hilbert schmidtness of some finitely generated submodules in h2 d2 abstract a closed subspace m of the hardy space h 2 d 2 over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions z 1 and z 2 whether every finitely generated submodule is hilbert schmidt is an unsolved problem this paper proves that every finitely generated submodule m containing z 1 z 2 is hilbert schmidt where is any finite blaschke product some other related topics such as fringe operator and fredholm index are also discussed

additive invariants on the hardy space over the polydisc abstract in recent years various advances have been made with respect to the nevanlinna pick kernels especially on the symmetric fock space while the development on the hardy space over the polydisc is relatively slow in this paper several results known on the symmetric fock space are proved for the hardy space over the polydisc the known proofs on the symmetric fock space make essential use of the nevanlinna pick properties specifically we study several integer valued numerical invariants which are defined on an arbitrary invariant subspace of the vector valued hardy spaces over the polydisc these invariants include the samuel multiplicity curvature fiber dimension and a few others a tool used to overcome the difficulty associated with non nevanlinna pick kernels is tauberian theory

invariant subspaces dilation theory and the structure of the predual of a dual algebra i on etudie la structure des preduaux d algebres d operateurs on obtient un nouveau theoreme de sous espace invariant on trouve des proprietes concernant le treillis sous espace invariant de l operateur decalage de bergman

markov snakes and superprocesses we suggest the name markov snakes for a class of path valued markov processes introduced recently by j f le gall in connection with the theory of branching measure valued processes le gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v v2 we establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of markov snakes and vice versa by using this theorem we get limit theorems for discrete markov snakes

a weighted occupation time for a class of measured valued branching processes a weighted occupation time is defined for measure valued processes and a representation for it is obtained for a class of measure valued branching random motions on rd considered as a process in its own right the first and second order asymptotics are found as time t specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d and when infinite a central limit type renormalization is considered yielding gaussian or asymmetric stable generalized random fields in the limit in one gaussian case the results are contrasted in high versus low dimensions

cores for parabolic operators with unbounded coefficients abstract let a i j 1 n q i j s x d i j i 1 n b i s x d i be a family of elliptic differential operators with unbounded coefficients defined in r n 1 in m kunze l lorenzi a lunardi nonautonomous kolmogorov parabolic equations with unbounded coefficients trans amer math soc in press under suitable assumptions it has been proved that the operator g a d s generates a semigroup of positive contractions t p t in l p r n 1 for every 1 p where is an infinitesimally invariant measure of t p t here under some additional conditions on the growth of the coefficients of a which cover also some growths with an exponential rate at we provide two different cores for the infinitesimal generator g p of t p t in l p r n 1 for p 1 and we also give a partial characterization of d g p finally we extend the results so far obtained to the case when the coefficients of the operator a are t periodic with respect to the variable s for some t 0

a countable representation of the fleming viot measure valued diffusion the fleming viot measure valued diffusion arises as the infinite population limit of various discrete genetic models with general type space the paper gives a countable construction of the process as the empirical measure carried by a certain interactive particle system this explicit representation facilitates the study of various properties of the fleming viot process the construction also carries versions of the familiar genealogical processes from population genetics in particular kingman s coalescent thus unifying the genealogical and measure valued approaches to the subject

birth delay of a markov process and the stopping distributions for regular processes we consider markov processes with a fixed transition functionp r x t b and with random birth times we show that a process tilde x t tilde p can be obtained from x t p by birth delay if and only if tilde p left tilde x t in b right leqq p left x t in b right for allt andb as an application we give a new version and a new proof of the results of rost r and fitzsimmons f2 on stopping distributions of markov processes the key lemma 1 1 replaces the filling scheme used by the previous authors birth delay was considered from a different prospective in f1

the spectral picture of reinhardt measures using a combination of techniques from the theory of groupoids the theory of several complex variables and single and multivariable operator theory we describe in detail the spectral picture of the commuting pair mz1 mz2 of multiplication operators by the coordinate functions z1 and z2 acting on the hilbert space p2 c z1 z2 l2 for an arbitrary reinhardt measure on c2

