geometric interpolation in p schatten class the aim of this work is to apply the complex interpolation method to norms of n tuples of operators in the p schatten classes of a hilbert space h the norms considered define finsler metrics in a certain manifold of positive operators and can be regarded as weighted p norms the weight being a positive invertible operator as a by product we obtain clarkson s type inequalities

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

topological aspects of chow quotients this paper studies the canonical chow quotient of a smooth projective variety by a reductive algebraic group the main purpose is to introduce the perturbation translation specialization relation that gives a computable characterization of the chow cycles of the chow quotient also we provide in the languages that are familiar to topologists and differential geometers many topological interpretations of chow quotient that have the advantage to be more intuitive and geometric more precisely over the field of complex numbers these interpretations are symplectically the moduli spaces of stable orbits with prescribed momentum charges and topologically the moduli space of stable action manifolds

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

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 resolvent average on symmetric cones of jb algebras abstract in this paper we introduce the resolvent averages on symmetric cones of jb algebra and derive important properties such as self duality monotonicity and subhomogeneity this leads to the conclusion that each resolvent average contracts the thompson metric

on the nonlinear matrix equation xp a mt x b m abstract the nonlinear matrix equation x p a m t x b m where p 1 is a positive integer m is an n n nonsingular matrix a is a positive semidefinite matrix and b is a positive definite matrix is considered we denote by c d the geometric mean of positive definite matrices c and d based on the properties of the thompson metric we prove that this nonlinear matrix equation always has a unique positive definite solution and that the fixed point iteration method can be efficiently employed to compute it in addition estimates of the positive definite solution and perturbation analysis are investigated numerical experiments are given to confirm the theoretical analysis

in betweenness a geometrical monotonicity property for operator means abstract 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 we exhibit two classes of non trivial means that are monotonic with respect to the euclidean metric we also show that all kubo ando means are monotonic with respect to the trace metric which is the natural metric for the geometric mean

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

a robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

geometry of midpoint sets for thompson s metric abstract it is well known that the convex cone p m of m m positive definite matrices is a cartan hadamard riemannian manifold with respect to the riemannian trace metric where the geometric mean of two positive definite matrices coincides with the unique metric midpoint between them in this paper we consider the thompson metric on p m inherited from the spectral norm and study some geometric structures of the thompson midpoints we prove that there is a unique midpoint minimal geodesic between a and b if and only if the spectrum of a 1 b is contained in a a 1 for some a 0 and the set of thompson midpoints between a and b is compact and is convex in both riemannian and euclidean sense it is further shown that the set of all weighted midpoints between a and b is compact and convex in riemannian sense

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

carlson s iterative mean algorithm of positive definite matrices abstract in this paper we propose an iterative mean algorithm involving arithmetic and geometric means of n positive definite matrices which generalizes the 3 dimensional algorithm of positive reals discovered by carlson 1970 10 we show that the iterative mean algorithm is convergent and the common limit satisfies multidimensional versions of all properties permutation symmetry concavity monotonicity homogeneity congruence invariancy duality mean inequalities that one would expect for the carlson mean of positive reals convergence and perturbation analysis with numerical experiments are presented in terms of the thompson metric and the spectral norm

norm inequalities in operator ideals abstract in this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm among the well known inequalities which can be proved with this technique are the lowner heinz inequality inequalities relating various operator means and the corach porta recht inequality we prove two general inequalities and from them we derive several inequalities by specialization many of them new we also show how some inequalities known to be valid for matrices or bounded operators can be extended with this technique to normed ideals in c algebras in particular to the noncommutative l p spaces of a semi finite von neumann algebra

arithmetic geometric mean and related inequalities for operators abstract in recent years certain arithmetic geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on we first point out that they are direct consequences of integral expressions of relevant operators furthermore we obtain related new inequalities theorems 4 5 and 6 based on our current approach

