on some matrix inequalities the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2sj ab sj a a b b j 1 2 for any matrices a b we first give new proofs of this inequality and its equivalent form then we use it to prove the following trace inequality let a0 be a positive definite matrix and a1 ak be positive semidefinite matrices then tr j 1k i 0jai 2aj tra0 1

contractive maps on operator ideals and norm inequalities ii abstract let i be a norm ideal of operators equipped with a unitarily invariant norm we exploit integral representations of certain functions to prove that certain ratios of linear operators acting on operators in i are contractive this leads to some new and old norm inequalities we also lift a variety of inequalities to the operator setting which were proved in the matrix setting earlier

on a question of bhatia and kittaneh abstract we settle in the affirmative a question of bhatia and kittaneh for p and q positive semidefinite n n matrices the inequality r pq 1 2 r p q holds for r 1 2 n

structure preserving parallel algorithms for solving the bethe salpeter eigenvalue problem abstract the bethe salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized bethe salpeter equation in the context of computing exciton energies and states a computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice we establish the equivalence between bethe salpeter eigenvalue problems and real hamiltonian eigenvalue problems based on theoretical analysis structure preserving algorithms for a class of bethe salpeter eigenvalue problems are proposed we also show that for this class of problems all eigenvalues obtained from the tamm dancoff approximation are overestimated in order to solve large scale problems of practical interest we discuss parallel implementations of our algorithms targeting distributed memory systems several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

mixed mean inequalities for several positive definite matrices on basis of the geometric mean proposed recently by t ando chi kwong li and roy mathias in this paper we present several kinds of mixed means for three or more positive definite matrices and prove some related mixed mean inequalities

norm bounds for hadamard products and an arithmetic geometric mean inequality for unitarily invariant norms abstract an arithmetic geometric mean inequality for unitarily invariant norms and matrices 2 a xb aa x xbb is an immediate consequence of a basic inequality for singular values of hadamard products

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

matrix young inequalities for the hilbert schmidt norm abstract let a b and x be n n complex matrices such that a and b are positive semidefinite if p q 1 with 1 p 1 q 1 it is shown that 1 p a p x 1 q xb q 2 2 1 r 2 a p x xb q 2 2 axb 2 2 where r max p q and 2 is the hilbert schmidt norm generalizations and applications of this inequality are also considered

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

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

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

a cauchy schwarz inequality for operators with applications abstract for any unitarily invariant norm on hilbert space operators it is shown that for all operators a b x and positive real numbers r we have a xb r 2 aa x r xbb r some consequences are then discussed a simple proof is given for the fact that for positive operators a b the function spr a t b t 1 t is monotone in t on the positive half line

on drury s solution of bhatia kittaneh s question abstract let a b be n n positive semidefinite matrices bhatia and kittaneh asked whether it is true j a b 1 2 j a b j 1 n where j j is the j th largest singular value eigenvalue respectively the question was recently solved by drury in the affirmative this article revisits drury s solution in particular we simplify the proof for a key auxiliary result in his solution

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

the spectrum of the billiard laplacian of a family of random billiards random billiards are billiard dynamical systems for which the reflection law giving the post collision direction of a billiard particle as a function of the pre collision direction is specified by a markov scattering operator p billiards with microstructure are random billiards whose markov operator is derived from a microscopic surface structure on the boundary of the billiard table the microstructure in turn is defined in terms of what we call a billiard cellq the shape of which completely determines the operator p this operator defined on an appropriate hilbert space is bounded self adjoint and for the examples considered here a hilbert schmidt operator a central problem in the statistical theory of such random billiards is to relate the geometric characteristics of q and the spectrum of p we show for a particular family of billiard cell shapes parametrized by a scale invariant curvature k fig 2 that the billiard laplacian p i is closely related to the ordinary spherical laplacian and indicate by partly analytical and partly numerical means how this provides asymptotic information about the spectrum of p for small values of k it is shown in particular that the second moment of scattering about the incidence angle closely approximates the spectral gap of p

eigenvalue inequalities for convex and log convex functions abstract we give a matrix version of the scalar inequality f a b f a f b for positive concave functions f on 0 we show that choi s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences some inequalities for log convex functions are presented and a new arithmetic geometric mean inequality for positive matrices is given we also point out a simple proof of the bhatia kittaneh arithmetic geometric mean inequality