superprocesses of stochastic flows we construct a continuous superprocess x on m r d which is the unique weak feller extension of the empirical process of consistent k point motions generated by a family of differential operators the process x differs from known dawson watanabe type fleming viot type and ornstein uhlenbeck type superprocesses this new type of superprocess provides a connection between stochastic flows and measure valued processes and determines a stochastic coalescence which is similar to those of smoluchowski moreover the support of x describes how an initial measure on r d is transported under the flow as an example the process realizes a viewpoint of darling about the isotropic stochastic flows under certain conditions

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

on transitive algebras containing a standard finite von neumann subalgebra abstract let m be a finite von neumann algebra acting on a hilbert space h and a be a transitive algebra containing m in this paper we prove that if a is 2 fold transitive then a is strongly dense in b h this implies that if a transitive algebra containing a standard finite von neumann algebra in the sense of u haagerup the standard form of von neumann algebras math scand 37 1975 271 283 is 2 fold transitive then a is strongly dense in b h non selfadjoint algebras related to free products of finite von neumann algebras e g l f n and m 2 c 1 2 tr m 2 c 1 2 tr are studied brown measures of certain operators in m 2 c 1 2 tr m 2 c 1 2 tr are explicitly computed

joint extensions in families of contractive commuting operator tuples abstract in this paper we systematically study extension questions in families of commuting operator tuples that are associated with the unit ball in c d

operator theory in the hardy space over the bidisk iii this paper is a continuation of a project of developing a systematic operator theory inh2 d2 a large part of it is devoted to a study ofevaluation operator which is a very useful tool in the theory a number of elementary properties of the evaluation operator are exhibited and these properties are used to derive results in other topics such as interpretation of characteristic opertor function inh2 d2 spectral equivalence compactness compressions of shift operators etc even though some results reflect the two variable nature ofh2 d2 the goal of this paper is to manifest a close tie between the operator theory inh2 d2 and classical single operator theory the unilateral shift of a finite multiplicity and the bergman shift will be used as examples to illustrate some of the results

free resolutions in multivariable operator theory let ad be the complex polynomial ring in d variables a contractive ad module is hilbert space h equipped with an ad action such that for any 1 2 d h z1 1 z2 zd d 2 1 2 2 2 d 2 such objects have been shown to be useful for modeling d tuples of mutually commuting operators acting on a hilbert space there is a subclass of the category of contractive ad modules whose members play the role of free objects given a contractive ad module one can construct a free resolution i e an exact sequence of partial isometries of the following form 2f1 1f0 0h 0 where fi is a free module for each i 0 the notion of a localization of a free resolution will be defined in which for each bd there is a vector space complex of linear maps derived from 3 c2 2 c1 1 c0 we shall show that the homology of this complex is isomorphic to the homology of the koszul complex of the d tuple 1 2 d of where i is the ith coordinate function of a mobius transform on bd such that 0

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

multiple algebraisations of an elliptic calogero sutherland model recently gomez ullate et al phys lett b 511 112 2001 have studied an n particle quantum problem with elliptic function potentials they have shown that the hamiltonian operator preserves a finite dimensional space of functions and as such is quasi exactly solvable qes in this article we show that other types of invariant function spaces exist which are in close relation to the algebraic properties of the elliptic functions accordingly series of new algebraic eigenfunctions can be constructed

compactness and invariance properties of evolution operators associated with kolmogorov operators with unbounded coefficients abstract in this paper we consider nonautonomous elliptic operators a with nontrivial potential term defined in i r d where i is a right halfline possibly i r we prove that we can associate an evolution operator g t s with a in the space of all bounded and continuous functions on r d we also study the compactness properties of the operator g t s finally we provide sufficient conditions guaranteeing that each operator g t s preserves the usual l p spaces and c 0 r d

