set valued versions of ky fan s inequality with application to variational inclusion theory in this paper we prove two set valued versions of ky fan s minimax inequality from these results versions of schauder s and kakutani s fixed point theorems can be deduced we formulate a variational inclusion problem for set valued maps and a differential inclusion problem concerning the contingent derivative sufficient conditions for the existence of solution for these inclusion problems are obtained generalizing classical variational inequality problems

toeplitz operators and solvable c algebras on hermitian symmetric spaces bounded symmetric domains cartan domains and exceptional domains are higher dimensional generalizations of the open unit disc in this note we give a structure theory for the c algebra t generated by all toeplitz operators tf h p fh with continuous symbol function g c s on the shilov boundary 5 of a bounded symmetric domain d of arbitrary rank r here h belongs to the hardy space h s and p l s h s is the szego projection for domains of rank 1 and tube domains of rank 2 the structure of t has been determined in 1 2 in these cases toeplitz operators are closely related to pseudodifferential operators for the open unit disc t is the c algebra generated by the unilateral shift the structure theory for the general case 12 is based on the fact that d can be realized as the open unit ball of a unique jordan triple system z 7 theorem 4 1 denoting the jordan triple product by uv w a tripotent e g z satisfies ee e e tripotents generalize the partial isometries of matrix algebras and determine the boundary structure of d c z cf 7 theorem 6 3 our principal result 12 cf also 3 4 8 is the following

hua operators poisson transform and relative discrete series on line bundles over bounded symmetric domains let d g k be a bounded symmetric domain and s k l be its shilov boundary we consider the action of g with weight nu in mathbb z on functions on d viewed as sections the line bundle and the corresponding eigenspace of g invariant di erential operators the poisson transform maps hyperfunctions on the s to the eigenspaces we characterize the image in terms of hua operators on the sections of the line bundle for some special parameter the poisson transform is of szego type mapping into the relative discrete series we compute the corresponding elements in the discrete series

variational conclusions of set valued bifunctions on convex subsets of banach spaces with applications in this paper variational conclusions of set valued bifunctions on convex subsets of banach spaces are investigated and several new results are obtained the results are applied to the fixed point theory and variational inequalities we obtain two fixed point theorems and existence theorems of solutions of variational inequalities

on the convergence in distribution of measurable multifunctions random sets normal integrands stochastic processes and stochastic infima the concept of the distribution function of a closed valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi convergence in distribution of normal integrands and stochastic processes in particular various compactness criteria are exhibited the connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of donsker s theorem brownian motion as a limit of normalized random walks we also suggest the potential application of the theory to the study of the convergence of stochastic infima

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

c algebras of multiplication operators on bergman spaces soit un domaine borne dans c soit a 2 l espace de bergman de toutes les fonctions analytiques sur qui sont de carre integrable par rapport a la mesure volumetrique et soit c la c algebre engendree par les operateurs de multiplication par les fonctions coordonnee sur on etudie les proprietes de c

homogeneous vector bundles and intertwining operators for symmetric domains the main features of homogeneous cowen douglas operators well known for the unit disk are generalised to the setting of hermitian bounded symmetric domains of arbitrary rank c 2016 elsevier inc all rights reserved

homogeneous multiplication operators on bounded symmetric domains let d g k be an irreducible bounded symmetric domain of dimension d and let h d be the analytic continuation of the weighted bergman spaces of holomorphic functions on d we consider the d tuple m m1 md of multiplication operators by coordinate functions and consider its spectral properties we find those parameters for which the tuple m is subnormal and we answer some open questions of bagchi and misra in particular we prove that when d bd is the unit ball in cd then bd is a k spectral set of m if and only if h bd is the hardy space or a weighted bergman space

analytic models for commuting operator tuples on bounded symmetric domains for a domain in c d and a hilbert space h of analytic functions on which satisfies certain conditions we characterize the commuting d tuples t t 1 t d of operators on a separable hilbert space h such that t is unitarily equivalent to the restriction of m to an invariant subspace where m is the operator d tuple z i on the hilbert space tensor product h h for the unit disc and h the hardy space h 2 this reduces to a well known theorem of sz nagy and foias for h a reproducing kernel hilbert space on c c d such that the reciprocal 1 k x y of its reproducing kernel is a polynomial in x and y this is a recent result of ambrozie muller and the second author in this paper we extend the latter result by treating spaces h for which 1 k ceases to be a polynomial or even has a pole namely the standard weighted bergman spaces or rather their analytic continuation h h on a cartan domain corresponding to the parameter v in the continuous wallach set and reproducing kernel hilbert spaces h for which 1 k is a rational function further we treat also the more general problem when the operator m is replaced by m w w being a certain generalization of a unitary operator tuple for the case of the spaces h on cartan domains our results are based on an analysis of the homogeneous multiplication operators on which seems to be of an independent interest

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

non vanishing functions and toeplitz operators on tube type domains abstract we prove an index theorem for toeplitz operators on irreducible tube type domains and we extend our results to toeplitz operators with matrix symbols in order to prove our index theorem we proved a result asserting that a non vanishing function on the shilov boundary of a tube type bounded symmetric domain not necessarily irreducible is equal to a unimodular function defined as the product of powers of generic norms times an exponential function

applications of ky fan s inequality on σ compact set to variational inclusion and n person game theory in this paper ky fan s inequality on compact set is applied to variational inclusions and n person game theory we give results of some variational inclusions and existence of non cooperative equilibrium in n person game on compact set

degenerate principal series representations and their holomorphic extensions let x h l be an irreducible real bounded symmetric domain realized as a real form in an hermitian symmetric domain d g k the intersection s of the shilov boundary of d with x defines a distinguished subset of the topological boundary of x and is invariant under h it can be realized as s h p for certain parabolic subgroup p of h we study the spherical representations of h induced from p we find formulas for the spherical functions in terms of the macdonald hypergeometric function this generalizes the earlier result of faraut koranyi for hermitian symmetric spaces d we consider a class of h invariant integral intertwining operators from the representations on l2 s to the holomorphic representations of g restricted to h we construct a new class of complementary series for the groups h so n m su n m with n m 2 and sp n m with n m 1 we realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of g su n m su n m su n m and su 2n 2m respectively

reproducing kernel functions and asymptotic expansions on jordan kepler manifolds abstract we study the complex geometry of generalized kepler manifolds defined in jordan theoretic terms introduce hilbert spaces of holomorphic functions defined by radial measures and find the complete asymptotic expansion of the corresponding reproducing kernels for kahler potentials both in the flat and bounded setting

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

dixmier trace for toeplitz operators on symmetric domains abstract for toeplitz operators on bounded symmetric domains of higher rank there is no obvious way to define the dixmier trace within the toeplitz c algebra since commutators are in general not compact in this paper we solve this problem by constructing a hilbert quotient module corresponding to partitions of length 1 which leads to commutators in the macaev class l n we also obtain an explicit formula for the dixmier trace of such commutators in terms of the underlying boundary geometry which involves a new type of flag manifold defined in jordan theoretic terms

spectral conditions for admissibility of evolution equations in hilbert space abstract we study properties of solutions of the evolution equation u t b u t f t where b is a closable operator on the space ap r h of almost periodic functions with values in a hilbert space h such that b commutes with translations the operator b generates a family b of closed operators on h such that b e i t x e i t b x whenever e i t x d b for a closed subset r we prove that the following properties i and ii are equivalent i for every function f ap r h such that f there exists a unique mild solution u ap r h of eq such that u ii b i is invertible for all and sup b i 1

the berezin transform and invariant differential operators the berezin calculus is important to quantum mechanics creation annihilation operators and operator theory toeplitz operators we study the basic berezin transform linking the contravariant and covariant symbol for all bounded symmetric domains and express it in terms of invariant differential operators

variational inclusions for contingent derivative of set valued maps in this paper we give two versions of ky fan s inequality for set valued maps acting between normed vector spaces and we consider sufficient conditions to solve a variational inclusion problem concerning derivatives of set valued maps a selection result for set valued maps between finite dimensional vector spaces and its contingent derivative is obtained as well from this result we derive some conditions for the existence of a solution of a generalized variational inequality problem

toeplitz quantization and asymptotic expansions for real bounded symmetric domains an analogue of the star product familiar from deformation quantization is studied in the setting of real bounded symmetric domains the analogue turns out to be a certain invariant operator which one might call star restriction from functions on the hermitification of the domain into functions on the domain itself in particular we establish the usual i e semiclassical asymptotic expansion of this star restriction and further obtain a real variable analogue of a theorem of arazy and orsted concerning the analogous expansion for the berezin transform

calculus rules on a new set valued second order derivative for convex functions etant donnee une fonction convexe f qui a ete construite a partir d autres fonctions f i dont les differentielles secondes sont aisees a determiner on cherche a exprimer les differentielles secondes de f en fonction des 2 f i

on the existence and stability of approximate solutions of perturbed vector equilibrium problems abstract in this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of painleve kuratowski to the initial set then we introduce the concept of approximate solution for equilibrium problem governed by set valued maps and we study the stability of these solutions the particular case of linear continuous operators is considered as well

toeplitz operators on bounded symmetric domains in this paper jordan algebraic methods are applied to study toeplitz operators on the hardy space h2 s associated with the shilov boundary s of a bounded symmetric domain d in c of arbitrary rank the jordan triple system z c associated with d is used to determine the relationship between toeplitz operators and differential operators further it is shown that each jordan triple idempotent e e z induces an irreducible representation e symbol of the c algebra generated by all toeplitz operators tf with continuous symbol function f 0 introduction toeplitz operators on the boundary t aa of the open unit disc a c c play an important role in function theory of one complex variable cf 9 chapter 7 in several dimensions toeplitz operators have been mainly studied for strictly pseudo convex domains d c c 4 13 14 24 28 in particular for the hilbert ball 7 using the relationship with pseudo differential operators another class of domains in c generalizing the unit disc is the class of bounded symmetric domains cartan domains and exceptional domains which have a more complicated boundary structure compared to the strictly pseudo convex case the aim of this paper is the study of toeplitz operators t1 with continuous symbol functionf e c s on the shilov boundary s of a bounded symmetric domain d of arbitrary rank r in the special cases of the hilbert ball r 1 and the lie ball r 2 the structure of the operators ti and of the toeplitz c algebra

index theory for toeplitz operators on bounded symmetric domains in this note we give an index theory for toeplitz operators on the hardy space of the shilov boundary of an arbitrary bounded symmetric domain our results generalize earlier work of gohberg krein and venugopalkrishna 12 for domains of rank 1 and of berger coburn koranyi 1 for domains of rank 2 bounded symmetric domains cartan domains classical or exceptional are the natural higher dimensional analogues of the open unit disk each such domain is homogeneous under a semisimple lie group of biholomorphic transformations and has an essentially unique realization as a convex circular domain which is conveniently described in jordan algebraic terms the underlying vector space z c n carries a jordan triple product denoted by uyv w uv w and the bounded symmetric domain d is the open unit ball of z with respect to the spectral norm 7 the basic example is the space z c of rectangular matrices with jordan triple product uv w uv w wv u 2 giving rise to the hyperbolic matrix ball d z g z spectrum z z 1 the symmetric domains of rank 1 are the smooth hilbert balls in c n for domains d of higher rank r the boundary dd forms a nonsmooth stratified space in jordan algebraic terms the structure of dd can be described as follows an element e z satisfying ee e e is called a tripotent for matrices the tripotents are just the partial isometries every tripotent e induces a splitting of z into the peirce spaces z e z g z ee z xz for a 0 1 the set

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

directional differentiability of metric projections onto moving sets at boundary points soient x y des espaces normes de dimension finie et x y une multifonction assignant a chaque xex un ensemble ferme x cy a x sont associees la fonction distance d x y r et la projection metrique a valeur d ensemble x y y on etudie les proprietes differentielles de au voisinage d un point x 0 y 0 avec y 0 point limite de l ensemble x 0

evolutionary semigroups and lyapunov theorems in banach spaces we present a spectral mapping theorem for continuous semigroups of operators on any banach space e the condition for the hyperbolicity of a semigroup on e is given in terms of the generator of an evolutionary semigroup acting in the space of e valued functions the evolutionary semigroup generated by the propagator of a nonautonomous differential equation in e is also studied a discrete technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity exponential dichotomy of the nonautonomous equation