the homogeneous shifts abstract a bounded linear operator t on a complex hilbert space is called homogeneous if the spectrum of t is contained in the closed unit disc and all bi holomorphic automorphisms of this disc lift to automorphisms of the operator modulo unitary equivalence we prove that all the irreducible homogeneous operators are block shifts therefore as a first step in classifying all of them it is natural to begin with the homogeneous scalar shifts in this paper we determine all the homogeneous scalar weighted shifts they consist of the unweighted bilateral shift two one parameter families of unilateral shifts adjoints of each other a one parameter family of bilateral shifts and a two parameter family of bilateral shifts this classification is obtained by a careful analysis of the possibilities for the projective representation of the mobius group associated with an irreducible homogeneous shift

semismoothness of solutions to generalized equations and the moreau yosida regularization we show that a locally lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point we present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible nonpolyhedral convex sets we prove that the semismoothness of solutions to the moreau yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function

existence of weakly efficient solutions in nonsmooth vector optimization in this paper we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems we consider problems whose objective functions are defined between infinite and finite dimensional banach spaces our results are stated under hypotheses of generalized convexity and make use of variational like inequalities

v subdifferentials of convex operators abstract for any convex operator from a convex set c of a topological vector space e into another one f endowed with a convex cone f a notion of v subdifferential v a of at a c is introduced although it is equivalent to the notion of e sub differential when f r it enjoys many important properties which are not satisfied by the e subdifferential whenever int f o the nonvacuity of v a is proved whenever v is a neighbourhood of zero in f and belongs to a large class of mappings analogous to the class of lower semicontinuous real valued convex functions other properties of v subdifferentials are studied and applications to differentiability of are made

shimura invariant differential operators and their eigenvalues let bar d be the covariant cauchy riemann operator and mathcal d the covariant holomorphic differential operator on a line bundle over a hermitian symmetric space g k we study the shimura invariant differential operators defined via bar d and mathcal d we find the eigenvalues of a family of the shimura operators and of the generators

on quotient modules the case of arbitrary multiplicity let m be a hilbert module of holomorphic functions defined on a bounded domain cm let m0 be the submodule of functions vanishing to order k on a hypersurface z in in this paper we describe the quotient module mq

prevalence an addendum to prevalence a translation invariant almost every on infinite dimensional spaces bull amer math soc n s 27 1992 no 2 217 238 mr1161274 93k 28018

the lagrange multiplier rule in set valued optimization the known lagrange multiplier rule is extended to set valued constrained optimization problems using the contingent epiderivative as differentiability notion a necessary optimality condition for weak minimizers is derived which is also a sufficient condition under generalized convexity assumptions

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

subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions abstract several kinds of behaviors of extended real valued lower semicontinuous functions are known to be equivalent to certain appropriate conditions in terms of the clarke subdifferential the paper provides a systematic study showing that any such condition with the clarke subdifferential is valid if and only if it holds with any operator representing the clarke subdifferential like in the subdifferential proximal formula

branching laws for minimal holomorphic representations abstract in this paper we study the branching law for the restriction from su n m to so n m of the minimal representation in the analytic continuation of the scalar holomorphic discrete series we identify the group decomposition with the spectral decomposition of the action of the casimir operator on the subspace of s o n o m invariants the plancherel measure of the decomposition defines an l 2 space of functions for which certain continuous dual hahn polynomials furnish an orthonormal basis it turns out that the measure has point masses precisely when n m 2 under these conditions we construct an irreducible representation of so n m identify it with a parabolically induced representation and construct a unitary embedding into the representation space for the minimal representation of su n m

homogeneous tuples of multiplication operators on twisted bergman spaces letbthe bergman kernel on the domain n mofn mcontractive complex matrices m n 1 let w wn mbe the associated wallach set consisting of the 0 for whichb m n is non negative definite and hence the reproducing kernel of a functional hilbert space h for w we examine themn tuplem of operators on h whose components are multiplications by themnco ordinate functions this tuple is homogeneous with respect to the group action ofpsu n m on the matrix ball utilising this group action we are able to determine the set of all w for which i m is bounded and for which ii m is bounded and jointly subnormal further the joint taylor spectrum ofm is determined for all as in i the subnormality ofm turns out to be closely tied with the representation theory ofpsu n m namely m is subnormal precisely when the natural projective representation ofpsu n m on the twisted bergman space h is a subrepresentation of an induced representation of multiplicity 1 finally we examine the values of for whichm admits its taylor spectrum as ak spectral set and obtain incomplete results on this question this question remains open and interesting onn 1 gaps that is for belonging to the union ofn 1 pairwise disjoint open intervals most of the techniques developed in this paper are applicable to all bounded cartan domains though we stick to the matrix domains n mfor concreteness