regular transition functions and regular superprocesses the class of regular markov processes is very close to the class of right processes studied by meyer getoor and others we say that a transition function p is regular if it is the transition function of a well defined regular markov process a characterization of regular transition functions is given which implies that if p is regular then the dawson watanabe and the flemingviot supertransition functions over p belong to the same class

hilbert polynomials and arveson s curvature invariant abstract we define and study hilbert polynomials for certain holomorphic hilbert spaces we obtain several estimates for these polynomials and their coefficients our estimates inspire us to investigate the connection between the leading coefficients of hilbert polynomials for invariant subspaces of the symmetric fock space and arveson s curvature invariant for coinvariant subspaces we are able to obtain some formulas relating the curvature invariant with other invariants in particular we prove that arveson s version of the gauss bonnet chern formula is true when the invariant subspaces are generated by any polynomials

superprocesses and their linear additive functionals let x x t p be a measure valued stochastic process linear functionals of x are the elements of the minimal closed subspace l of l 2 p which contains all x t b with x t b 2 dp we represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their laplace transforms we consider markov processes with nonstationary transition functions to reveal better the principal role of the backward equations

conditioned super brownian motion we investigate classes of conditioned super brownian motions namely h transformsp h with non negative finitely based space time harmonic functionsh t we prove thath h is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching brownian motions we identify the limit behaviour of h transforms with functionsh t h t 1 depending only on the total mass 1 using the palm measures of the super brownian motion we describe for an additive spacetime harmonic functionh t h t x dx theh transformp h as a conditioned super brownian motion in which an immortal particle moves like an h transform of brownian motion

dilations wandering subspaces and inner functions the objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on hilbert spaces it is shown that for a large class of analytic functional hilbert spaces h k on the unit ball in c n wandering subspaces for restrictions of the multiplication tuple m z m z1 m zn can be described in terms of suitable h k inner functions we prove that h k inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi homogeneous polynomials as an application along the way we prove a refinement of a result of arveson on the uniqueness of minimal dilations of pure row contractions c 2017 elsevier inc all rights reserved

invariant domains in the hardy space over the unit disk abstract inspired by some recent developments in operator theory on vector valued function spaces we identify and study a class of linear manifolds called invariant domains in hilbert spaces which implements a natural generalization of invariant subspaces for bounded operators for coordinate multiplication operator on the hardy space over the unit disc a beurling type characterization of invariant domains together with some algebraic properties is obtained we also go beyond the hardy space with some related discussions on the structure of graph invariant subspaces gis

lame equation sl 2 algebra and isospectral deformations it is proven that the n zone lame equation is equivalent to a spectral problem for a quadratic element of a universal enveloping sl 2 algebra the riemannian surface corresponding to the eigenstates in a parametric space forms the 2n 1 sheet surface which splits in four subsurfaces three of them contain the same number of sheets

a generalization of the cellular indecomposable property via fiber dimension abstract the cellular indecomposable property introduced by olin and thomson in 1984 11 is well known for the dirichlet space but it fails trivially for the vector valued case the purpose of this paper is to use the fiber dimension to reformulate the property such that it naturally extends the scalar valued case yet fix the vector valued case in a meaningful way using the new formulation we are able to generalize several previous results to the vector valued setting in particular we extend a theorem of bourdon relating the cellular indecomposable property and the codimension one property to codimension n several of our results appear to be new even for the hardy space over the unit disc

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

the green kubo formula for general markov processes with a continuous time parameter for general markov processes the green kubo formula is shown to be valid under a mild condition a class of stochastic evolution equations on a separable hilbert space and three typical infinite systems of locally interacting diffusions on irreversible in most cases are shown to satisfy the green kubo formula and the einstein relations for these stochastic evolution equations are shown explicitly as a corollary