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

young s inequality in trace class operators if a and b are trace class operators and if u is a partial isometry then where 1 denotes the norm in the trace class the present paper characterises the cases of equality in this young inequality and the characterisation is examined in the context of both the operator and the hilbert schmidt forms of young s inequality

on an operator kantorovich inequality for positive linear maps abstract we improve the operator kantorovich inequality as follows let a be a positive operator on a hilbert space with 0 m a m then for every unital positive linear map a 1 2 m m 2 4 m m 2 a 2 as a consequence a 1 a a a 1 m m 2 2 m m

inequalities for commutators of positive operators abstract it is shown that if a b and x are operators on a complex separable hilbert space such that a and b are compact and positive then the singular values of the generalized commutator a x x b are dominated by those of x a b where is the usual operator norm consequently for every unitarily invariant norm we have a x x b x a b it is also shown that if a and b are positive and x is compact then a x x b max a b x for every unitarily invariant norm moreover if x is positive then the singular values of the commutator a x x a are dominated by those of 1 2 a x x consequently a x x a 1 2 a x x for every unitarily invariant norm for the usual operator norm these norm inequalities hold without the compactness conditions and in this case the first two norm inequalities are the same our inequalities include and improve upon earlier inequalities proved in this context and they seem natural enough and applicable to be widely useful

purely imaginary eigenvalues of zakharov shabat systems zakharov shabat systems with single hump and real but not necessarily symmetric potentials are shown to have purely imaginary eigenvalues only coupled with examples of double hump potentials with nonimaginary eigenvalues this establishes that confinement of zakharov shabat eigenvalues to the imaginary axis is a characteristic of potentials whose energy is concentrated in a single region of the time axis

new inequalities for the generalized karcher mean abstract recently palfia introduced a generalized karcher mean as a solution of an operator equation in this article we present several relations for this new mean in particular we investigate the behavior of this generalized mean when filtered through operator monotone functions and positive linear maps revealing its information monotonicity

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

squaring operator polya szego and diaz metcalf type inequalities abstract we square operator polya szego and diaz metcalf type inequalities as follows if operator inequalities 0 m 1 2 a m 1 2 and 0 m 2 2 b m 2 2 hold for some positive real numbers m 1 m 1 and m 2 m 2 then for every unital positive linear map the following inequalities hold a b 2 m 1 m 2 m 1 m 2 2 m 1 m 2 m 1 m 2 4 a b 2 and m 2 m 2 m 1 m 1 a b 2 m 1 m 1 m 2 2 m 2 2 m 2 m 2 m 1 2 m 1 2 2 8 m 2 m 1 m 1 m 2 m 1 2 m 1 2 m 2 m 2 2 a b 2

inequalities for sums and direct sums of hilbert space operators abstract we prove several singular value inequalities and norm inequalities involving sums and direct sums of hilbert space operators it is shown among other inequalities that if x and y are compact operators then the singular values of x y 2 are dominated by those of x y applications of these inequalities are also given

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

the hua matrix and inequalities related to contractive matrices abstract we first deny a conjecture raised in xu et al 2011 14 and then we present some eigenvalue or singular value inequalities related to contractive matrices

matrix power means and the information monotonicity abstract lim and palfia established the notion of the matrix power means for k positive definite matrices k 3 and showed that the matrix power means have the information monotonicity for a unital positive linear mapping in this note by virtue of the generalized kantorovich constant we show counterparts to the information monotonicity of the matrix power means

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

krein signatures for the faddeev takhtajan eigenvalue problem one of the difficulties in analyzing eigenvalue problems that arise in connection with integrable systems is that they are frequently non self adjoint making it difficult to determine where the spectrum lies in this paper we consider the problem of locating and counting the discrete eigenvalues associated with the faddeev takhtajan eigenvalue problem for which the sine gordon equation is the isospectral flow in particular we show that for potentials having either zero topological charge or topological charge 1 and satisfying certain monotonicity conditions the point spectrum lies on the unit circle and is simple furthermore we give an exact count of the number of eigenvalues this result is an analog of that of klaus and shaw for the zakharov shabat eigenvalue problem we also relate our results as well as those of klaus and shaw to the krein stability theory for j unitary matrices in particular we show that the eigenvalue problem associated to the sine gordon equation has a j unitary structure and under the above conditions the point eigenvalues have a definite krein signature and are thus simple and lie on the unit circle

norm inequalities for commutators of positive operators and applications let x y and z be operators on a hilbert space such that x and z are positive it is shown that left xy yz right leqslant rm max left left x right left z right right left y right applications of this commutator inequality are given