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

existence theorems for generalized noncoercive equilibrium problems the quasi convex case we provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems an extension of generalized quasi variational inequalities defined in reflexive banach spaces in the quasi convex case in addition several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems in fact we find some particular cones as estimates for the recession cone of the solution set these cones for the ones containing the latter set are proved to be sharp enough to encompass several special situations found in the literature

rankin cohen brackets transvectants and covariant differential operators we give a geometric construction of the transvectant on a hermitian symmetric space which in the case of the unit disk is also called the rankin cohen bracket in terms of the covariant differentiation

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

necessary conditions for the bolza problem in this paper we provide necessary conditions for the bolza problem which are obtained by using a generalization of frechet derivative and by reducing the original problem to a convex problem the investigation of this convex problem is carried out by approximating the given functions by smooth ones

a classification of homogeneous operators in the cowen douglas class a complete list of homogeneous operators in the cowen douglas class b n mathbb d is given this classification is obtained from an explicit realization of all the homogeneous hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi holomorphic automorphism group of the unit disc

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

branching problems in reproducing kernel spaces for a semisimple lie group g satisfying the equal rank condition the most basic family of unitary irreducible representations is the discrete series found by harish chandra in our work here we study some of the branching laws for discrete series when restricted to a subgroup h of the same type by combining classical results with recent work of kobayashi in particular we prove discrete decomposability under harish chandra s condition of cusp form on the reproducing kernel we show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators

classification and time evolution of density matrices for a n state system a possible method of classifying all density matrices for a n state quantum system into different types using the sameness of the characteristic polynomial of density matrices belonging to a given type is discussed the method employs the su n generator expansion of density matrices it is demonstrated that density matrices belonging to a given type although different share many common properties including the same eigenvalues the same shape of the region in the parametric space in which lie the allowed parameter values and the same global averages of the pertinent observables furthermore we discuss the time evolution of a given type of density matrices and show that in the appropriately defined time dependent comoving matrix basis a density matrix of a given type belongs during entire unitary evolution to the same type in this way one is able to group and organize the multitude of different density matrices with certain analogous properties into a smaller number of types thus systematizin

local monotonicity and full stability for parametric variational systems this paper introduces and characterizes new notions of lipschitzian and holderian full stability of solutions to general parametric variational systems defined via partial subdifferential of prox regular functions acting in finite dimensional and hilbert spaces these notions which postulate certain quantitative properties of single valued localizations of solution maps are closely related to local strong maximal monotonicity of associated set valued mappings based on advanced tools of variational analysis and generalized differentiation we derive verifiable characterizations of the local strong maximal monotonicity and full stability notions under consideration via some positive definiteness conditions involving second order constructions of variational analysis

berger measure for s a b c d abstract in this note we get the berger measure of the weighted shift s a b c d with weights n a n b c n d a b c d 0 and n 0 as well as its p subshift then we will give examples from analytic function theory to illustrate that the necessary condition given by curto and yoon 2006 12 for subnormality of 2 variable weighted shift is not sufficient furthermore we will do some calculations about subnormal backward extension of s a b c d

the classification of three parameter density matrices for a qutrit the properties and structure of the eight dimensional parameter space of three state density matrices are examined using the su 3 generator expansion three dimensional cross sections of the space of generalized bloch vectors are determined by the monte carlo sampling method and an appropriate classification of 56 different three parameter density matrices into ten types according to the shape of the corresponding region of allowed values of three relevant parameters is introduced positions of the representative points corresponding to the pure states are determined for all ten types

