symmetric coherent choquet capacities choquet capacities are a generalization of probability measures that arise in robustness decision theory and game theory many capacities that arise in robustness are symmetric or can be transformed into sym metric capacities we characterize the extreme points of the set of upper distribution functions corresponding to coherent symmetric choquet ca pacities on 0 1 we also show that the set of 2 altemating capacities is a simplex and we give a choquet representation of this set

multilinear schur multipliers and schatten properties of operator taylor remainders abstract we establish sharp conditions on scalar functions and perturbations that guarantee schatten summability of n th order operator taylor remainders in the special case of dimension one our estimates of these remainders deliver well known classical estimates of scalar taylor remainders we prove that if a scalar function f is in the set c n and a perturbation is in the p th schatten class s p p n then the respective n th order operator taylor remainder is an element of s p n and has an estimate like the one in 16 we construct examples of f c n and perturbations in s n such that the n th order taylor remainder of the respective operator function is not in s 1 our construction relies in particular on novel dimension dependent estimates for schatten norms of multilinear schur multipliers from below that are of interest in their own right our results apply to both self adjoint and unitary operators

trace formulas for relative schatten class perturbations abstract we derive trace formulas for a pair of self adjoint operators h v and h under the assumption that h i 1 v is in a schatten class this extends the trace formulas of 8 where v alone is assumed to be in a schatten class our trace formulas apply in particular in the setting of differential operators and are based on taylor like approximations of operator functions this significantly improves non taylor based trace formulas of 10

on positivity of spectral shift functions abstract we show that every spectral shift function of an even order 2 k is nonnegative outside the convex hull of the spectrum of an initial operator cvh h every spectral shift function of an odd order 2 k 1 is nonnegative respectively nonpositive outside cvh h whenever a perturbation is nonnegative respectively nonpositive we also derive several sufficient conditions for positivity of 2 k and 2 k 1 on the whole real line

trace formulas for additive and non additive perturbations abstract trace formulas for pairs of self adjoint maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained in particular the existence of a complex valued spectral shift function for a pair h h of maximal accumulative operators has been proved we investigate also the existence of a real valued spectral shift function moreover we treat in detail the case of additive trace class perturbations assuming that h and h h v are maximal accumulative and v is trace class we prove the existence of a summable complex valued spectral shift function we also obtain trace formulas for pairs h h assuming only that h and h are resolvent comparable in this case the determinant of the characteristic function of h is involved in trace formulas the results rely on the technique of boundary triplets for densely and non densely defined closed symmetric operators the approach allows to express the spectral shift function of a pair of extensions in terms of the abstract weyl function and the boundary operator we improve and generalize certain classical results of m g kre n for pairs of self adjoint and dissipative operators results of a v rybkin for such pairs as well as of v m adamjan b s pavlov and m g kre n for pairs h h with a maximal dissipative operator h

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

stability and uniqueness properties of taylor approximations of matrix functions abstract we consider taylor approximations of matrix functions and establish bounds controlling the size of a perturbation in terms of the size of the taylor remainder or equivalently the associated spectral shift function s this can be viewed as a stability property for the inverse problem of recovering the perturbation from the spectral shift function s from our bounds we derive a uniqueness result for spectral sums if the trace of the taylor remainder of order n 2 equals zero for a finite number of monomials then the perturbation is zero this uniqueness result is in contrast with nonuniqueness of the first order taylor remainder as an application we characterize equality of two graphs in terms of self returning walks

bayes theorem for choquet capacities we give an upper bound for the posterior probability of a measurable set a when the prior lies in a class of probability measures p the bound is a rational function of two choquet integrals if p is weakly compact and is closed with respect to majorization then the bound is sharp if and only if the upper prior probability is 2 alternating the result is used to compute bounds for several sets of priors used in robust bayesian inference the result may be regarded as a characterization of 2 alternating choquet capacities