spectral gap for a class of random billiards a random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post collision distribution of velocities for any given pre collision velocity we consider such collision operators for certain random billiards that we call billiards with microstructure collisions modeled by these operators can still be thought of as elastic and time reversible the operators are canonically determined by a second deterministic billiard system that models microscopic roughness on the billiard table boundary our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards among the main results we give geometric conditions for these operators to be hilbert schmidt and relate their spectrum and speed of convergence to stationary markov chains with geometric features of the microscopic billiard structure the relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples

an extension of a matrix inequality of thompson abstract a matrix inequality of thompson is extended as an application a partial affirmative answer to a recent question of lin is provided

coupling constant behavior of eigenvalues of zakharov shabat systems we consider the eigenvalues of the non self adjoint zakharov shabat systems as the coupling constant of the potential is varied in particular we are interested in eigenvalue collisions and eigenvalue trajectories in the complex plane we identify shape features in the potential that are responsible for the occurrence of collisions and we prove asymptotic formulas for large coupling constants that tell us where eigenvalues collide or where they emerge from the continuous spectrum some examples are provided which show that the asymptotic methods yield results that compare well with exact numerical computations

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

a note on the arithmetic geometric mean inequality for matrices abstract we give a simple proof of the inequality aa x xbb 2 a ab where a b and x are arbitrary n n matrices and is any unitarily invariant norm for the schatten p norms p with 1 p aa x xbb p 2 a xb p if and only if aa x xbb

more inequalities for positive semidefinite 2 2 block matrices and their blocks abstract it is obtained that for positive semidefinite block matrix h m k k n if a b m n c such that m a x a 2 b 2 2 and f is a nonnegative increasing convex function on 0 satisfying f 0 0 then 2 s j f a k b m a x a 2 b 2 s j f h for j 1 2 n also it is shown that if h m k k n is a positive semidefinite 2 2 block matrix with square matrices m and n of the same size then h r 2 a b r 1 m n for r 1 where a and b are the largest eigenvalues of the matrices m and n respectively in addition if r 3 2 then h r 2 4 m 2 n 2 m 0 n 0 2 r 2 for every unitarily invariant norm

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

on some operator inequalities abstract for hilbert space operators s t with s invertible and self adjoint corach porta and recht recently proved that sts 1 s 1 ts 2 t a generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic geometric mean inequality some related inequalities are also discussed

an arithmetic geometric mean inequality for singular values and its applications abstract in this short note we give a new equivalent form of the arithmetic geometric mean inequality for singular values as applications of our result we give a new proof of an inequality due to bhatia and davis 1993 4 and we obtain a singular value inequality for matrix means which is similar to one proved by drury 2012 9 finally we present a log majorization inequality for singular values

multiple scattering in random mechanical systems and diffusion approximation this paper is concerned with stochastic processes that model multiple or iterated scattering in classical mechanical systems of billiard type defined below from a given deterministic system of billiard type a random process with transition probabilities operator p is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions of particular interest are systems with weak scattering which are associated to parametric families of operators ph depending on a geometric or mechanical parameter h that approaches the identity as h goes to 0 it is shown that ph i h converges for small h to a second order elliptic differential operator mathcal l on compactly supported functions and that the markov chain process associated to ph converges to a diffusion with infinitesimal generator mathcal l both ph and mathcal l are self adjoint densely defined on the space l 2 mathbb h eta of square integrable functions over the lower half space mathbb h in mathbb r m where is a stationary measure this measure s density is either post collision maxwell boltzmann distribution or knudsen cosine law and the random processes with infinitesimal generator mathcal l respectively correspond to what we call mb diffusion and generalized legendre diffusion concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes

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

on singular value inequalities for matrix means abstract for positive semidefinite n n matrices a and b the singular value inequality 2 t s j a r b 2 r a 2 r b r 2 s j a 2 t a b b 2 is shown to hold for r 1 2 1 3 2 and all 2 t 2

geometric operator inequalities abstract the geometrical meaning of several well known inequalities is discussed they include the so called loewner heinz mcintosh and segal inequalities it is shown that some of them can be deduced from the others even for unitarily invariant forms some spectral properties of the elementary operators associated to the inequalities are studied

new properties for certain positive semidefinite matrices abstract we bring in some new notions associated with 2 2 block positive semidefinite matrices these notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or geometric mean of the diagonal blocks we investigate some relations between them many examples are included to illustrate these relations

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed