the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

geometric mean flows and the cartan barycenter on the wasserstein space over positive definite matrices abstract we introduce a class of flows on the wasserstein space of probability measures with finite first moment on the cartan hadamard riemannian manifold of positive definite matrices and consider the problem of differentiability of the corresponding cartan barycentric trajectory as a consequence we have a version of lie trotter formula and a related unitarily invariant norm inequality furthermore a fixed point theorem related to the karcher equation and the cartan barycentric trajectory is also presented as an application

an araki lieb thirring inequality for geometrically concave and geometrically convex functions for positive definite matrices a and b the araki lieb thirring inequality amounts to an eigenvalue log submajorisation relation for fractional powers atbt w log t ab 0 t 1 while for t 1 the reversed inequality holds in this paper i generalise this inequality replacing the fractional powers xt by a larger class of functions i prove that a continuous non negative function f with domain dom f 0 x0 for some positive x0 possibly infinity satisfies f a f b w log f2 1 2 ab for all positive semidefinite a and b with spectrum in dom f if and only if f is geometrically concave and 0 xf x f x for all x dom f the reversed inequality holds for a continuous non negative function f if and only if f is geometrically convex and xf x f x for all x dom f as an application i obtain a complementary inequality to the golden thompson inequality

majorization relations for hadamard products abstract we settle affirmatively the conjecture of johnson and bapat on the hadamard product a b of positive definite matrices a b i k n i a b ab k 1 2 n

convergence in a quasilinear parabolic equation with time almost periodic boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t g t t 0 where g t is an almost periodic function we prove the following convergence results if a time global solution u is bounded then it converges to an almost periodic solution which is unique up to space shift stable and asymptotically stable if u goes to infinity with positive lower average speed then it converges to an almost periodic traveling wave which is unique up to space shift stable asymptotically stable and has positive average speed

matrix inequalities in statistical mechanics some matrix inequalities used in statistical mechanics are presented a straightforward proof of the thermodynamic inequality is given and its equivalence to the peierls bogoliubov inequality is shown

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

shannon like inequalities for f connections of positive linear maps and positive operators abstract in this paper we investigate a map t b t f t b called an f connection induced by an operator convex concave function f where t denotes a reflexive generalized inverse of a positive linear map t between unital c algebras a and b of hilbert space operators and b b some special cases of f connection with invertible t are related to geometric operator mean relative operator entropy and csiszar operator f divergence we formulate conditions under which the inequality f 1 i k 1 n t k f t k b k holds where t k a b are positive linear maps and b k ran t k in particular the following shannon like inequality 0 k 1 n t k f t k b k is shown for an operator convex function f with f 1 0 the obtained results are specified for csiszar operator f divergence some recent results by isa et al 2013 15 are recovered

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

operator entropy and fidelity associated with the geometric mean abstract we consider operator versions of quantum tsallis relative entropy that involve the weighted matrix geometric mean and establish relationships with the fidelity of a pure state and an arbitrary state we define the symmetrized operator entropy based on the tsallis relative operator entropy and study its properties

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

inequalities for quantum relative entropy some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log majorization result the thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the peierls bogoliubov inequality is obtained

applications of geometric means on symmetric cones let v be a euclidean jordan algebra and let omega be the corresponding symmetric cone the geometric mean a b of two elements a and b in omega is defined as a unique solution which belongs to omega of the quadratic equation p x a 1 b where p is the quadratic representation of v in this paper we show that for any a in omega the sequence of n mathrm th iterate f a n x of the function f a omega to omega defined by f a x 1 over 2 x p a x 1 converges to a as applications we obtain that the geometric mean a b of omega can be represented as a limit of successive iteration of arithmetic means and harmonic means and we derive the lowner heinz inequality on the symmetric cone omega 0 leq a leq b mathrm implies a p leq b p mathrm for 0 leq p leq 1 furthermore we obtain a formula mathrm exp x y lim n to infty mathrm exp frac 2 n x exp frac 2 n y n which leads a golden thompson type inequality mathrm exp x y leq p mathrm exp x over 2 mathrm exp y for the spectral norm on v

