the center of the generic division ring and twisted multiplicative group actions abstract let f be a field and let p be a prime the problem we study is whether the center cp of the division ring of p p generic matrices is stably rational over f given a finite group g and a z g lattice we let f m be the quotient field of the group algebra of the abelian group m procesi and formanek linear multilinear algebra 7 1979 203 212 have shown that for all n there is a z s n lattice gn such that cn is stably isomorphic to the fixed field under the action of sn of f gn let h be a p sylow subgroup of sp let a be the root lattice and let l f z s p h we show that there exists an action of sp on l z s p z h a twisted by an element ext 1 s p z s p z h a l such that l z s p z h a s p is stably isomorphic to cp the extension corresponds to an element of the relative brauer group of l over lh since z s p z h a and z s p h are quasi permutations l z s p z h a s p is stably rational over f however it is not known whether l z s p z h a s p is stably rational over f thus the result represents a reduction on the problem since z s p z h a is quasi permutation however the twist introduces a new level of complexity

reduced commutativity of moduli of operators abstract in this paper we investigate the question of when the equations a s a s a a s s s where s is a finite set of positive integers imply the quasinormality or normality of a in particular it is proved that if s p m m p n n p where p 1 and 2 m n then a is quasinormal moreover if a is invertible and s m n n m with m n then a is normal the case when s m m n and a n a n a a n is also discussed

monotone matrix functions of two variables abstract monotone operator functions were studied by koranyi who generalised lowner s theorem on monotone matrix functions of arbitrary high order n to two variables an alternate proof of koranyi s representation of monotone operator functions is provided

a counterexample to the question proposed by yanagi furuichi kuriyama on matrix inequalities and related counterexamples abstract we shall give an counterexample to some questions proposed by yanagi furuichi kuriyama on matrix inequalities and we consider some problems on matrix inequalities related to the questions proposed by them and we shall give several counterexamples to these questions

operator means of probability measures abstract let 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 p in parallel with kubo and ando s definition of two variable operator means and show that every operator mean is contractive for the 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

quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

retract rational fields abstract let k be an infinite field the notion of retract k rationality was introduced by saltman in the study of noether s problem and other rationality problems we will investigate the retract rationality of a field in this paper theorem 1 let k k l be fields if k is retract k rational and l is retract k rational then l is retract k rational theorem 2 for any finite group g containing an abelian normal subgroup h such that g h is a cyclic group for any complex representation g gl v the fixed field c v g is retract c rational theorem 3 if g is a finite group then all the sylow subgroups of g are cyclic if and only if c m g is retract c rational for all g lattices m for all short exact sequences 0 c m m 0 because the unramified brauer group of a retract c rational field is trivial theorems 2 and 3 generalize previous results of bogomolov and barge respectively see theorems 5 9 and 6 1

least energy solutions for a weakly coupled fractional schrodinger system abstract in this paper we consider the following system of two weakly coupled fractional nonlinear schrodinger equations s u u u 2 p b x u p 1 v p 1 u x r n s v 2 s v v 2 p b x v p 1 u p 1 v x r n by use of the s harmonic extension technique we establish the existence of a nontrivial least energy solution of the system via variational methods especially in the autonomous case i e b x b a positive least energy solution with both nontrivial components is obtained

existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional kirchhoff type systems abstract this paper is concerned with the following linearly coupled fractional kirchhoff type system a b r 3 2 u 2 d x u u f u v in r 3 c d r 3 2 v 2 d x v v g v u in r 3 u v h r 3 where a c 0 b d 0 are constants 3 4 1 and 0 is a coupling parameter under the general berestycki lions conditions on the nonlinear terms f and g we prove the existence of positive vector ground state solutions of poho aev type for the above system via variational methods moreover the asymptotic behavior of these solutions as 0 is explored as well recent results from the literature are generally improved and extended

a matrix convexity approach to some celebrated quantum inequalities abstract some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the hansen pedersen jensen inequalities a general argument is given using matrix perspectives of operator convex functions a matrix analogue of marechal s extended perspectives provides additional inequalities including a p q 1 result of lieb