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

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

an arithmetic geometric harmonic mean inequality involving hadamard products abstract given matrices of the same size a a ij and b b ij we define their hadamard product to be a b a ij b ij we show that if x i 0 and q p 0 then the n n matrices x i x j x i x j x i 1 x j 1 x i x j and x i p x j p x i q x j q are positive definite and we relate these facts to some matrix valued arithmetic geometric harmonic mean inequalities some of which involve hadamard products and others unitarily invariant norms it is known that if a is positive semidefinite then max a b b 1 max a ij where denotes the spectral norm we show that the converse of this statement is false and give a useful partial converse

operator power means due to lawson lim palfia for 1 t 2 abstract for 1 t 1 lim palfia defined a new family of operator power means of positive definite matrices and subsequently by lawson lim their notion and most of their results extend to the setting of positive invertible operators on a hilbert space each of these means except t 0 arises as a unique positive invertible solution of a non linear operator equation and satisfies all desirable properties of power arithmetic means of positive real numbers the purpose of this paper is to extend the range in which operator power means due to lawson lim palfia are defined we investigate some properties of operator power means for t 2 2 1 1

on exact solutions of the schafer wayne short pulse equation wki eigenvalue problem we consider a new equation recently found by sch afer and wayne hereafter named schshort pulse equation swspe describing the propagation of an ultrashort pulse in nonlinear media using some vanishing boundary conditions we construct and discuss the n soliton solutions to the previous equation by means of the wadati konno ichikawa wki method which is arguably more direct than the map through the sine gordon equation investigated much earlier by sakovich and sakovich 2005 j phys soc japan 74 239 2006 j phys a math gen 39 l361 we particularly focus our attention on the two soliton solution as an application as a result the collision process for two soliton solutions with similar amplitudes exhibits very different behaviour from the case when the amplitudes are dissimilar

symmetric γ submanifolds of positive definite matrices and the sylvester equation xm nx abstract in this paper we consider the special sylvester equation xm nx 0 for fixed n n matrices m and n where a positive definite solution x is sought we show that the solution sets varying over m n provide a new family of geodesic submanifolds in the symmetric riemannian manifold p n of positive definite matrices which is stable under congruence transformations it consists of geodesically complete convex cones of p n invariant under cartan symmetries it is further shown that the solution set is stable under the iterative means obtained by the weighted arithmetic harmonic and geometric means

spaces of nonpositive curvature arising from a finite algebra fil conde cristian marcelo universidad nacional de general sarmiento instituto de ciencias argentina consejo nacional de investigaciones cientificas y tecnicas oficina de coordinacion administrativa saavedra 15 instituto argentino de matematica alberto calderon argentina

complex analytic quotients of algebraic g varieties it is shown that any compact semistable quotient of a normal variety by a complex reductive lie group g in the sense of heinzner and snow is a good quotient this reduces the investigation and classification of such complex analytic quotients to the corresponding questions in the algebraic category as a consequence of our main result we show that every compact space in nemirovski s class fancyscript q g has a realisation as a good quotient and that every complete algebraic variety in fancyscript q g is unirational with finitely generated cox ring and at worst rational singularities in particular every compact space in class fancyscript q t where t is an algebraic torus is a toric variety

the short pulse equation and associated constraints the short pulse equation spe is considered as an initial boundary value problem it is found that the solutions of the spe must satisfy an integral relation otherwise the temporal derivative exhibits discontinuities this integral relation is not necessary for a solution to exist an infinite number of such constraints can be dynamically generated by the evolution equation

strong monotonicity for various means abstract point wise monotonicity in parameters for various one parameter families of scalar means such as power difference means binomial means and stolarsky means is well known but norm comparison for corresponding operator means requires monotonicity in the sense of positive definiteness among other things we obtain monotonicity in the sense of infinite divisibility which is much stronger than that in the sense of positive definiteness these strong monotonicity results are proved based on explicit computations for measures in relevant levy khintchine or actually kolmogorov formulas