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

α z renyi relative entropies we consider a two parameter family of renyi relative entropies d z that are quantum generalisations of the classical renyi divergence d p q this family includes many known relative entropies or divergences such as the quantum relative entropy the recently defined quantum renyi divergences as well as the quantum renyi relative entropies all its members satisfy the quantum generalizations of renyi s axioms for a divergence we consider the range of the parameters z for which the data processing inequality holds we also investigate a variety of limiting cases for the two parameters obtaining explicit formulas for each one of them

order automorphisms on positive definite operators and a few applications abstract we first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so called chaotic order we then apply those results to the following problems 1 description of all bijective transformations on the space of nonsingular density operators quantum states which preserve the umegaki or the belavkin staszewski relative entropy 2 characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order

two reverse inequalities associated with tsallis relative operator entropy via generalized kantorovich constant and their applications recently tsallis relative operator entropy tp a b and tsallis relative entropy dp a b are discussed by furuichi yanagi kuriyama we shall show two reverse inequalities involving tsallis relative operator entropy tp a b via generalized kantorovich constant k p as some applications of two reverse inequalities we shall show two trace reverse inequalities involving tr tp a b and dp a b and also a known reverse trace inequality involving the relative operator entropy s a b by fujii kamei and the umegaki relative entropy s a b is shown as a simple corollary we show the following result let a and b be strictly positive operators on a hilbert space h such that m1 i a m1 i 0 and m2 i b m2 i 0 put m m2m1 m m2m1 h mm m1m2m1m2 1 and p 0 1 let be normalized positive linear map on b h then the following inequalities hold i 1 k p p a p b tp a b tp a b tp a b and ii f p a tp a b tp a b tp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and k p 0 1 and f p mpphp hh 11 k p 1p 1 0 in addition let a and b be strictly positive definite matrices iii 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b and iv f p tr a dp a b tr tp a b dp a b in particular both iii and iv yield the following known result logs 1 tr a s a b tr s a b s a b where s 1 h1h 1elogh1h 1 is said to be the specht ratio and s 1 1

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

anderson localization for the discrete one dimensional quasi periodic schrodinger operator with potential defined by a gevrey class function in this paper we consider the discrete one dimensional schrodinger operator with quasi periodic potential vn v x n we assume that the frequency satisfies a strong diophantine condition and that the function v belongs to a gevrey class and it satisfies a transversality condition under these assumptions we prove in the perturbative regime that for large disorder and for most frequencies the operator satisfies anderson localization moreover we show that the associated lyapunov exponent is positive for all energies and that the lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity we also prove a partial nonperturbative result assuming that the function v belongs to some particular gevrey classes

lie trotter means of positive definite operators abstract as extension of the lie trotter product formula we define the two variable and multivariate lie trotter means with several examples including the sagae tanabe and hansen inductive means we show that the weighted means satisfying the arithmetic geometric harmonic mean inequalities are the multivariate lie trotter means we finally give a few interesting versions of the extended lie trotter formula derived from multivariate lie trotter means

anderson localization for generic deterministic operators abstract we consider a class of ensembles of lattice schrodinger operators with deterministic potentials including quasi periodic potentials with diophantine frequencies depending upon an infinite number of parameters in an auxiliary measurable space using a variant of the multi scale analysis we prove anderson localization for generic ensembles in the strong disorder regime and establish an analog of minami type bounds for spectral spacings

convexity inequalities for estimating free energy and relative entropy for quantum systems it is shown that the relative entropy s p q tr p log p tr p log q of two positive semi definite operators p and q satisfies gamma 1 tr p p1 gamma q gamma or s p q or gamma 1 tr p1 gamma qgamma p for 0 gamma or 1 and that these bounds become exact in the limit gamma to phi analogous inequalities hold for states in classical statistical mechanics or information theory with trace replaced by integration or summation furthermore the average of these bounds is in general a better approximation to s p q than either bound alone and the average is amenable to further improvement via repeated richardson extrapolation if p and q are gibbs equilibrium states then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

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

a variational expression for the relative entropy we prove that for the relative entropy of faithful normal states and on the von neumann algebram the formula s varphi omega sup omega h log varphi h i h h in m holds

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

trace inequalities for logarithms and powers of j hermitian matrices abstract spectral inequalities are stated for the trace of the exponential or the logarithmic of certain j hermitian matrices j i r i n r 0 r n the obtained inequalities are established in the context of indefinite inner product spaces and they are known to be valid for hilbert space operators or operator algebras