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

operator monotone functions jacobi operators and orthogonal polynomials abstract we reveal a connection between operator monotone functions and orthogonal polynomials especially we express an operator monotone function with a jacobi operator and show that it is a limit of rational operator monotone functions further we prove that the principal inverse of an orthogonal polynomial is operator monotone and hence it has a holomorphic extension to the open upper half plane namely a pick function or nevanlinna function

matrix convexity of functions of two variables abstract let i j be intervals such that 0 i j let mm be the algebra of all m m complex matrices and let mm i be the set of all hermitian members of mm whose spectrum is contained in i let m be the set of all hermitian projections in mm we give a complete characterization of the set of functions f i j r which satisfy the inequality p q f pap qbq p q p q f a b p q for all a m m i b mn j p m q n we also generalize some results on matrix convexity of functions of one variable to the case of functions of two variables

negative solutions to three dimensional monomial noether problem three dimensional monomial noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients we find the necessary and sufficient condition for the coefficients to have a negative solution the results are obtained by two criteria of irrationality using galois cohomology

the convex concave characteristics of gaussian channel capacity functions in this correspondence we give several inherent properties of the capacity function of a gaussian channel with and without feedback by using some operator inequalities and matrix analysis we give a new proof method which is different from the method appearing in k yanagi and h w chen operator inequality and its application to information theory taiwanese j math vol 4 no 3 pp 407 416 sep 2000 we obtain the following results c sub n z p and c sub n fb z p are both concave functions of p c sub n z p is a convex function of the noise covariance matrix and c sub n fb z p is a convex like function of the noise covariance matrix this new proof method is very elementary and the results shall help study the capacity of gaussian channel finally we state a conjecture concerning the convexity of c sub n fb spl middot p

inverse functions of polynomials and orthogonal polynomials as operator monotone functions we study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient let p n n o be a sequence of orthonormal polynomials and p n the restriction of p n to a n where a n is the maximum zero of p n then p 1 n and the composite p n 1 o p 1 n are operator monotone on 0 furthermore for every polynomial p with a positive leading coefficient there is a real number a so that the inverse function of p t a p a defined on 0 oo is semi operator monotone that is for matrices a b 0 p a a p a 2 p b a p a 2 implies a 2 b 2

concavity of some operator functions and frechet differential mappings abstract by using operator perspectives of regular operator mappings we obtain some operator concave convex functions and derive the concavity convexity of some trace functions associated with the frechet differential mapping of certain power functions especially we explore the concavity of the frechet differential mapping x d g x d f x 1 in positive definite invertible operators with f t t p for 0 p 1 and g t t q for p q p 1 and also with f t log t and g t t q for 0 q 1

high energy solutions of systems of kirchhoff type equations in rn in the present paper five new critical point theorems on the product spaces are given and three existence theorems for the sequence of high energy solutions for the systems of kirchhoff type equations are studied

modulus of convexity for operator convex functions given an operator convex function f x we obtain an operator valued lower bound for cf x 1 c f y f cx 1 c y c 0 1 the lower bound is expressed in terms of the matrix bregman divergence a similar inequality is shown to be false for functions that are convex but not operator convex

an order inequality characterizing invariant barycenters on symmetric cones abstract this paper is concerned with invariant and contractive barycenters on the wasserstein space of probability measures on metric spaces of non positive curvature where the center of gravity also called the cartan barycenter is the canonical barycenter on hadamard spaces we establish an order inequality of probability measures on partially ordered symmetric spaces of non compact type namely symmetric cones self dual homogeneous cones characterizing the cartan barycenter among other invariant and contractive barycenters the derived inequality and partially ordered structures on the probability measure space lead also to significant results on norm inequalities including the ando hiai inequality for probability measures on symmetric cones