nonsmooth milyutin dubovitskii theory and clarke s tangent cone in optimization theory the idea of approximating nonconvex sets by convex cones which satisfy an abstract condition the intersection principle is due to milyutin and dubovitskii this approach has been successfully applied to optimization problems with differentiable data the validity of the intersection principle for the clarke tangent cone which is the apposite approximant for nonsmooth constraints is established via an intersection theorem which relates the clarke tangent cone of an intersection of sets to the intersection of the individual clarke tangent cones provided the latter are inseparable this allows an extension of the milyutin dubovitskii approach to nonsmooth optimization theory a multiplier rule of the type obtained by clarke and hiriart urruty for a nonsmooth inequality constrained problem is thereby encompassed within milyutin dubovitskii theory

tensor products of holomorphic representations and bilinear differential operators let h be the weighted bergman space on a bounded symmetric domain d g k it has analytic continuation in the weight and for in the so called wallach set h still forms unitary irreducible projective representations of g we give the irreducible decomposition of the tensor product h 1 h 2 of the representations for any two unitary weights and we find the highest weight vectors of the irreducible components we find also certain bilinear differential intertwining operators realizing the decomposition and they generalize the classical transvectants in invariant theory of sl 2 c as applications we find a generalization of the bol s lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product h 1 h 2 modulo the subspace of functions vanishing of certain degree on the diagonal

classification of quasi homogeneous holomorphic curves and operators in the cowen douglas class abstract in this paper we study quasi homogeneous operators which include the homogeneous operators in the cowen douglas class we give two separate theorems describing canonical models with respect to equivalence under unitary and invertible operators respectively for these operators using techniques from complex geometry this considerably extends the similarity and unitary classification of homogeneous operators in the cowen douglas class obtained recently by the last author and a koranyi in a significant generalization of the properties of the homogeneous operators we show that quasi homogeneous operators are irreducible and determine which of them are strongly irreducible applications include the equality of the topological and algebraic k group of a quasi homogeneous operator and an affirmative answer to a well known question of halmos

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

on the local representation of piecewise smooth equations as a lipschitz manifold we study systems of equations f x 0 given by piecewise differentiable functions f from r n to r k k n the focus is on the representability of the solution set locally as an n k dimensional lipschitz manifold for that nonsmooth versions of inverse function theorems are applied it turns out that their applicability depends on the choice of a particular basis to overcome this obstacle we introduce a strong full rank assumption sfra in terms of clarke s generalized jacobians the sfra claims the existence of a basis in which clarke s inverse function theorem can be applied aiming at a characterization of sfra we consider also a full rank assumption fra the fra insures the full rank of all matrices from the clarke s generalized jacobian the article is devoted to the conjectured equivalence of sfra and fra for min type functions we give reformulations of sfra and fra using orthogonal projections basis enlargements cross products dual variables as well as via exponentially many convex cones the equivalence of sfra and fra is shown to be true for min type functions in the new case k 3

on reducing submodules of hilbert modules with g n invariant kernels fix a bounded domain omega subset of c n and a positive definite kernel k on omega both invariant under g n the permutation group on n symbols let h subset of hol omega be the hilbert module determined by k we show that h splits into orthogonal direct sum of subspaces pph indexed by the partitions p of n we prove that each submodule pph is a locally free hilbert module of rank equal to square of the dimension chi p 1 of the irreducible representation corresponding to p given two partitions p and q we show that if chi p 1 not equal chi q 1 then the sub modules pph and pqh are not unitarily equivalent we prove that if h is a contractive analytic hilbert module on omega then the taylor joint spectrum of the n tuple of multiplication operators by elementary symmetric polynomials on pph is clos s omega where s c n c n is the symmetrization map it is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials cz gn in n variables equipped with the sup norm on clos s omega c 2018 elsevier inc all rights reserved

branching laws for discrete wallach points we consider the projective representations of the group of holomorphic automorphisms of a symmetric tube domain v i that are obtained by analytic continuation of the holomorphic discrete series for a representation corresponding to a discrete point in the wallach set we find the decomposition under restriction to the identity component of gl using riesz distributions an explicit intertwining operator is constructed as an analytic continuation of an integral operator the density of the plancherel measure involves quotients of functions and the c function for a symmetric cone of smaller rank

homogeneous operators on hilbert spaces of holomorphic functions i in this paper we construct a large class of multiplication operators on reproducing kernel hilbert spaces which are homogeneous with respect to the action of the m o bius group consisting of bi holomorphic automorphisms of the unit disc mathbb d for every m in n we have a family of operators depending on m 1 positive real parameters the kernel function is calculated explicitly it is proved that each of these operators is bounded lies in the cowen douglas class of mathbb d and is irreducible these operators are shown to be mutually pairwise unitarily inequivalent