path processes and historical superprocesses a superprocessx over a markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles the historical process hat xi for is the process whose state at timet is the path of over time interval 0 t the superprocess hat x over hat xi the historical superprocess over reflects not only the particle distribution at any fixed time but also the structure of family trees the principal property of a historical process hat xi is that hat xi s is a function of hat xi t for alls t every process with this property is calleda path process we develop a theory of superprocesses over path processes whose core is the integration with respect to measure functionals by applying this theory to historical superprocesses we construct the first hitting distributions and prove a special markov property for superprocesses

the dirac operator of a commuting d tuple abstract given a commuting d tuple t t1 td of otherwise arbitrary operators on a hilbert space there is an associated dirac operator dt significant attributes of the d tuple are best expressed in terms of dt including the taylor spectrum and the notion of fredholmness in fact all properties of t derive from its dirac operator we introduce a general notion of dirac operator in dimension d 1 2 that is appropriate for multivariable operator theory we show that every abstract dirac operator is associated with a commuting d tuple and that two dirac operators are isomorphic iff their associated operator d tuples are unitarily equivalent by relating the curvature invariant introduced in a previous paper to the index of a dirac operator we establish a stability result for the curvature invariant for pure d contractions of finite rank it is shown that for the subcategory of all such t that are a fredholm and and b graded the curvature invariant k t is stable under compact perturbations we do not know if this stability persists when t is fredholm but ungraded although there is concrete evidence that it does

spectral properties for a class of continuous state branching processes with immigration abstract we explicitly construct nontrivial invariant probability measures for a class of continuous state branching processes with immigration the class of these measures include random gamma measures and path space measures of levy subordinators as particular examples using the explicit construction we study long time behaviour and hypercontractivity of the transition semigroups in corresponding l 2 spaces

an invariant subspace of the bergman space having the codimension two property nutzungsbedingungen digizeitschriften e v gewahrt ein nicht exklusives nicht ubertragbares personliches und beschranktes recht auf nutzung dieses dokuments dieses dokument ist ausschlieslich fur den personlichen nicht kommerziellen gebrauch bestimmt das copyright bleibt bei den herausgebern oder sonstigen rechteinhabern als nutzer sind sie sind nicht dazu berechtigt eine lizenz zu ubertragen zu transferieren oder an dritte weiter zu geben die nutzung stellt keine ubertragung des eigentumsrechts an diesem dokument dar und gilt vorbehaltlich der folgenden einschrankungen sie mussen auf samtlichen kopien dieses dokuments alle urheberrechtshinweise und sonstigen hinweise auf gesetzlichen schutz beibehalten und sie durfen dieses dokument nicht in irgend einer weise abandern noch durfen sie dieses dokument fur offentliche oder kommerzielle zwecke vervielfaltigen offentlich ausstellen auffuhren vertreiben oder anderweitig nutzen es sei denn es liegt ihnen eine schriftliche genehmigung von digizeitschriften e v und vom herausgeber oder sonstigen rechteinhaber vor mit dem gebrauch von digizeitschriften e v und der verwendung dieses dokuments erkennen sie die nutzungsbedingungen an

the curvature of a hilbert module over ℂ z1 zd a notion of curvature is introduced in multivariable operator theory the curvature invariant of a hilbert module over z1 zd is a nonnegative real number which has significant extremal properties which tends to be an integer and which is hard to compute directly it is shown that for graded hilbert modules the curvature agrees with the euler characteristic of a certain finitely generated algebraic module over the appropriate polynomial ring this result is a higher dimensional operator theoretic counterpart of the gauss bonnet formula which expresses the average gaussian curvature of a compact oriented riemann surface as the alternating sum of the betti numbers of the surface and it solves the problem of calculating the curvature of graded hilbert modules the proof of that result is based on an asymptotic formula which expresses the curvature of a hilbert module in terms that allow its comparison to a corresponding asymptotic expression for the euler characteristic

spherical contractions and interpolation problems on the unit ball in this note fractional representations of multipliers on vector valued functional hilbert spaces are used to give a proof of arveson s version of von neumann s inequality for n contractions on the unit ball we prove a commutant lifting theorem for operators on the classical hardy space over the unit ball in mathbb c n as applications we obtain interpolation results for functions in the schur class we deduce a toeplitz corona theorem on the unit ball and we give a simplified definition of arveson s curvature invariant for n contractions with finite dimensional defect space in the final part we describe a solution of the operator valued nevanlinna pick problem with uniform bounds on uniqueness sets in the unit ball