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

singular traces and perturbation formulae of higher order abstract let h v be self adjoint operators such that v belongs to the weak trace class ideal we prove higher order perturbation formula f h v j 0 n 1 1 j d j d t j f h t v t 0 r f n t d m n t where is a trace on the weak trace class ideal and m n is a finite measure that is not necessarily absolutely continuous this result extends the first and second order perturbation formulas of dykema and shripka who generalised the krein and koplienko trace formulas to the weak trace class ideal we also establish the perturbation formulae when the perturbation v belongs to the quasi banach ideal weak l n for any n 1

koplienko spectral shift function on the unit circle the note shows that for every pair of contractions a trace formula of koplienko holds with a summable spectral shift function see theorem 1 below this answers a question by f gesztesy a pushnitski b simon see zh mat fiz anal geom 4 1 63 107 202 2008 open question 11 2

generic type results for chaotic dynamics in equations with periodic forcing terms abstract we prove that a class of equations containing the classical periodically forced pendulum problem displays the main features of chaotic dynamics for a set of forcing terms open and dense in suitable spaces the approach is based on global variational methods

multiple operator integrals and higher operator derivatives abstract in this paper we consider the problem of the existence of higher derivatives of the function t a tk where is a function on the real line a is a self adjoint operator and k is a bounded self adjoint operator we improve earlier results by sten kin in order to do this we give a new approach to multiple operator integrals this approach improves the earlier approach given by sten kin we also consider a similar problem for unitary operators

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

symmetric upper probabilities as a first step toward developing statistical models based on upper and lower probabilities we study upper probabilities and upper expectations on the unit interval that are symmetric by which we mean invariant with respect to equimeasurability these upper probabilities are generalizations of uniform probability measures we give some characterizations of these upper probabilities specifically we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization we also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star shaped with respect to the origin and to the point 1 1 we derive inner and outer approximations to symmetric classes of probabilities based on the upper probability the class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized we provide a geometric characterization of a hierarchy of upper probabilities including fine s generalized upper probabilities and 2 alternating choquet capacities in particular we establish a 1 1 correspondence between symmetric 2 alternating capacities and nonincreasing density functions we prove that undominated generalized upper probabilities do not exist in the symmetric case examples from robust statistics are considered an example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations a corollary is that symmetry of the choquet integral does not imply symmetry of the upper expectation functional

simultaneously least favorable experiments a concept of worst case sufficiency is defined generalizing le cam s approximate sufficiency instead of using total variation norm as did le cam 1964 neighborhoods are described by upper expectations a corresponding version of the theorem of le cam blackwell sherman stein is proved in the case of finite parameter space as a main tool serve standard experiments and their upper limits here to be called upper standard functionals a characterization of simultaneously least favorable experiments dominated by a family of upper expectations is proved it says that least favorable experiments exist if and only if the upper standard functional acts additively on a cone of concave functions

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

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

tracial bounds for multilinear schur multipliers abstract we derive new types of two sided bounds for traces of multilinear schur multipliers that sharpen existing bounds in a number of situations we also characterize the tracial positivity of the n linear multipliers on symmetric tuples assuming their symbols are mildly sparse when n 5

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

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

solution set splitting at low energy levels in schrodinger equations with periodic and symmetric potential abstract the time independent superlinear schrodinger equation with spatially periodic and positive potential admits sign changing two bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set moreover we provide a recipe to explicitly verify the condition and we carry out the calculation in dimension one for a specific class of potentials

special capacities the hunt stein theorem and transformation groups the problem of maximin testing between families of probability measures generated by special capacities and transformation groups is solved by showing that the hunt stein theorem is applicable to such generally nondominated families of probability measures and that the problem reduces to maximin testing between neighbourhoods of distributions of maximal invariant defined by special capacities closely related to the initial ones an application to testing approximate binormality is also sketched

on the huber strassen theorem we show that the huber strassen theorem can be applied to many common robust neighborhood models without restrictive compactness assumptions our method is to replace the original polish topology with a compact topology which is possible according to the kuratowski isomorphism theorem