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Register a new userMatrix limit theorems of Kato type related to positive linear maps and operator means
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Authors
Fumio Hiai
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We obtain limit theorems for $Phi(A^p)^{1/p}$ and $(A^psigma B)^{1/p}$ as $ptoinfty$ for positive matrices $A,B$, where $Phi$ is a positive linear map between matrix algebras (in particular, $Phi(A)=KAK^*$) and $sigma$ is an operator mean (in particular, the weighted geometric mean), which are considered as certain reciprocal Lie-Trotter formulas and also a generalization of Katos limit to the supremum $Avee B$ with respect to the spectral order.
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We improve the existing Ando-Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie-Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.
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 $mathbb{P}$, in parallel with Kubo and Andos definition of two-variable operator means, and show that every operator mean is contractive for the $infty$-Wasserstein distance. By means of a fixed point method we consider deformation of such operator means, and show that the deformation of any operator mean becomes again an operator mean in our sense. Based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures.
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $phi$ in the form $K_D^phi(H,K)=sum_{i,j}phi(lambda_i,lambda_j)^{-1} Tr P_iHP_jK$ when $sum_ilambda_iP_i$ is the spectral decomposition of the foot point $D$ and the Hermitian matrices $H,K$ are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping $Dmapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distributions and the quantum Fisher information. In the paper the case $phi(x,y)=M(x,y)^theta$ is mostly studied when $M(x,y)$ is a mean of the positive numbers $x$ and $y$. There are results about the geodesic curves and geodesic distances. The geometric mean, the logarithmic mean and the root mean are important cases.
In this paper, the notion of operator means in the setting of JB-algebras is introduced and their properties are studied. Many identities and inequalities are established, most of them have origins from operators on Hilbert space but they have different forms and connotations, and their proofs require techniques in JB-algebras.
This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support? We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by Dykema, Figiel, Weiss, and Wodzicki, and the analogous am operations at infinity that we develop in this article, the conjecture is proven for all ideals not contained in the largest am-infinity stable ideal and not containing the smallest am-stable ideal. It is also proven for all soft-edged ideals (i.e., I= IK(H)) and all soft-complemented ideals (i.e., I= I/K(H)), which include many classical operator ideals. In the process, we prove that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-infinity stable and that, for a principal ideal, am-infinity stability is equivalent to regularity at infinity of the sequence of s-numbers of the generator. Furthermore, we apply trace extension methods to two problems on elementary operators studied by V. Shulman and to Fuglede-Putnam type problems of the second author.
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