existence and asymptotic behavior of vector solutions for coupled nonlinear kirchhoff type systems abstract this paper deals with the following linearly coupled nonlinear kirchhoff type system a 1 b 1 r 3 u 2 d x u 1 u f u v in r 3 a 2 b 2 r 3 v 2 d x v 2 v g v u in r 3 u v h 1 r 3 where a i 0 b i 0 i 0 are constants for i 1 2 0 is a parameter and f g c r r under the general berestycki lions type assumptions on f and g we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods we also study the asymptotic behavior of these solutions as 0

reduction in the rationality problem for multiplicative invariant fields abstract for a faithful zg lattice a and a field k on which the group g acts by field automorphisms let r be the normal subgroup generated by the elements of g which act trivially on k and act as reflections on a we prove that the rationality of the multiplicative invariant field k a g over k ar g is equivalent to the rationality of k a g over k ar g where g is a particular subgroup of g such that g r g we then use this reduction result to prove that k a g is rational over k where g is the automorphism group of a crystallographic root system g acts trivially on k and a is any lattice on the space q

positive solutions of asymptotically linear equations via pohozaev manifold abstract we present a new method for finding positive solutions of nonlinear elliptic equations which are non homogeneous and asymptotically linear at infinity by using projections on a pohozaev manifold rather than the nehari manifold associated with the problem

higher order extension of lowner s theory operator k tone functions the new notion of operator matrix k tone functions is introduced which is a higher order extension of operator matrix monotone and convex functions differential properties of matrix k tone functions are shown characterizations properties and examples of operator k tone functions are presented in particular integral representations of operator k tone functions are given generalizing familiar representations of operator monotone and convex functions

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

high energy solutions of systems of kirchhoff type equations on rn the main purpose of this paper is to unify and improve theorems 3 1 3 3 in wu 2012 8 as well as some results in other literature such as theorems 1 4 in wu 2011 9 we present a new proof technique to conclude the existence of high energy solutions for the kirchhoff type system under some assumptions that are weaker than those in wu 2012 8

a unified approach to operator monotone functions abstract the notion of operator monotonicity dates back to a work by lowner in 1934 a map f s n s m is called operator monotone if a b implies f a f b here s n is the space of symmetric matrices with the semidefinite partial order often the function f is defined in terms of an underlying simpler function f of main interest is to find the properties of f that characterize operator monotonicity of f in that case it is said that f is also operator monotone classical examples are the lowner operators and the spectral scalar valued isotropic functions operator monotonicity for these two classes of functions is characterized in seemingly very different ways this work extends the notion of operator monotonicity to symmetric functions f on k arguments the latter is used to define generated k isotropic maps f s n s n k for any n k necessary and sufficient conditions are given for f to generate an operator monotone k isotropic map f when k 1 the k isotropic map becomes a lowner operator and when k n it becomes a spectral function this allows us to reconcile and explain the differences between the conditions for monotonicity for the lowner operators and the spectral functions

strong converse to the quantum channel coding theorem a lower bound on the probability of decoding error for a quantum communication channel is presented from which the strong converse to the quantum channel coding theorem is immediately shown the results and their derivations are mostly straightforward extensions of the classical counterparts which were established by arimoto 1973 except that more careful treatment is necessary here due to the noncommutativity of operators

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

on noether s problem for central extensions of symmetric and alternating groups for a field k and a finite group g acting regularly on a set of indeterminates x xg g g let k g denote the invariant field k x g we first prove for the alternating group an that if n is odd then q an is rational over q an 1 we then obtain an analogous result where an is replaced by an arbitrary finite central extension of either an or sn valid over q n for suitable n concrete applications of our results yield 1 a new proof of maeda s result on the rationality of q x1 x5 a5 q 2 an affirmative answer to noether s problem over q for both a5 and s5 3 an affirmative answer to noether s problem over c for every finite central extension group of either an or sn with n 5

