The metric increasing property of the exponential map is known to be equivalent to the fact that the set of positive definite matrices is a Riemannian manifold of nonpositive curvature. We show that this property is an easy consequence of the logarithmic-geometric mean inequality for positive numbers. Operator versions of this inequality lead to a generalisation of the exponential metric increa...

We show that the quadratic matrix equation VW + η(W )W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iteration underlying our proofs. This work bears on operator-valued free probability theory, in particular on the ...

It is known that the norm of the solution to a weighted linear least-squares problem is uniformly bounded for the set of diagonally dominant symmetric positive definite weight matrices. This result is extended to weight matrices that are nonnegative linear combinations of symmetric positive semidefinite matrices. Further, results are given concerning the strong connection between the boundednes...

This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance o...

In this paper, by using Mond-Pečarić method we provide some inequalities for connections of positive definite matrices. Next, we discuss specifications of the obtained results for some special cases. In doing so, we use α-arithmetic, α-geometric and α-harmonic operator means.

We prove that for any one-dimensional Schrr odinger operator with potential V (x) satisfying decay condition jV (x)j C x ?3=4? ; the absolutely continuous spectrum lls the whole positive semi-axis. The description of the set in R + on which the singular part of the spectral measure might be supported is also given. Analogous results hold for Jacobi matrices.

In the present work we show that joint probability distribution of eigenvalues can be expressed in terms a differential operator acting on some other matrix quantities. Those quantities might diagonal or pseudo-diagonal entries as it is case for Hermitian matrices. These representations are called derivative principles. We them additive spaces Hermitian, antisymmetric, anti-self-dual, and compl...

The main goal of this article is to establish several new \({\mathbb {A}}\)-numerical radius equalities for \(n\times n\) circulant, skew imaginary tridiagonal, and anti-tridiagonal operator matrices, where {A}}\) the diagonal matrix whose entries are positive bounded A. Some special cases our results lead earlier works in literature, which shows that more general. Further, some pinching type i...