isolated singularities of some semilinear elliptic equations abstract let be an open subset of r n n 3 containing 0 we consider the solutions of u x g u x f x in 0 where g is nondecreasing and f is bounded and we study the possible singularities at 0 when u x o x 1 n we prove that u is isotropic near 0 and show that either it is a c1 function in removable singularity or x n 2 u x c c 0 weak singularity or x n 2 u x strong singularity we also characterize the g s for which solutions with a weak singularity exist and improve a previous removability result of h brezis and l veron arch rational mech anal 23 1979 153 166

fiber dimension for invariant subspaces abstract in this paper we study the fiber dimension of invariant subspaces for a large class of operators we define a class of invariant subspaces called cf subspaces which are related to the codimension one property we obtain several characterizations of cf subspaces including one in terms of samuel multiplicity other new findings include 1 a lattice additive formula and its applications section 4 2 a new concept of absorbance which describes a rough containment relation for invariant subspaces section 5 3 the existence of a unique smallest cf subspace containing an arbitrary invariant subspace and preserving the fiber dimension section 6

essentially normal hilbert modules and k homology this paper mainly concerns the essential normality of graded submodules essentially all of the basic hilbert modules that have received attention over the years are p essentially normal including the d shift hilbert module the hardy and bergman modules of the unit ball arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer some positive results have been obtained by arveson and douglas however the problem has been resistant in dimensions d 2 3 this paper shows that the arveson s conjecture is true in any dimension the paper also gives an affirmative answer in the case of the graded principal submodule finally the paper is associated with k homology invariants arising from graded quotient modules by which geometry of the quotient modules and geometry of algebraic varieties are connected in dimensions d 2 3 it is shown that k homology invariants determined by graded quotients are nontrivial the paper also establishes results on p smoothness of k homology elements and gives an explicit expression for k homology invariant in dimension d 2

samuel multiplicity and the structure of semi fredholm operators abstract two numerical invariants refining the fredholm index are introduced for any semi fredholm operator in such a way that their difference calculates the fredholm index these two invariants are inspired by samuel multiplicity in commutative algebra and can be regarded as the stabilized dimension of the kernel and cokernel a geometric interpretation of these invariants leads naturally to a 4 4 uptriangular matrix model for any semi fredholm operator on a separable hilbert space this model can be regarded as a refined local version of the apostol s 3 3 triangular representation for arbitrary operators some classical results such as gohberg s punctured neighborhood theorem can be read off directly from our matrix model banach space operators are also considered

quotients of standard hilbert modules we initiate a study of hilbert modules over the polynomial algebra that are obtained by completing with respect to an inner product having certain natural properties a standard hilbert module is a finite multiplicity version of one of these standard hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra and their quotients by submodules give rise to universal solutions of nonlinear relations essentially all of the basic hilbert modules that have received attention over the years are standard including the hilbert module of the shift the hardy and bergman modules of the unit ball modules associated with more general domains in and those associated with projective algebraic varieties we address the general problem of determining when a quotient of an essentially normal standard hilbert module is essentially normal this problem has been resistant our main result is that it can be linearized in that the nonlinear relations defining the submodule can be reduced appropriately to linear relations through an iteration procedure and we give a concrete description of linearized quotients