parametric extensions of shannon inequality and its reverse one in hilbert space operators we shall state the following parametric extensions of shannon inequality and its reverse one in hilbert space operators let p 0 1 and also let a1 a2 an and b1 b2 bn be two sequences of strictly positive operators on a hilbert space h such that j 1naj pbj i then j 1nsp 1 aj bj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp aj bj log j 1n aj p 1bj i j 1naj pbj j 1n aj p 1bj i j 1naj pbj log j 1n aj p 1bj i j 1naj pbj j 1nsp 1 aj bj where sq a b a12a 12ba 12qloga 12ba 12a12 for a 0 b 0 and any real number q and a qb a12a 12ba 12qa12 for a 0 b 0 and any real number q in particular if j 1naj j 1nbj i then j 1ns2 aj bj j 1nbjaj 1bjlog j 1nbjaj 1bj log j 1nbjaj 1bj j 1ns1 aj bj 0 j 1ns aj bj log j 1najbj 1aj j 1najbj 1ajlog j 1najbj 1aj j 1ns 1 aj bj where s a b s0 a b a12loga 12ba 12a12 which is the relative operator entropy of a 0 and b 0 our results can be considered as parametric extensions of the following celebrated shannon inequality ann math statistics 22 1951 79 bull syst tech j 27 1948 379 pitman monographs and surveys in pure and applied mathematics 97 addison wesley longman 1998 p 233 which is very useful and so famous in information theory let a1 a2 an and b1 b2 bn be two probability vectors then 0 j 1najlogbj j 1najlogaj see inequalities 2 4 of corollary 2 4

almost periodic dynamics of perturbed infinite dimensional dynamical systems abstract this paper is concerned with the dynamics of an infinite dimensional gradient system under small almost periodic perturbations under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions we prove that the perturbed non autonomous system has exactly the same number of almost periodic solutions as a consequence the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions an application of the result to the chafee infante equation is discussed

limit periodic schrodinger operators in the regime of positive lyapunov exponents abstract we investigate the spectral properties of discrete one dimensional schrodinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a cantor group we show that for given cantor group and minimal translation there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero hausdorff dimension and all spectral measures are purely singular continuous the associated lyapunov exponent is a continuous strictly positive function of the energy it is possible to include a coupling constant in the model and these results then hold for every non zero value of the coupling constant

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

convex multivariate operator means abstract we prove that a trace function generated by the functional calculus is geodesically convex in the riemannian manifold of positive definite matrices if and only if it is geodesically convex in positive numbers the analysis of multivariate operator means is facilitated by the study of classes of means that are fix points under a contraction with respect to the thompson metric although such methods are powerful they crucially depend on monotonicity we develop new techniques to prove existence of multivariate operator means that are not necessarily monotone

on operator inequalities of some relative operator entropies abstract in our recent paper we introduced the notions of relative operator entropy and tsallis relative operator entropy as a parameter extensions of relative operator entropy and tsallis relative operator entropy in this paper we give upper and lower bounds of these new notions according to operator geometric mean introduced in nikoufar et al 2013 14

generic singular spectrum for ergodic schrodinger operators we consider schrodinger operators with ergodic potential v n f t n n z where t is a nonperiodic homeomorphism we show that for generic f c the spectrum has no absolutely continuous component the proof is based on approximation by discontinuous potentials which can be treated via kotani theory

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

spectral approximation for ergodic cmv operators with an application to quantum walks abstract we establish concrete criteria for fully supported absolutely continuous spectrum for ergodic cmv matrices and purely absolutely continuous spectrum for limit periodic cmv matrices we proceed by proving several variational estimates on the measure of the spectrum and the vanishing set of the lyapunov exponent for cmv matrices which represent cmv analogues of results obtained for schrodinger operators due to y last in the early 1990s having done so we combine those estimates with results from inverse spectral theory to obtain purely absolutely continuous spectrum

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

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

eigenvalue inequalities for products of matrix exponentials motivated by models from stochastic population biology and statistical mechanics we prove new inequalities of the form cp eae 2 cp eatr where a and b are n x n complex matrices 1 n 00 and cp is a real valued continuous function of the eigenvalues of its matrix argument for example if a is essentially nonnegative b is diagonal real and cp is the spectral radius then holds if in addition a is irreducible and b has at least two different diagonal elements then the inequality is strict the proof uses kingman s theorem on the logconvexity of the spectral radius lie s product formula and perturbation theory we conclude with conjectures

structure of ω limit sets for almost periodic parabolic equations on s1 with reflection symmetry abstract the structure of the limit sets is thoroughly investigated for the skew product semiflow which is generated by a scalar reaction diffusion equation u t u x x f t u u x t 0 x s 1 r 2 z where f is uniformly almost periodic in t and satisfies f t u u x f t u u x we show that any limit set contains at most two minimal sets moreover any hyperbolic limit set is a spatially homogeneous 1 cover of hull h f when dim v c 1 v c is the center space associated with it is proved that either is a spatially homogeneous or is a spatially inhomogeneous 1 cover of h f