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

toeplitz c algebras on bounded symmetric domains in this paper we develop a structure theory for the toeplitz c algebra y for an arbitrary bounded symmetric domain d in c classical or exceptional as the main result it is shown that 5y is a c algebra of type i which is solvable i e admits a finite composition sequence of closed ideals sk in 7 such that the subquotients fk i k are continuous fields of elementary c algebras of compact operators the length of this composition sequence coincides with the rank r of d as a hermitian symmetric space for 0 k r the spectral components sk of ay parametrizing the irreducible representations of y are realized in terms of the boundary components complex analytic faces of d the structure theory for y is based on the fact that the tangent space z c c of d carries a jordan algebraic structure z is a jordan triple system with triple product uv w describing the geometry of d in algebraic terms for example d is the open unit ball with respect to the so called spectral norm on z and the boundary components of d are in 1 1 correspondence with the tripotents e e z partial isometries satisfying ee e 4 e in particular the shilov boundary s of d consists of all tripotents of maximal rank r the jordan theoretic description of d was used in 17 to determine an explicit peter weyl decomposition of the hardy space h2 s cf 15 in terms of the norm functions generalized determinants associated with the jordan triple

ideal weakly efficient solutions for vector optimization problems we establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded the solutions to be referred are ideal strong weakly efficient weakly pareto also several necessary and or sufficient conditions for the solution set to be non empty and compact are established moreover some characterizations of the non emptiness boundedness of the convex solution set in case the solutions are searched in a subset of the real line are also given however the solution set fails to be convex in general in addition special attention is addressed when the underlying cone is the non negative orthant and when the semi strict quasiconvexity of each component of the vector valued function is assumed our approach is based on the asymptotic description of the functions and sets some examples illustrating such an approach are also exhibited

about contingent epiderivatives abstract in this paper we analyze some questions concerning the contingent epiderivative when the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives we also investigate the relationship between the contingent derivative and the generalized epiderivative furthermore we give existence theorems with respect to daniell cones as a particular case we study set valued maps that take values in r n

frechet differentiability of regular locally lipschitzian functions abstract this paper considers frechet differentiability almost everywhere in the sense of category of regular locally lipschitzian real valued functions defined on open subsets of a banach space it is first shown that for separable banach spaces clarke s generalized gradient of such a function is a minimal convex and compact valued upper semicontinuous multifunction using a theorem of christensen and kenderov it is then shown that for separable asplund spaces such a function is frechet differentiable on a dense g subset of its domain

infinite dimensional generalized jacobian properties and calculus rules the extension to infinite dimensional domains of clarke s generalized jacobian is the focus of this paper first a generalization of a fabian preiss theorem to the infinite dimensional setting is obtained as a consequence a new formula relating the clarke s generalized jacobians corresponding to finite dimensional spaces k l with k l is established furthermore in the infinite dimensional case basic properties pertaining the generalized jacobian are developed and then an identification of this set valued map is produced applications of these results in the form of chain rules including sum and product rules and a computational formula for continuous selections are derived

tangent sets calculus and necessary conditions for extremality in this paper we deduce some relations concerning the tangent sets to the inverse image of a set under a function and we use these relations to derive necessary conditions for extremality in the form of a multiplier rule

higher order radial derivatives and optimality conditions in nonsmooth vector optimization abstract we propose notions of higher order outer and inner radial derivatives of set valued maps and obtain main calculus rules some direct applications of these rules in proving optimality conditions for particular optimization problems are provided then we establish higher order optimality necessary conditions and sufficient ones for a general set valued vector optimization problem with inequality constraints a number of examples illustrate both the calculus rules and the optimality conditions in particular they explain some advantages of our results over earlier existing ones and why we need higher order radial derivatives