the structure of inner multipliers on spaces with complete nevanlinna pick kernels let k be the reproducing kernel for a hilbert space h k of analytic functions on bd the open unit ball in cd d 1 k is called a complete np kernel if k0 1 and if 1 1 k z is positive definite on bd bd let d be a separable hilbert space and consider h k d h k d and think of it as a space of d valued h k functions a theorem of mccullough and trent j funct anal 178 2000 226 249 partially extends the beurling lax halmos theorem for the invariant subspaces of the hardy space h2 d they show that if k is a complete np kernel and if d is a separable hilbert space then for any scalar multiplier invariant subspace m of h k d there exists an auxiliary hilbert space e and a multiplication operator h k e h k d such that is a partial isometry and m h k e such multiplication operators are called inner multiplication operators and they satisfy the orthogonal projection onto m in this paper we shall show that for many interesting complete np kernels the analogy with the beurling lax halmos theorem can be strengthened we show that for almost every z bd the nontangential limit z of the multiplier bd b e d associated with an inner multiplication operator is a partial isometry and that rank z is equal to a constant almost everywhere the result applies to certain weighted dirichlet spaces and to the space h2d which is determined by the kernel k z 11 z d z bd in particular our result implies that the curvature invariant of arveson proc natl acad sci usa96 1999 11 096 11 099 of a pure contractive hilbert module of finite rank is an integer this answers a question of w arveson proc natl acad sci usa96 1999 11096 11099

fredholm and invertible tuples of operators the deformation problem using j l taylor s definition of joint spectrum we study fredholm and invertible tuples of operators on a hilbert space we give the foundations for a several variables theory including a natural generalization of atkinson s theorem and an index which well behaves we obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length we then consider the deformation problem and solve it for the class of almost doubly commuting fredholm pairs with a semi fredholm coordinate 1 introduction 1 let t be a bounded linear operator on a banach space t is said to be invertible if there exists an operator s on such that ts st 1 the identity operator on 9c by the open mapping theorem this is equivalent to ker t 0 and r t range of t the last formulation does not rely upon the existence of an inverse for t but rather on the action of the operator t when t is replaced by an tuple of commuting operators several definitions of nonsingular ity exist j l taylor 19 has obtained one which reflects the actions of the operators by considering the koszul complex associated with the tuple 2 in this paper we develop a general several variables theory on the basis of taylor s work and study commuting and almost commuting commuting mod ulo the compacts tuples of operators on a hilbert space we obtain a characterization of joint invertibility in terms of the invertibility of a single operator which is essential for our approach from that we get a number of corollaries which generalize nicely the known elementary results in one variable at the same time the referred characterization allows us to define a continuous invariant under compact perturbations integer valued index on the class of fred holm tuples those almost commuting tuples which are invertible in the calkin algebra this index extends the classical one for fredholm operators we prove that an almost commuting tuple of essentially normal operators with all commu tators in trace class has index zero 2 and that a natural generalization of atkinson s theorem holds for tuples

linear graph transformations on spaces of analytic functions let h be a hilbert space of analytic functions with multiplier algebra m h and let m f t 1 f t n 1 f f is an element of d be an invariant graph subspace for m h n here n 2 d subset of h is a vector subspace t i d h are linear transformations that commute with each multiplication operator m phi is an element of m h and m is closed in h n in this paper we investigate the existence of non trivial common invariant subspaces of operator algebras of the type a m a is an element of b h ad subset of d at i f t i af for all f is an element of d in particular for the bergman space l 0 2 we exhibit examples of invariant graph subspaces of fiber dimension 2 such that a m does not have any nontrivial invariant subspaces that are defined by linear relations of the graph transformations for m c 2015 elsevier inc all rights reserved less

invariant subspaces in banach spaces of analytic functions we study the invariant subspace structure of the operator of multiplication by z mz on a class of banach spaces of analytic functions for operators on hilbert spaces our class coincides with the adjoints of the operators in the cowendouglas class 1 q we say that an invariant subspace x satisfies cod x 1 if zx has codimension one in x we give various conditions on invariant subspaces which imply that codx 1 in particular we give a necessary and sufficient condition on two invariant subspaces x x with cod x cod x 1 so that their span again satisfies cod x v x 1 this result will be used to show that any invariant subspace of the bergman space l p 1 which is generated by functions in l2p must satisfy cod x 1 for an invariant subspace x we then consider the operator s m 1xl under some extra assumption on the domain of holomorphy we show that the spectrum of s coincides with the approximate point spectrum iff cod x 1 finally in the last section we obtain a structure theorem for invariant subspaces with cod x 1 this theorem applies to dirichlet type spaces