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard

les espaces lp d une algebre de von neumann definies par la derivee spatiale abstract this work answers a question raised by a connes on the spatial theory of von neumann algebras preprint inst hautes etudes sci france and generalizes for a general von neumann algebra the theory of non commutative integration of j dixmier bull soc math france 81 1953 and i segal ann of math 57 1973

an extension of the kunz souillard approach to localization in one dimension and applications to almost periodic schrodinger operators abstract we generalize the approach to localization in one dimension introduced by kunz souillard and refined by delyon kunz souillard and simon in the early 1980 s in such a way that certain correlations are allowed several applications of this generalized kunz souillard method to almost periodic schrodinger operators are presented on the one hand we show that the schrodinger operators on l 2 z with limit periodic potential that have pure point spectrum form a dense subset in the space of all limit periodic schrodinger operators on l 2 z more generally for any bounded potential one can find an arbitrarily small limit periodic perturbation so that the resulting operator has pure point spectrum our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit periodic schrodinger operators on l 2 z on the other hand we show that schrodinger operators on l 2 z with arbitrarily small one frequency quasi periodic potential may have pure point spectrum for some phases this was previously known only for one frequency quasi periodic potentials with norm exceeding 2 namely the super critical almost mathieu operator with a typical frequency and phase moreover this phenomenon can occur for any frequency whereas no previous quasi periodic potential with liouville frequency was known that may admit eigenvalues for any phase

generalized shannon inequalities based on tsallis relative operator entropy abstract tsallis relative operator entropy is defined and then its properties are given shannon inequality and its reverse one in hilbert space operators derived by furuta linear algebra apll 381 2004 219 are extended in terms of the parameter of the tsallis relative operator entropy moreover the generalized tsallis relative operator entropy is introduced and then several operator inequalities are derived

trace inequalities in nonextensive statistical mechanics abstract in this short paper we establish a variational expression of the tsallis relative entropy in addition we derive a generalized thermodynamic inequality and a generalized peierls bogoliubov inequality finally we give a generalized golden thompson inequality

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

reverse inequalities involving two relative operator entropies and two relative entropies we shall discuss relation among tsallis relative operator entropy tp a b the relative operator entropy s a b by j i fujii kamei the tsallis relative entropy dp a b by furuichi yanagi kuriyama and the umegaki relative entropy s a b we show the following result let a and b be strictly positive definite matrices such that m1i a m1i 0 and m2i b m2i 0 put h m1m2m1m2 1 and p 0 1 then the following inequalities hold 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and the first inequality is the reverse one of the second known inequalty in particular logs 1 tr a s a b tr s a b s a b where s 1 is the specht ratio defined by s 1 h1h 1elogh1h 1 and the first inequality is the reverse one of the second known inequalty it is known that k p 0 1 for p 0 1 and s 1 1

tsallis entropies and matrix trace inequalities in quantum statistical mechanics extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the tsallis entropy namely one parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized peierls bogoliubov inequality is stated in the sense that one may be obtained from the other and vice versa

generic continuous spectrum for ergodic schrodinger operators we consider families of discrete schrodinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function we introduce the concepts of topological and metric repetition property assuming that the underlying dynamical system satisfies one of these repetition properties we show using gordon s lemma that for a generic continuous sampling function the set of elements in the associated family of schrodinger operators that have no eigenvalues is large in a topological or metric sense respectively we present a number of applications particularly to shifts and skew shifts on the torus

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

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

strictly ordered minimal subsets of a class of convex monotone skew product semiflows we study the topological and ergodic structure of a class of convex and monotone skew product semiflows we assume the existence of two strongly ordered minimal subsets k1 k2 and we obtain an ergodic representation of their upper lyapunov exponents in the case of null upper lyapunov exponents we obtain a lamination into minimal subsets of an intermediate region where the restriction of the semiflow is affine in the hyperbolic case we deduce the long time behaviour of every trajectory ordered with k2 some examples of skew product semiflows generated by non autonomous differential equations and satisfying the assumptions of monotonicity and convexity are also presented

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

generalizations of operator shannon inequality based on tsallis and renyi relative entropies abstract recently we obtained relations among relative operator entropies of sequences for example for relative operator entropy s a b renyi relative operator entropy i t a b and tsallis relative operator entropy t t a b of sequences of strictly positive operators a a 1 a n and b b 1 b n such that i 1 n a i i 1 n b i i s a b i t a b t t a b 0 holds for 0 t 1 this is an extension of operator version of shannon inequality briefly operator shannon inequality discussed by furuta and yanagi kuriyama furuichi in this paper we shall obtain two generalizations of this inequality by considering generalizations of relative operator entropies of sequences

quantumness of bipartite states in terms of conditional entropies quantum discord as defined by olliver and zurek 2002 phys rev lett 88 017901 as the difference of two natural quantum extensions of the classical mutual information plays an interesting role in characterizing quantumness of correlations inspired by this idea we will study quantumness of bipartite states arising from different quantum analogs of the classical conditional entropy our approach is intrinsic in contrast to the olliver zurek method that involves extrinsic local measurements for this purpose we introduce two alternative variants of quantum conditional entropies via conditional density operators which in turn are intuitive quantum extensions of equivalent classical expressions for the conditional probability the significance of these quantum conditional entropies in characterizing quantumness of bipartite states is illustrated through several examples