the curvature invariant for a class of homogeneous operators for an operator t in the class b n introduced by cowen and douglas the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle e t determine the unitary equivalence class of the operator t in a subsequent paper the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant derivatives are necessary to determine the unitary equivalence class of the operator t is an element of b n here we show that some of the covariant derivatives are necessary our examples consist of homogeneous operators in b n for homogeneous operators the simultaneous unitary equivalence class of the curvature and all its covariant derivatives at any point w in the unit disc are determined from the simultaneous unitary equivalence class at 0 this shows that it is enough to calculate all the invariants and compare them at just one point say 0 these calculations are then carried out in number of examples one of our main results is that the curvature along with its covariant derivative of order 0 1 at 0 determines the equivalence class of generic homogeneous hermitian holomorphic vector bundles over the unit disc

equivalence of quotient hilbert modules ii for any open connected and bounded set c m let a be a natural function algebra consisting of functions holomorphic on let m be a hilbert module over the algebra a and m0 m be the submodule of functions vanishing to order k on a hypersurface z recently the authors have obtained an explicit complete set of unitary invariants for the quotient module q m m0 in the case of k 2 in this paper we relate these invariants to familiar notions from complex geometry we also find a complete set of unitary invariants for the general case we discuss many concrete examples in this setting as an application of our equivalence results we characterise certain homogeneous hilbert modules over the bi disc algebra one source of fascination in the study of operator theory is the wide variety of connections made with other branches of mathematics techniques from algebra topology geometry and analysis are used to understand bounded linear operators on hilbert space and in many instances the behavior and properties of the operators can be used to illustrate critical features and aspects of the other fields this is particularly true in the case of multivariate operator theory that is when several operators or an algebra of operators is studied here the setting and results from these other areas can be quite sophisticated and the techniques used to understand multivariate operator theory often require additional development such is the focus of this paper although the spectral theorem is a key tool in the study of self adjoint and normal opera tors there are large and important classes of naturally occurring operators to which this theory doesn t apply examples illustrating such phenomena can be obtained by considering multipli cation operators on spaces of holomorphic functions on some domain in c m for domains in c one is in the realm of single operator theory while it is multivariate operator theory for m 1 if one considers the unit ball b m in c m and the bergman space a 2 b m for it one obtains a

positivity of the n n density matrix expressed in terms of polarization operators we use polarization operators known from the quantum theory of angular momentum to expand the n n dimensional density operators thereby we construct generalized bloch vectors representing density matrices we study their properties and derive positivity conditions for any n we also apply the procedure to study bloch vector space for a qubit and a qutrit

chain rules for nonsmooth functions abstract a systematic method is presented for the derivation of chain rules for compositions of functions f f where f is nondecreasing this method is valid for directional derivatives and subgradients associated with any tangent cone having a short list of properties some major special cases are examined in detail in particular calculus rules are derived for rockafellar s epi derivatives and clarke generalized gradients

hermitian symmetric spaces and their unitary highest weight modules abstract the purpose of this article is to determine the set of unitarizable highest weight modules corresponding to hermitian symmetric spaces of the noncompact type the major step is that of proving unitarity at the last possible place with this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem bernstein gelfand gelfand and a diagramatic representation of the set of positive noncompact roots

lipschitz functions with prescribed derivatives and subderivatives in general it is difficult to construct lipschitz functions which are not directly built up from either convex or distance functions one impediment to such constructions is that outside of the real line it is difficult to find anti derivatives the main result of this paper provides under suitable circumstances a technique for constructing such anti derivatives more precisely we show that if f 1 f 2 dots f n are continuously differentiable real valued locally lipschitz functions defined on a non empty open subset a of a separable banach space x then there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co nabla f 1 x nabla f 2 x dots nabla f n x this same construction also shows that for any finite family t 1 t 2 dots t n of maximal cyclically monotone mappings from a into non empty subsets of x there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co t 1 x t 2 x dots t n x moreover we show that g is convex if and only if t 1 t 2 cdots t n

parametrizations of kojima s system and relations to penalty and barrier functions we investigate two homotopies that perturb kojima s system for describing critical points of a nonlinear optimization problem in finite dimension each of them characterizes stationary points of a usual penalty and a new barrier function the latter is a continuous deformation of the objective symmetric to the penalty from a formal point of view stationary points of these functions appear as perturbed critical points and vice versa this permits new interpretations of the related solution methods and allows estimates of the solutions by using implicit function theorems for lipschitzian equations

bloch vectors for qudits we present three different matrix bases that can be used to decompose density matrices of d dimensional quantum systems so called qudits the generalized gell mann matrix basis the polarization operator basis and the weyl operator basis such a decomposition can be identified with a vector the bloch vector i e a generalization of the well known qubit case and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices we present a new method to decompose density matrices via so called standard matrices consider the important case of an isotropic two qudit state and decompose it according to each basis in the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements which is appropriate for an experimental realization

subdifferentials of nonconvex vector valued functions abstract the formula of clarke s subdifferential for the sum of two real valued locally lipschitz functions has been extended by rockafellar to the case where one of the two functions is directionally lipschitzian we introduce in this paper a notion of cone related to the epigraph of a function and with the aid of this cone we extend the directionally lipschitzian behaviour to vector valued functions and we study the sum of two vector valued functions

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

implicit multifunction theorems with positively homogeneous maps abstract we extend robinson s and ledyaev and zhu s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps allowing us to obtain results that more closely resemble the classical single valued implicit function theorem we highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results as part of our analysis we study perturbations of generalized linear openness and metric regularity finally we discuss how our methods may also be adapted to study generalized calmness

toeplitz algebras subnormal tuples and rigidity on reproducing c z1 zd modules this paper mainly considers toeplitz algebras subnormal tuples and rigidity concerning reproducing c z1 zd modules by making use of arveson s boundary representation theory we find there is more rigidity in several variables than there is in single variable we specialize our attention to reproducing c z1 zd modules with u invariant kernels by examining the spectrum and the essential spectrum of the d tuple mz1 mzd and deducing an exact sequence of c algebras associated with toeplitz algebra finally we deal with toeplitz algebras defined on arveson submodules and rigidity of arveson submodules

which subgradients have sum formulas on donne une formule de somme generale valable pour un cone tangent avec une courte liste de proprietes

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

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

radial epiderivatives and asymptotic functions in nonconvex vector optimization the notions of lower and upper radial epiderivatives a radial notion different from that generally discussed for nonconvex vector valued functions are introduced many of their properties are established and some of the optimality conditions for a point to be an ideal pareto or weak pareto minimum involving these epiderivatives are also presented in particular a characterization for the ideal minima in terms of the lower radial epiderivative is proved such a result appears to be new in the literature under some mild assumptions on the given function it is proved that the asymptotic cone of its epigraph is the epigraph of its upper radial epiderivative moreover given a vector minimization problem we describe the asymptotic behavior of its solution set by introducing some cones of asymptotic directions of the function involved finally we define the lower and upper radial subdifferentials and express the optimality conditions by means of these subdifferentials certainly these optimality conditions subsume various necessary or sufficient conditions found in the literature for convex or nonconvex functions

a chain rule for first and second order epiderivatives and hypoderivatives a chain rule is established for contingent and adjacent epiderivatives and hypoderivatives of compositions g h where h is assumed to be hadamard directionally differentiable corollaries include a formula for the contingent and adjacent cones of an equality constraint set defined by a hadamard directionally differentiable function an analogous chain rule for second order contingent and adjacent epiderivatives and hypoderivatives is also developed

nonsmooth calculus in finite dimensions the notion of subgradient originally defined for convex functions has in recent years been extended via the upper subderivative to cover functions that are not necessarily convex or even continuous a number of calculus rules have been proven for these generalized subgradients this paper develops the finite dimensional generalized subdifferential calculus for strictly lower semicontinuous functions under considerably weaker hypotheses than those previously used the most general finite dimensional convex subdifferential calculus results are recovered as corollaries other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program various chain rule formulations are considered equality in the subdifferential calculus formulae is proven under hypotheses weaker than the usual subdifferential regularity assumptions