the renyi power means of matrices abstract in this article we introduce a new family of power means for m positive definite matrices called renyi power means these means are defined in terms of the matrix function q z a b b 1 2 z a z b 1 2 z z related to the renyi divergence we study several properties of these means and some related norm inequalities

an approximation theoretic approach to maximum entropy spectral analysis it is shown that the theory of best approximation in weighted mean square can be used to provide a mathematical foundation for the analysis and extension of the maximum entropy method of spectral analysis

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

the ando hiai inequalities for the solution of the generalized karcher equation and related results abstract in this paper we shall give a concrete relation between the generalized karcher equation and operator means as its solution next we shall show two types of the ando hiai inequalities for the solution of the generalized karcher equation in this discussion we also give a property of relative operator entropy

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

on some inequalities for the matrix power and karcher means abstract we establish a useful inequality for matrix power means of positive definite matrices it eventually provides not only a new characterization of the karcher mean cartan centroid riemannian geometric mean but also a simple proof of the convergence of power means to the karcher mean some new matrix mean inequalities on the karcher and power means are presented

on an eigenvalue inequality involving the hadamard product abstract let a b be n n positive definite matrices then for 0 t 1 i k n i a b i k n i a t b a 1 t b i k n i a b k 1 n this gives a weighted extension of a result of ando 1 the case where the eigenvalues i are replaced with the singular values i is also considered

levinson s algorithm wold s decomposition and spectral estimation levinson s algorithm is developed in the context of mean square estimation and is applied to a variety of topics related to wiener filtering and spectral estimation the study includes the innovations approach to prediction theory wold s decomposition lattice filters autoregressive processes the method of maximum entropy and the general class of extrapolating spectra

riemannian metrics on positive definite matrices related to means abstract the riemannian metric on the manifold of positive definite matrices is defined by a kernel function in the form k d h k i j i j 1 tr p i hp j k when i i p 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 d g d is a kernel metric as well several riemannian geometries of the literature are particular cases for example the statistical metric for multivariate gaussian distributions and the quantum fisher information in the paper the case x y m x y 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

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

on a conjecture of bhatia lim and yamazaki abstract in this short note we prove a conjecture due to bhatia lim and yamazaki on the matrix power means as a consequence we obtain a related inequality for the matrix heron mean and its non kubo ando extension conjectured by bhatia et al

the additive causal part of burg s maximum entropy estimator and data consistency the additive causal part of burg s maximum entropy me estimator spectrum is calculated in closed form an immediate corollary for data consistency is shown

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

spectral inequalities for kubo ando operator means abstract eigenvalue and singular value inequalities involving kubo ando operator connections or means are established previous results from lemos and soares in 2018 are generalized or complemented some log majorizations are included as a consequence a refinement of an independent result by ando and by visick in 1995 on the eigenvalues of the hadamard product is derived some singular value inequalities by zou in 2017 are further extended

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

some log majorizations and an extension of a determinantal inequality abstract an eigenvalue inequality involving a matrix connection and its dual is established and some log majorization type results are obtained in particular some eigenvalues inequalities considered by f hiai and m lin 9 an associated conjecture and a singular values inequality by l zou 20 are revisited a reformulation of the inequality det a u b det a b for positive semidefinite matrices a b with u a unitary matrix that appears in the polar decomposition of b a is also extended using some known norm inequalities associated to furuta inequality and araki cordes inequality

multi variable weighted geometric means of positive definite matrices abstract we define a family of weighted geometric means g t a t 0 1 n where and a vary over all positive probability vectors in r n and n tuples of positive definite matrices resp each of these weighted geometric means interpolates between the weighted alm t 0 n and bmp t 1 n geometric means alm and bmp geometric means have been defined by ando li mathias and bini meini poloni respectively we show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two variable weighted geometric mean

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

on ando li mathias geometric mean equations abstract in this paper we consider a family of nonlinear matrix equations based on the higher order geometric means of positive definite matrices that proposed by ando li mathias we prove that the geometric mean equation x b g a 1 a 2 a m x x x n has a unique positive definite solution depending continuously on the parameters of positive definite ai and positive semidefinite b it is shown that the unique positive definite solutions gn a1 a2 am for b 0 satisfy the minimum properties of geometric means yielding a sequence of higher order geometric means of positive definite matrices

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

inequalities for the wasserstein mean of positive definite matrices abstract we prove majorization inequalities for different means of positive definite matrices these include the cartan mean the karcher mean the log euclidean mean the wasserstein mean and the power mean

the a property of elliptic measures for operators with coefficients supported in whitney type cubes abstract it is proved the preservation of the a property with respect to surface measure of harmonic measure associated to second order divergence form operators with bounded measurable coefficients when the discrepancy between the main coefficients of the operators is supported on whitney type rectangles of the unit ball of r n in particular neither the discrepancy between the coefficients satisfies a carleson measure condition nor the coefficients of the operators involved are radially independent as in previous well known works as an application of the perturbation result we provide a class of operators with coefficients supported on whitney type rectangles for which the a property holds

some inequalities for the matrix heron mean abstract in this paper we prove some norm inequalities for the matrix heron mean of two positive definite matrices we also prove a determinant inequality for the t power mean of two positive definite matrices as a consequence we obtain a determinant inequality for the matrix heron mean

some inequalities for unitarily invariant norm abstract we shall prove the inequalities a b a b aa bb 2 ab a b a b 4 ab for all n n complex matrices a b and all unitarily invariant norms if further a b are positive definite it is proved that j 1 k j a b j 1 k j a 1 b 1 k n 0 1 where denotes the operator means considered by kubo and ando and j x 1 j n denote the eigenvalues of x arranged in the decreasing order whenever these all are real a number of inequalities are obtained as applications

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations