spectral flow is the integral of one forms on the banach manifold of self adjoint fredholm operators abstract one may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 vancouver icm address of i m singer our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded breuer fredholm operators in a semifinite von neumann algebra these formulae have a geometric interpretation which derives from the proof namely we define a family of banach submanifolds of all bounded self adjoint breuer fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path we emphasise that our methods do not give a single globally defined one form on the self adjoint breuer fredholms whose integral along all paths is spectral flow rather as the choice of the plural forms in the title suggests we need a family of such one forms in order to confirm singer s idea the original context for this result concerned paths of unbounded self adjoint fredholm operators we therefore prove analogous formulae for spectral flow in the unbounded case as well the proof is a synthesis of key contributions by previous authors whom we acknowledge in detail in the introduction combined with an additional important recent advance in the differential calculus of functions of non commuting operators

finite approximations of switched homogeneous systems for controller synthesis in this note we demonstrate the use of a control oriented notion of finite state input output approximation to synthesize correct by design controllers for hybrid plants under sensor limitations specifically we consider the problem of designing stabilizing switching controllers for a pair of unstable homogeneous second order systems with binary output feedback in addition to yielding a deterministic finite state approximate model of the hybrid plant our approach allows one to efficiently establish a useable upper bound on the quality of approximation and leads to a discrete optimization problem whose solution immediately provides a certified finite state controller for the plant the resulting controller consists of a deterministic finite state observer and a corresponding full state feedback control law

escape rates in hamiltonian systems particles escape from the vicinity of invariant tori is macroscopically modeled by a nonuniform diffusive process the space dependence of transport coefficients is fixed by using perturbation theory scalings this leads to universal predictions for the escape rates that are then observed in numerical simulations

trace formulas for resolvent comparable operators abstract we derive trace formulas for self adjoint perturbations of self adjoint operators such that the difference of the resolvents of the initial and perturbed operators is an element of a schatten von neumann ideal s n n n this substantially extends the result of 11 for perturbations corresponding to the resolvent difference in the hilbert schmidt class s 2 and the result of 13 for perturbations in s n n n

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

first integrals of autonomous systems of differential equations and the prelle singer procedure the prelle singer procedure for determining elementary first integrals of two dimensional autonomous systems of ordinary differential equations is introduced and how it can be generalized to higher dimensions is discussed the application of this procedure in several dynamical systems is reported

the banach saks p property we obtain general theorems which enable the calculation of the banach saks index set for rearrangement invariant function spaces in terms of their boyd indices for many important classes of rearrangement invariant spaces the index set is non trivial if and only if the boyd indices of the underlying space are non trivial although the latter equivalence does not hold in full generality we also study banach saks index sets defined for weakly null sequences of identically distributed independent random variables and their connections with the classical banach saks index set

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

weak type commutator and lipschitz estimates resolution of the nazarov peller conjecture let m be a semi finite von neumann algebra and let f r c be a lipschitz function if a b m are self adjoint operators such that a b l1 m then

resolution of peller s problem concerning koplienko neidhardt trace formulae a formula for the norm of a bilinear schur multiplier acting from the cartesian product s 2 s 2 of two copies of the hilbert schmidt classes into the trace class s 1 is established in terms of linear schur multipliers acting on the space s 1 of all compact operators using this formula we resolve peller s problem on koplienko neidhardt trace formulae namely we prove that there exist a twice continuously differentiable function f with a bounded second derivative a self adjoint unbounded operator a and a self adjoint operator b 2 s 2 such that f a b f a d dt f a tb t 0 2 s 1

higher order spectral shift abstract we construct higher order spectral shift functions extending the perturbation theory results of m g krein m g krein on a trace formula in perturbation theory mat sb 33 1953 597 626 in russian and l s koplienko l s koplienko trace formula for perturbations of nonnuclear type sibirsk mat zh 25 1984 62 71 in russian translation in trace formula for nontrace class perturbations siberian math j 25 1984 735 743 on representations for the remainders of the first and second order taylor type approximations of operator functions the higher order spectral shift functions represent the remainders of higher order taylor type approximations they can be expressed recursively via the lower order in particular krein s and koplienko s ones we also obtain higher order spectral averaging formulas generalizing the birman solomyak spectral averaging formula the results are obtained in the semi finite von neumann algebra setting with the perturbation taken in the hilbert schmidt class of the algebra

on a connection between kam and nekhoroshev s theorems abstract using the iteration of nekhoroshev s theorem as a basic tool we point out the existence of a hierarchic structure of nested domains underlying the phenomenon of diffusion at each level we find that the diffusions speed is exponentially small with respect to the previous level the set of kam tori is the domain characterized by a zero diffusion speed

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

the best constants for operator lipschitz functions on schatten classes suppose that f is a lipschitz function on r with kfklip 1 let a be a bounded self adjoint operator on a hilbert space h let p 1 and suppose that x b h is an operator such that the commutator a x is contained in the schatten class sp it is proved by the last two authors that then also f a x sp and there exists a constant cp independent of x and f such that k f a x kp cpk a x kp the main result of this paper is to give a sharp estimate for cp in terms of p namely we show that cp p 2 p 1 in particular this gives the best estimates for operator lipschitz inequalities we treat this result in a more general setting this involves commutators of n self adjoint operators a1 an for which we prove the analogous result the case described here in the abstract follows as a special case

on the role of high order resonances in normal forms and in separatrix splitting we discuss the role of high order resonances in the construction of normal forms for e close to integrable hamiltonian systems by heuristic considerations based on standard estimates we show that the remainder of normal forms is dominated by the terms corresponding to the main high order resonances and we provide a general argument to show that the size of such leading terms is exponentially small in 1e we apply this method to the problem of estimating the splitting of separatrices in resonant perturbed systems

efficient solution of optimal control problems using hybrid systems we consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem synthesize optimal enabling conditions for switching between locations in which the control is constant an algorithmic solution is obtained by translating the hybrid automaton to a finite automaton using a bisimulation and formulating a dynamic programming problem with extra conditions to ensure non zenoness of trajectories we show that the discrete value function converges to the viscosity solution of the hamilton jacobi bellman equation as a discretization parameter tends to zero

a dixmier trace formula for the density of states a version of connes trace formula allows to associate a measure on the essential spectrum of a schrodinger operator with bounded potential in solid state physics there is another celebrated measure associated with such operators the density of states in this paper we demonstrate that these two measures coincide we show how this equality can be used to give explicit formulae for the density of states in some circumstances

spectral shift function of higher order this paper resolves affirmatively koplienko s sib mat zh 25 62 71 1984 conjecture on existence of higher order spectral shift measures moreover the paper establishes absolute continuity of these measures and thus existence of the higher order spectral shift functions a spectral shift function of order n is the function n n h v such that operatorname tr biggl f h v sum k 0 n 1 frac 1 k frac d k dt k bigl f h tv bigr bigg t 0 biggr int mathbb r f n t eta n t dt for every sufficiently smooth function f where h is a self adjoint operator defined in a separable hilbert space and v is a self adjoint operator in the n th schatten von neumann ideal sn existence and summability of 1 and 2 were established by krein mat sb 33 597 626 1953 and koplienko sib mat zh 25 62 71 1984 respectively whereas for n 2 the problem was unresolved we show that n h v exists integrable and vert eta n vert l 1 mathbb r leq c n vert v vert s n n for some constant cn depending only on n our results for n rely on estimates for multiple operator integrals obtained in this paper our method also applies to the general semi finite von neumann algebra setting of the perturbation theory

differentiation of operator functions in non commutative lp spaces the principal results in this paper are concerned with the description of differentiable operator functions in the non commutative lp spaces 1 p associated with semifinite von neumann algebras for example it is established that if f r r is a lipschitz function then the operator function f is g teaux differentiable in l2 m for any semifinite von neumann algebra m if and only if it has a continuous derivative furthermore if f r r has a continuous derivative which is of bounded variation then the operator function f is g teaux differentiable in any lp m 1 p

chaotic hamiltonian systems survival probability we consider the dynamical system described by the area preserving standard mapping it is known for this system that p t the normalized number of recurrences staying in some given domain of the phase space at time t so called survival probability has the power law asymptotics p t ensuremath sim t ensuremath ensuremath nu we present new semiphenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective ultrametric diffusion on the boundary of a treelike space with hierarchically organized transition rates in the framework of our approach we have estimated the exponent ensuremath nu as ensuremath nu text ln text 2 text ln 1 r g ensuremath approx 1 44 where r g sqrt 5 ensuremath 1 2 is the critical rotation number

on deterministic control problems an approximation procedure for the optimal cost i the stationary problem we study deterministic optimal control problems having stopping time continuous and impulse controls in each strategy we obtain the optimal cost considered as the maximum element of a suitable set of subsolutions of the associated hamilton jacobi equation using an approximation method a particular derivative discretization scheme is employed convergence of approximate solutions is shown taking advantage of a discrete maximum principle which is also proved for the numerical solutions of approximate problems we use a method of relaxation type the algorithm is very simple it can be run on computers with small central memory in part i we study the stationary case in part ii siam j control optim 23 1985 pp 267 285 we study the nonstationary case

the optimal range of the calderon operator and its applications abstract we identify the optimal range of the calderon operator and that of the classical hilbert transform in the class of symmetric quasi banach spaces further consequences of our approach concern the optimal range of the triangular truncation operator operator lipschitz functions and commutator estimates in ideals of compact operators