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

an extension of inequalities by ando abstract we give variations on ando s result comparing f b f a and f b a with respect to unitarily invariant norms on matrices

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

jensen and minkowski inequalities for operator means and anti norms abstract jensen inequalities for positive linear maps of choi and hansen pedersen type are established for a large class of operator matrix means such as some p means and some kubo ando means these results are also extensions of the minkowski determinantal inequality to this end we develop the study of anti norms a notion parallel to the symmetric norms in matrix analysis including functionals like schatten q norms for a parameter q 1 and the minkowski functional det 1 n a

reliability function of general classical quantum channel in information theory the reliability function and its bounds describing the exponential behavior of the error probability are important quantitative characteristics of the channel performance from a more general point of view these bounds provide certain measures of distinguishability of a given set of classical states in a previous paper quantum analogs of the random coding and expurgation lower bounds for the case of pure signal states were introduced here we discuss the case of general quantum states in particular we prove the expurgation bound conjectured previously and find the quantum cutoff rate for arbitrary mixed signal states

on trace inequalities and their applications to noncommutative communication theory abstract certain trace inequalities related to matrix logarithm are shown these results enable us to give a partial answer of the open problem conjectured by a s holevo that is concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proven in the case of 0 s 1

existence and concentration result for the kirchhoff type equations with general nonlinearities in this paper we study the existence and concentration behaviors of positive solutions to the kirchhoff type equations varepsilon 2 m left varepsilon 2 n int mathbf r n nabla u 2 mathrm d x right delta u v x u f u quad rm in mathbf r n quad u in h 1 mathbf r n n geqq 1 where m and v are continuous functions under suitable conditions on m and general conditions on f we construct a family of positive solutions u varepsilon varepsilon in 0 tilde varepsilon which concentrates at a local minimum of v after extracting a subsequence ek

quasi monomial actions and some 4 dimensional rationality problems abstract let g be a finite group acting on k x 1 x n the rational function field of n variables over a field k the action is called a purely monomial action if x j 1 i n x i a i j for all g for 1 j n where a i j 1 i j n gl n z the main question is that under what situations the fixed field k x 1 x n g is rational purely transcendental over k this rationality problem has been studied by hajja kang hoshi rikuna when n 3 in this paper we will prove that k x 1 x 2 x 3 x 4 g is rational over k provided that the purely monomial action is decomposable to prove this result we introduce a new notion the quasi monomial action which is a generalization of previous notions of multiplicative group actions moreover we determine the rationality problem of purely quasi monomial actions of k x y g over k where k k g

perspectives of matrix convex functions in this paper we generalize the main results of effros eg 2009 proc natl acad sci usa 106 1006 1008 namely we provide the necessary and sufficient conditions for jointly convexity of perspective functions and generalized perspective functions

superadditivity and derivative of operator functions abstract we will show that if i j a i a j 0 for bounded operators a i 0 i 1 2 n then g i a i i g a i for every operator convex function g t on 0 with g 0 0 in particular i a i log i a i i a i log a i if each a i is invertible let a b 0 and a be invertible then we will observe that the frechet derivative d g s a b is increasing on 0 s for every operator convex function g t on 0 if and only if a b b a 0

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

the fast track to lowner s theorem abstract the operator monotone functions defined in the positive half line are of particular importance we give a version of the theory in which integral representations for these functions can be established directly without invoking lowner s detailed analysis of matrix monotone functions of a fixed order or the theory of analytic functions we found a canonical relationship between positive and arbitrary operator monotone functions defined in the positive half line and this result effectively reduces the theory to the case of positive functions

uncertainty principle and quantum fisher information ii heisenberg and schrodinger uncertainty principles give lower bounds for the product of variances var a var b if the observables a b are not compatible namely if the commutator a b is not zero in this paper we prove an uncertainty principle in schrodinger form where the bound for the product of variances var a var b depends on the area spanned by the commutators i a and i b with respect to an arbitrary quantum version of the fisher information

finite blocklength and moderate deviation analysis of hypothesis testing of correlated quantum states and application to classical quantum channels with memory martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning here we apply techniques borrowed from this field to quantum hypothesis testing which is the problem of discriminating quantum states belonging to two different sequences rho n n and sigma n n we obtain achievability bounds on the finite blocklength type ii stein and hoeffding errors which for i i d states are in general tighter than the corresponding bounds obtained by audenaert mosonyi and verstraete we also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a non homogeneous factorization property examples of such sequences include gibbs states of spin chains with translation invariant finite range interaction as well as finitely correlated quantum states we apply our results to find bounds on the capacity of a certain class of classical quantum channels with memory which satisfy a so called channel factorization property both in the finite blocklength and moderate deviation regimes

a characterization of extended monotone metrics abstract classical information geometry has emerged from the study of geometrical aspect of the statistical estimation cencov characterized the fisher metric as a canonical metric on probability simplexes s n 1 x 1 x n r n x i 1 from a statistical point of view and campbell extended the characterization of the fisher metric from probability simplexes to positive cone r n in quantum information geometry quantum states which are represented by positive hermitian matrices with trace one are regarded as an extension of probability distributions a quantum version of the fisher metric is introduced and is called a monotone metric petz characterized the monotone metrics on the space of all quantum states in terms of operator monotone functions a purpose of the present paper is to extend a characterization of monotone metrics from the space of all states to the space of all positive hermitian matrices on finite dimensional hilbert space this characterization corresponds quantum modification of campbell s work

trace functions with applications in quantum physics we consider both known and not previously studied trace functions with applications in quantum physics by using perspectives we obtain convexity statements for different notions of residual entropy including the entropy gain of a quantum channel studied by holevo and others

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

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

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

existence and multiplicity of solutions for asymptotically linear nonperiodic hamiltonian elliptic system abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v v v x v h u x u v x r n u x v x 0 as x where z u v r n r r n 3 assuming the potential v is nonperiodic and sign changing h x z is nonperiodic in x and asymptotically quadratic in z u v existence and multiplicity of solutions are obtained via a variational approach

some convex and monotone matrix functions we prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections as a consequence we prove that a certain product of matrix monotone functions is a matrix monotone function similar results are proved for matrix convex functions these results include some known results for matrix monotone and matrix convex functions

twisted multiplicative field invariants noether s problem and galois extensions noether s problem concerns fields f xg 1 g e g g f g where f is a field g is a finite group g acts by permuting the indeterminants x in the obvious way and f g refers to the field of invariant elements in showing the f g s were nonrational for certain g the author was led to consider multiplicative invariant fields ss which are defined as follows let g be a finite group and q a g lattice that is q is a finitely generated z g module which is free as an abelian group form the group algebra f g and its field of fractions f q g acts naturally on f q and the multiplicative invariant field is f q in ss it was shown that for certain q f q g was essentially equivalent to actually stably isomorphic to f g for g a split extension of g with abelian kernel this leads one to ask whether f g can be expressed as a kind of invariant field for g when g is a nonsplit extension of g the answer is yes and is the core of this paper in fact f g will be shown to be a so called a twisted multiplicative invariant field of g the goal of this paper is to present this fact and then draw a series of conclusions using it such a twisted invariant fields have appeared explicitly in previous works in ss these fields arise as another way of describing the invariant fields of reductive algebraic groups note that in s6 the finite group is the weyl group and the lattice is derived from the root lattice in s7 the unramified brauer groups of a twisted invariant fields are calculated