procrustes problems in riemannian manifolds of positive definite matrices abstract we consider the manifold of positive definite matrices endowed with the fisher riemannian metric and some other distances commonly used in information theory we show that for each of them the best approximant to a from the unitary orbit of another matrix b commutes with a

stopping criteria for the ando li mathias and bini meini poloni geometric means abstract we provide an upper bound for the number of iterations necessary to achieve a desired level of accuracy for the ando li mathias linear algebra appl 385 2004 305 334 and bini meini poloni math comput 79 2010 437 452 symmetrization procedures for computing the geometric mean of n positive definite matrices where accuracy is measured by the spectral norm and the thompson metric on the convex cone of positive definite matrices it is shown that the upper bound for the number of iterations depends only on the diameter of the set of n matrices and the desired convergence tolerance a striking result is that the upper bound decreases as n increases on any bounded region of positive definite matrices

structure preserving matrix means on the marcus minc stochastic matrices abstract in this paper we show that the set ps m of all m m positive definite stochastic matrices with diagonal entries bounded above by 1 m 1 is stable under the weighted geometric mean operation it is further shown that ps m is stable for some well known multivariable matrix means the least squares means for the riemannian trace metric and the kullback leibler divergence on the convex cone of m m positive definite matrices and alm ando li mathias and bmp bini meini poloni geometric means

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

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

an approach of majorization in spaces with a curved geometry abstract the hardy littlewood polya majorization theorem is extended to the framework of some spaces with a curved geometry such as the global npc spaces and the wasserstein spaces we also discuss the connection between our concept of majorization and the subject of schur convexity several applications are included

factorizations and geometric means of positive definite matrices abstract in this paper we provide a new class of metric geometric means of positive definite matrices varying over hermitian unitary matrices we show that each hermitian unitary matrix induces a factorization of the cone p m of m m positive definite hermitian matrices into geodesically convex subsets and a hadamard metric structure on p m an explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two variable mean is different to the standard geometric mean some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied

nonlinear equations based on jointly homogeneous mappings abstract in this paper we consider a nonlinear equation based on two variable monotone and jointly homogeneous functions on an open convex normal cone x g h a x k b x and show that it has a unique positive solution and furthermore the solution map is again monotone and jointly homogeneous we apply our results to nonlinear matrix equations involving the geometric mean of positive definite matrices which include some known nonlinear matrix equations

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

γ commuting triples of positive definite matrices and midpoint equations on unitary matrices triples of positive definite matrices that can be transformed to a commuting triple by a congruence transformation are called commuting in this paper we give various descriptions and characterizations of commuting triples and particularly focus on those involving the geometric mean and midpoint equations on a generating set of the special unitary group

group actions on s6 and complex structures on ℙ3 it is proved that if s 6 possesses an integrable complex structure then there exists a 1 dimensional family of pairwise different exotic complex structures on p3 c this follows immediately from the main result of the paper s 6 is not the underlying differentiable manifold of an almost homoge neous complex manifold x via elementary lie theoretic techniques this is reduced to ruling out the possibility of a c action on a certain non normal surface ex a contradiction is reached by analyzing combinatorial aspects of the non normal locus n of e and its preimage n in the normalization e

comparison of various means for operators for hilbert space operatorsh k xwithh k 0 the norm inequality h1 2xk1 2 12 hx xk is known where is an arbitrary unitarily invariant norm a refinement of this arithmetic geometric mean inequality is studied similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean

minimal curves in u n and gl n with respect to the spectral and the trace norms abstract consider the lie group of n n complex unitary matrices u n endowed with the bi invariant finsler metric given by the spectral norm x u u x x for any x tangent to a unitary operator u given two points in u n in general there exist infinitely many curves of minimal length in this paper we provide a complete description of such curves and as a consequence we give an equivalent condition for uniqueness similar studies are done for the grassmann manifolds on the other hand consider the cone of n n positive invertible matrices g l n endowed with the bi invariant finsler metric given by the trace norm x 1 a a 1 2 x a 1 2 1 for any x tangent to a g l n in this context also exist infinitely many curves of minimal length in this paper we provide a complete description of such curves proving first a characterization of the minimal curves joining two hermitian matrices x y h n the last description is also used to construct minimal paths in the group of unitary matrices u n endowed with the bi invariant finsler metric x 1 u u x 1 x 1 for any x tangent to u u n we also study the set of intermediate points in all the previous contexts

solving the nonlinear matrix equation x q i 1mmixδimi via a contraction principle abstract in this paper we consider the nonlinear matrix equation x q i 1 m m i x i m i where q is positive resp semidefinite definite and m i s are arbitrary resp nonsingular matrices we prove that if max i 1 i m 1 then the equation has a unique positive definite solution which is realized as the unique fixed point of a strict contraction with the lipschitz constant less than or equal to furthermore we show that the solution map varying over the determining coefficient matrices is continuous

convex geometric means abstract a class of multivariable weighted geometric means of positive definite matrices admitting jensen type inequalities for geodesically convex functions is considered it is shown that there are infinitely many such geometric means including the weighted inductive bini meini poloni and karcher means and each of these means provides a geometric mean majorization on the space of positive definite matrices some connections between our geometric mean majorizations and classical results of the standard majorization of real numbers are discussed in particular we establish the hardy littlewood polya majorization theorem and also rado s theorem and schur s convexity theorem for the weighted karcher mean

the contraction rate in thompson s part metric of order preserving flows on a cone application to generalized riccati equations abstract we give a formula for the lipschitz constant in thompson s part metric of any order preserving flow on the interior of a possibly infinite dimensional closed convex pointed cone this shows that in the special case of order preserving flows a general characterization of the contraction rate in thompson s part metric given by nussbaum leads to an explicit formula as an application we show that the flow of the generalized riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its lipschitz constant by a matrix inequality we also show that the same flow is no longer a contraction in other invariant finsler metrics on this cone including the standard invariant riemannian metric this is motivated by a series of contraction properties concerning the standard riccati equation established by bougerol liverani wojtkowski lawson lee and lim we show that some of these properties do and that some other do not carry over to the generalized riccati equation

cauchy schwarz norm inequalities for weak integrals of operator valued functions for a finite measures on and weakly measurable families at t and bt t of hilbert space operators we have the non commutative cauchy schwarz inequalities in schatten p ideals axbd p a aa d q 1ad 2qx b b bd r 1b d 2rp for all x p h and for all p q r 1 such that 1q 1r 2p if both at t and bt t consists of commuting normal operators then axbd a ad x b bd for all unitarily invariant norms and all x h if additionally a ad i and b bd i then i a ad xi b bd h and i a ad xi b bd x axbd applications include young s and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions

solving symmetric matrix word equations via symmetric space machinery abstract it has recently been proved by hillar and johnson c j hillar c r johnson symmetric word equations in two positive definite letters proc amer math soc 132 2004 945 953 that every symmetric word equation in positive definite matrices has a positive definite solution existence theorem in this paper we partially solve the uniqueness conjecture via the symmetric space and non positive curvature machinery existing in the open convex cone of positive definite matrices unique positive definite solutions are obtained in terms of geometric and weighted means and as fixed points of explicit strict contractions on the cone of positive definite matrices

solitary wave solutions of the short pulse equation an exact nonsingular solitary wave solution of the schafer wayne short pulse equation is derived from the breather solution of the sine gordon equation by means of a transformation between these two integrable equations

geometric interpolation in symmetrically normed ideals abstract the aim of this work is to apply the complex interpolation method to norms of n tuples of operators in a symmetrically normed ideal j b h defined by a symmetric norming function s n f the norms considered define finsler metrics in a certain manifold of positive operators and can be regarded as weighted norms the weight being a positive invertible operator