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

the index formula and the spectral shift function for relatively trace class perturbations we compute the fredholm index index da of the operator da d dt a on l2 r h associated with the operator path a t t where af t a t f t for a e t r and appropriate f l2 r h via the spectral shift function a a associated with the pair a a of asymptotic operators a a on the separable complex hilbert space h in the case when a t is generally an unbounded relatively trace class perturbation of the unbounded self adjoint operator a we derive a formula an extension of a formula due to pushnitski relating the spectral shift function a a for the pair a a and the corresponding spectral shift function h2 h1 for the pair of operators h2 h1 dada da da in this relative trace class context h2 h1 1 1 2 1 2 a a d 2 1 2for a e 0 this formula is then used to identify the fredholm index of da with 0 a a in addition we prove that index da coincides with the spectral flow spflow a t t of the family a t t r and also relate it to the fredholm perturbation determinant for the pair a a index da spflow a t t 0 a a 1lime 0im ln deth a iei a iei 1 0 h2 h1 with the choice of the branch of ln deth on c such that limim z ln deth a zi a zi 1 0 we also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta invariant

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

fractional kinetic model for chaotic transport in nonintegrable hamiltonian systems we propose a kinetic model of transport in nonintegrable hamiltonian systems based on a fractional kinetic equation with spatially dependent diffusion coefficient the diffusion coefficient is estimated from the remainder of the optimal normal form for the given region of the phase space after partitioning the phase space into building blocks a separate equation can be constructed for each block solving the kinetic equations approximately and estimating the diffusion time scales we convolve the solutions to get the description of the macroscopic behavior we show that in the limit of infinitely many blocks one can expect an approximate scaling relation between the lyapunov time and the diffusion or escape time which is either an exponential or a power law we check our results numerically on over a dozen hamiltonians and find a good agreement

double operator integrals this paper is concerned with perturbation formulae of the form f a f b lp m k a b lp m with k 0 being a constant depending on p and f only where f is a real valued lipschitz function and a b are self adjoint measurable operators affiliated with a semifinite von neumann algebra m such that the difference a b belongs to lp m 1 p in order to treat the situation where the von neumann algebra m is not necessarily hyperfinite we first develop an integration theory with respect to finitely additive spectral measures in a banach space applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to birman and solomyak to describe the class of integrable functions we employ our recent study of multiplier theory in umd spaces our perturbation formulae extend those of davies and birman solomyak for the case when m is a hyperfinite i factor i e for the schatten p classes we also discuss analogous perturbation results in the setting of symmetric operator spaces associated with m

higher order s2 differentiability and application to koplienko trace formula abstract let a be a selfadjoint operator in a separable hilbert space k a selfadjoint hilbert schmidt operator and f c n r we establish that t f a t k f a is n times continuously differentiable on r in the hilbert schmidt norm provided either a is bounded or the derivatives f i i 1 n are bounded this substantially extends the results of 3 on higher order differentiability of in the hilbert schmidt norm for f in a certain wiener class as an application of the second order s 2 differentiability we extend the koplienko trace formula from the besov class b 1 2 r 20 to functions f for which the divided difference f 2 admits a certain hilbert space factorization

bmo estimates for non commutative vector valued lipschitz functions abstract we construct markov semi groups t and associated bmo spaces on a finite von neumann algebra m and obtain results for perturbations of commutators and non commutative lipschitz estimates in particular we prove that for any a m self adjoint and f r r lipschitz there is a markov semi group t such that for x m f a x bmo m t c a b s f a x we obtain an analogue of this result for more general von neumann valued functions f r n n by imposing hormander mikhlin type assumptions on f in establishing these result we show that markov dilations of markov semi groups have certain automatic continuity properties we also show that markov semi groups of double operator integrals admit standard and reversed markov dilations