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

concavity of certain matrix trace and norm functions ii abstract we refine epstein s method to prove joint concavity convexity of matrix trace functions of lieb type tr f a p 1 2 b q a p 1 2 and symmetric anti norm functions of the form f a p b q where and are positive linear maps is an operator mean and f x with a certain power is an operator monotone function on 0 moreover the variational method of carlen frank and lieb is extended to general non decreasing convex concave functions on 0 so that we prove joint concavity convexity of more trace functions of lieb type

hadamard products and golden thompson type inequalities abstract by using hadamard products we give some reasonable upper and lower bounds of golden thompson type for e h 1 h m where h i i 1 2 m are arbitrary hermitian matrices and is an arbitrary unitarily invariant norm

a generalization of araki s log majorization abstract we generalize araki s log majorization to the log convexity theorem for the eigenvalues of a p 1 2 b p a p 1 2 as a function of p 0 where a b are positive semidefinite matrices and are positive linear maps between matrix algebras similar generalizations of the log majorization of ando hiai type for the weighted geometric mean a b are also given including a lemma on general operator means a e of a and a projection e

uniform positivity and continuity of lyapunov exponents for a class of c 2 quasiperiodic schrodinger cocycles abstract we show that for a class of c 2 quasiperiodic potentials and for any fixed diophantine frequency the lyapunov exponent of the corresponding schrodinger cocycles as a function of energies are uniformly positive and weakly holder continuous as a corollary we obtain that the corresponding integrated density of states is weakly holder continuous as well our approach is of purely dynamical systems which depends on a detailed analysis of asymptotic stable and unstable directions we also apply it to more general sl 2 r cocycles which in turn can be applied to get uniform positivity and continuity of lyapunov exponents around unique nondegenerate extremal points of any smooth potential and to a certain class of c 2 szeg cocycles

a characterization of the helmholtz free energy let h be a finite dimensional complex hilbert space b h be the set of all positive semi definite operators matrices on h and is a not necessarily linear map of b h preserving the generalized helmholtz free energy in this paper under suitable conditions we prove that there exists either a unitary or an anti unitary operator u on h such that a uau for any a b h

path of quasi means as a geodesic abstract as a generalization of the hiai petz geometries we discuss two types of them where the geodesics are the quasi arithmetic means and the quasi geometric means respectively each derivative of such a geodesic might determine a new relative operator entropy also in these cases the finsler metric can be induced by each unitarily invariant norm if the norm is strictly convex then the geodesic is the shortest we also give examples of the shortest paths which are not the geodesics when the finsler metrics are induced by the ky fan k norms

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

approximation dynamics and the stability of invariant sets abstract we introduce the concept of a weakly normally hyperbolic set for a system of ordinary differential equations this concept includes the notion of a hyperbolic flow as well as that of a normally hyperbolic invariant manifold moreover it has the property that it is closed under finite set products consequently the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly normally hyperbolic sets our main objective is to show that under a small c 1 perturbation a weakly normally hyperbolic set k is preserved by a homeomorphism where the image k y is a compact invariant set with a related hyperbolic structure for the perturbed equation in addition the homeomorphism is close to the identity in c 0 1 and the perturbed dynamics on k y are close to the original dynamics on k

matrix trace inequalities on tsallis relative entropy of negative order abstract in this paper we show matrix trace inequalities on the quantum tsallis relative entropy of negative order which includes the quasi geometric mean of positive definite matrices moreover we show an estimate of the difference between two tsallis relative entropies of negative order

a note on operator inequalities of tsallis relative operator entropy tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized shannon inequalities were shown in the previous paper after the review of some fundamental properties of tsallis relative operator entropy some operator inequalities related to tsallis relative operator entropy are shown in the present paper our inequalities give the upper and lower bounds of tsallis relative operator entropy the operator equality on tsallis relative operator entropy is also shown by considering the tensor product this relation generalizes the pseudoadditivity for tsallis entropy as a corollary of our operator equality derived from the tensor product manipulation we show several operator inequalities including the superadditivity and the subadditivity for tsallis relative operator entropy our results are generalizations of the superadditivity and the subadditivity for tsallis entropy