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

a note on superprocesses subject to a mild restriction ona generator of the one particle motion we show thea fleming viot superprocess can be obtained from thea dawson watanabe superprocess by conditioning the latter to have constant total mass

invariant probability distributions for measure valued diffusions we investigate the set of invariant probability distributions for measure valued diffusion processes corresponding to semilinear operators of the form u t l 0 u u u 2 where l 0 1 2 d i j 1 a i j 2 d i 1 b i

on the supports of measure valued critical branching brownian motion on etudie la taille globale et la localisation des supports des etats du processus de markov a valeur de mesure qui est un analogue du mouvement brownien ramifie critique classique sur l espace euclidien a d dimensions

extended calogero models a construction for exactly solvable kn body systems we propose a systematic procedure for the construction of exactly solvable kn body systems which are natural generalisations of calogero models as examples we present two new 3n body models and determine explicit expressions for their eigenvalues and eigenfunctions

the fredholm index of a pair of commuting operators ii this paper concerns fredholm theory in several variables and its applications to hilbert spaces of analytic functions one feature is the introduction of ideas from commutative algebra to operator theory specifically we introduce a method to calculate the fredholm index of a pair of commuting operators to achieve this we define and study the hilbert space analogs of samuel multiplicities in commutative algebra then the theory is applied to the symmetric fock space in particular our results imply a satisfactory answer to arveson s program on developing a fredholm theory for pure d contractions when d 2 including both the fredholmness problem and the calculation of indices we also show that arveson s curvature invariant is in fact always equal to the samuel multiplicity for an arbitrary pure d contraction with finite defect rank it follows that the curvature is a similarity invariant

natural linear additive functionals of superprocesses we investigate natural linear additive nla functionals of a general critical xi k psi superprocess x we prove that all of them have only fixed discontinuities all homogeneous nla functionals of time homogeneous superprocesses are continuous this was known before only in the case of quadratic branching we introduce an operator mathscr e u defined in terms of xi k psi and we prove that the potential h and the log potential u of a nla functional a are connected by the equation u mathscr e u h the potential is always an exit rule for xi and the condition h mathscr e h infty a e is sufficient for an exit rule h to be a potential in an accompanying paper these results are applied to boundary value problems for partial differential equations involving nonlinear operator lu u alpha where l is a second order elliptic differential operator and 1 alpha leq 2

transitive algebras and reductive algebras on reproducing analytic hilbert spaces in this paper we consider the well known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic hilbert spaces for an analytic hilbert space h k with complete nevanlinna pick kernel k it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of h k c m have positive answer if the algebras contain all analytic multiplication operators this extends several known results on the problems

curvature invariant on noncommutative polyballs abstract in this paper we develop a theory of curvature resp multiplicity invariant for tensor products of full fock spaces f 2 h n 1 f 2 h n k and also for tensor products of symmetric fock spaces f s 2 h n 1 f s 2 h n k this is an attempt to find a more general framework for these invariants and extend some of the results obtained by arveson for the symmetric fock space by the author and kribs for the full fock space and by fang for the hardy space h 2 d k over the polydisc to prove the existence of the curvature and its basic properties in these settings requires a new approach based on noncommutative berezin transforms and multivariable operator theory on polyballs and varieties as well as summability results for completely positive maps the results are presented in the more general setting of regular polyballs

coupling and ergodic theorems for fleming viot processes fleming viot processes are probability measure valued diffusion processes that can be used as stochastic models in population genetics here we use duality methods to prove ergodic theorems for fleming viot processes including those with recombination coupling methods are also used to establish ergodicity of fleming viot processes first without and then with selection a special type of selection known as symmetric overdominance is treated by other methods

cyclic row contractions and rigidity of invariant subspaces abstract it is known that pure row contractions with one dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard d shift acting on the full fock space upon settling for a softer relation than unitary equivalence we relax the defect condition and simply require the row contraction to admit a cyclic vector we show that cyclic pure row contractions can be transformed in a precise technical sense into compressions of the standard d shift cyclic decompositions of the underlying hilbert spaces are the natural tool to extend this fact to higher multiplicities we show that such decompositions face multivariate obstacles of an algebraic nature nevertheless decompositions are obtained for nilpotent commuting row contractions by analyzing function theoretic rigidity properties of their invariant subspaces