beurling fourier algebras operator amenability and arens regularity abstract we introduce the class of beurling fourier algebras on locally compact groups and show that they are non commutative analogs of classical beurling algebras we obtain various results with regard to the operator amenability operator weak amenability and arens regularity of beurling fourier algebras on compact groups and show that they behave very similarly to the classical beurling algebras of discrete groups we then apply our results to study explicitly the beurling fourier algebras on su 2 the 2 2 special unitary group we demonstrate that how beurling fourier algebras are closely connected to the amenability of the fourier algebra of su 2 another major consequence of our results is that our investigation allows us to construct families of unital infinite dimensional closed arens regular subalgebras of the fourier algebra of certain products of su 2

a numerical study of the topology of normally hyperbolic invariant manifolds supporting arnold diffusion in quasi integrable systems abstract we investigate numerically the stable and unstable manifolds of the hyperbolic manifolds of the phase space related to the resonances of quasi integrable systems in the regime of validity of the nekhoroshev and kam theorems using a model of weakly interacting resonances we explain the qualitative features of these manifolds characterized by peculiar flower like structures we detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies we find numerically a correlation among these transitions and the speed of arnold diffusion

on the combinatorial and algebraic complexity of quantifier elimination in this paper a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given this algorithm improves the complexity of the asymptotically fastest algorithm for this problem known to this data a new feature of this algorithm is that the role of the algebraic part the dependence on the degrees of the imput polynomials and the combinatorial part the dependence on the number of polynomials are sparated another new feature is that the degrees of the polynomials in the equivalent quantifier free formula that is output are independent of the number of input polynomials as special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields and also for solving the existential problem in the first order theory over real closed fields are obtained

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

time optimal swing up of the planar pendulum this paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart this mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open loop controls relying on the theory of optimal synthesis we provide a discontinuous feedback giving optimal solutions for any initial data the approach is that of geometric control theory

an extension of the koplienko neidhardt trace formulae abstract koplienko sibirsk mat zh 25 5 1984 62 found a trace formula for perturbations of self adjoint operators by operators of hilbert schmidt class s 2 a similar formula in the case of unitary operators was obtained by neidhardt math nachr 138 1988 7 in this paper we improve their results and obtain sharp conditions under which the koplienko neidhardt trace formulae hold

preserving algebraic invariants with runge kutta methods abstract we study runge kutta methods for the integration of ordinary differential equations and the retention of algebraic invariants as a general rule we derive two conditions for the retention of such invariants the first is a condition on the coefficients of the methods the second is a pair of partial differential equations that otherwise must be obeyed by the invariant this paper extends previous work on multistep methods in iserles technical report na1997 13 damtp university of cambridge 1997 the cases related to the retention of quadratic and cubic invariants perhaps of greatest relevance in applications are thoroughly discussed we conclude recommending a generalized class of runge kutta schemes namely lie group type runge kutta methods these are schemes for the solution of odes on lie groups but can be employed together with group actions to preserve a larger class of algebraic invariants without restrictions on the coefficients

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

lipschitz continuity of the absolute value and riesz projections in symmetric operator spaces a principal result of the paper is that ifeis a symmetric banach function space on the positive half line with the fatou property then for all semifinite von neumann algebras m the absolute value mapping is lipschitz continuous on the associated symmetric operator spacee m with lipschitz constant depending only oneif and only ifehas non trivial boyd indices it follows that if m is any von neumann algebra then the absolute value map is lipschitz continuous on the corresponding haageruplp space provided 1 p

frechet differentiability of sp norms abstract one of the long standing questions in the theory of schatten von neumann ideals of compact operators is whether their norms have the same differentiability properties as the norms of their commutative counterparts we answer this question in the affirmative a key technical observation underlying our proof is a discovery of connection between this question and recent affirmative resolution of l s koplienko s conjecture concerning existence of higher order spectral shift functions

functions of unitary operators derivatives and trace formulas abstract let a be a bounded self adjoint operator on a separable hilbert space h and u 0 an arbitrary unitary operator we derive formulas for higher order derivatives of the function s e i s a u 0 for sufficiently smooth functions we compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of krein koplienko neidhardt

on the arazy conjecture concerning schur multipliers on schatten ideals abstract we confirm in the affirmative the conjecture made by arazy in 2 suppose that f is a continuously differentiable function on 1 1 such that f t c t t 1 1 for some constant c 0 and 0 let j j 1 l r for some 0 r if 0 p and 1 q satisfy 1 p r 1 q then the matrix f j f k j k j k 1 is a schur multiplier from the schatten class s q into s p we also consider a general version of this conjecture for arbitrary quasi banach ideals of compact operators

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p