some results on strongly operator convex functions and operator monotone functions abstract this paper concerns three classes of real valued functions on intervals operator monotone functions operator convex functions and strongly operator convex functions strongly operator convex functions were previously treated in 3 and 4 where operator algebraic semicontinuity theory or operator theory were substantially used in this paper we provide an alternate treatment that uses only operator inequalities or even just matrix inequalities we show also that if t 0 is a point in the domain of a continuous function f then f is operator monotone if and only if f t f t 0 t t 0 is strongly operator convex using this and previously known results we provide some methods for constructing new functions in one of the three classes from old ones we also include some discussion of completely monotone functions in this context and some results on the operator convexity or strong operator convexity of f when f is operator convex or strongly operator convex

loewner s theorem in several variables abstract in this paper we establish a multivariable non commutative generalization of loewner s theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half plane into itself the non commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples not a priori assumed to be analytic or continuous as a schur complement of a positive linear matrix pencil to achieve this we establish the abstract integral formula for these functions using only matrix convexity and lmis that represents operator monotone and operator concave free functions as a conditional expectation of a schur complement of a linear matrix pencil on a tensor product operator algebra the results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so including the karcher mean to obtain an explicit closed formula for these operator means

a trace inequality arising from quantum information theory abstract we introduce a skew information of lieb s type s f g a x tr f a xg a x tr f a g a x 2 for selfadjoint matrices a x we give conditions for f and g so that sf g is positive or negative as another important application we settle the problem posed by yanagi furuichi and kuriyama from quantum information theory

multiple solutions for asymptotically quadratic and superquadratic elliptic system of hamiltonian type this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u h v x u v x r n v v x v h u x u v x r n u x 0 v x 0 as x where z u v r n r i r n 3 and the potential v x is nonperiodic and sign changing by applying a generalized linking theorem for strongly indefinite functionals we establish the existence of multiple solutions for asymptotically quadratic nonlinearity as well as the existence of infinitely many solutions for superquadratic nonlinearity

on matrix inequalities between the power means counterexamples abstract we prove that the known sufficient conditions on the real parameters p q for which the matrix power mean inequality a p b p 2 1 p a q b q 2 1 q holds for every pair of matrices a b 0 are indeed best possible the proof proceeds by constructing 2 2 counterexamples the best possible conditions on p q for which a p 1 p a q 1 q holds for every unital positive linear map and a 0 are also clarified

operator means and positivity of block operators the aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators it is shown that for positive operators a b c and for every non negative operator monotone function f on 0 infty the block matrix begin aligned left begin array cc f a f c f c f b end array right end aligned is positive if and only if c le a b in particular if c le a b then begin aligned left begin array cc a c c b end array right end aligned is positive

mixed matrix operator inequalities we will show that if 0 a b c and s s t t 1 then a s bs t ct implies that ar 2 s bss t cst ar 2 ar 2 s bts t ctt ar 2 s r t r fort 1 t s 0 r 0 and a1 r ar 2 s bts t ctt ar 2 1 r t r fort 1 r 0 these are the extensions of inequalities shown by t furuta proc amer math soc 101 1987 85 111 1991 511 moreover we will give an inequality say mixed inequality which includes the above inequalities and the inequality due to hansen pedersen math ann 258 1982 229

on a reverse heinz kato furuta inequality abstract in the set up of minkowski spaces the schwarz inequality holds with the reverse inequality sign as a consequence the same occurs with the triangle inequality in this note extensions of this indefinite version of the schwarz inequality are presented namely a reverse heinz kato furuta inequality valid for timelike vectors is included and related inequalities that also hold with the reverse sign are investigated

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

concavity of the auxiliary function appearing in quantum reliability function reliability functions characterize the asymptotic behavior of the error probability for transmission of data on a channel holevo introduced the quantum channel and gave an expression for a random coding lower bound involving an auxiliary function holevo ogawa and nagaoka conjectured that this auxiliary function is concave here we give a proof of this conjecture

quantum data processing we prove a data processing inequality for quantum communication channels which states that processing a received quantum state may never increase the mutual information between input and output states