuss the recent advancements of operator means for accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean,…
In this paper we consider power means of positive Hilbert space operators both in the conventional and in the Kubo-Ando senses. We describe the corresponding isomorphisms (bijective transformations respecting those means as binary…
We consider in this paper two different types of the weighted geometric means of positive definite operators. We show the component-wise bijection of these geometric means and give a geometric property of the spectral geometric mean as a…
By means of a fixed point method we discuss the deformation of operator means and multivariate means of positive definite matrices/operators. It is shown that the deformation of an operator mean becomes again an operator mean. The means…
It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence…
The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of…
Let $\mathbb{P}$ be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion of operator means of probability measures on $\mathbb{P}$,…
A connection in Kubo-Ando sense is a binary operation for positive operators on a Hilbert space satisfying the monotonicity, the transformer inequality and the continuity from above. A mean is a connection $\sigma$ such that $A \sigma A =A$…
An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality and the fixed-point property. It is well known that there are one-to-one…
We introduce the notions of in-betweenness and monotonicity with respect to a metric, for operator means. These notions can be seen as generalising their natural counterpart for scalar means, and as a relaxation of the notion of geodesity.…
Means are used in several applications from electronic engeneering to information theory, however there is no general theorem on how to extend a given M(x, y) mean function to multiple variable forms. In this article we would like to…
We give an upper bound for the weighted geometric mean using the weighted arithmetic mean and the weighted harmonic mean. We also give a lower bound for the weighted geometric mean. These inequalities are proven for two invertible positive…
If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one…
We provide a complete characterization of a subclass of weakly associative means of positive operators in the class of symmetric Kubo-Ando means. This class, which includes the geometric mean, was first introduced and studied in L.…
We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo--Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the…