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

large deviations for the three dimensional super brownian motion let t dx denote a three dimensional super brownian motion with deterministic initial state 0 dx dx the lebesgue measure let v r 3 r be holder continuous with compact support not identically zero and such that r 3v x dx 0 we show that log p 0 t r 3v x s dx ds bt 3 4 is of order t 1 2 as t for b 0 this should be compared with the known result for the case r 3v x dx 0 in that case the normalization bt 3 4 b 0 must be replaced by bt b r 3v x dx in order that the same statement hold true while this result only captures the logarithmic order the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems

occupation time large deviations for critical branching brownian motion super brownian motion and related processes we derive a large deviation principle for the occupation time functional acting on functions with zero lebesgue integral for both super brownian motion and critical branching brownian motion in three dimensions our technique based on a moment formula of dynkin allows us to compute the exact rate functions which differ for the two processes obtaining the exact rate function for the super brownian motion solves a conjecture of lee and remillard we also show the corresponding clt and obtain similar results for the superprocesses and critical branching process built over the symmetric stable process of index in r d with d 2 2 d

on regularity of superprocesses three theorems on regularity of measure valued processesx with branching property are established which improve earlier results of fitzsimmons f1 and the author d5 the main difference is that we treatx as a family of random measures associated with finely open setsq in time space heuristically x describes an evolution of a cloud of infinitesimal particles to everyq there corresponds a random measurex which arises if each particle is observed at its first exit time fromq the statex t at a fixed timet is a particular case we consider a monotone increasing familyq t of finely open sets and we establish regularity properties of bar x t x tau t as a function oft the results are used in d6 d7 and d10 for investigating the relations between superprocesses and non linear partial differential equations basic definitions on markov processes and superprocesses are introduced in sect 1 the next three sections are devoted to proving the regularity theorems they are applied in sect 5 to study parts of superprocess the relation to the previous work is discussed in more detail in the concluding section it may be helpful to look briefly through this section before reading sects 2 5

polylinear additive functionals of superprocesses abstract letxbe a superdiffusion in a domaineof r d a polylinear additive functional ofxcorresponding to a positive borel function is given by the formula a b b dt 1 dt k e k t 1 z 1 t k z k x t 1 dz 1 x t k dz k by a passage to the limit we extend this definition to a certain class of generalized functions more precisely we associate an additive functionala of xt p with every measure subject to the condition dt 1 dz 1 dt k dz k dt 1 dz 1 dt k dz k emsp q t 1 z 1 t k dz k t 1 z 1 t k dz k for a certain kernelq we prove that if t ti s 0 fori 1 kand for alls then measurea has a s the same property this result is applicable in particular to self intersection local times ofx

invariant subspaces and nevanlinna pick kernels abstract a theorem of beurling lax halmos represents a subspace m of h2 c d the hardy space of analytic functions with values in the hilbert space e and square summable power series invariant for multiplication by z as h2 f where f is a subspace of e and is an inner function with values in l f e when the hardy space is replaced by the hilbert space h k determined by a nevanlinna pick kernel k such as the dirichlet kernel or the row contraction kernel on the ball in c d the blh theorem survives with f an auxiliary hilbert space and a l f e valued function which is inner in the sense that the operator m of multiplication by is a partial isometry under mild additional hypotheses when e c mz the operator of multiplication by z is cellularly indecomposable and has the codimension one property however if m is invariant for mz m mz m need not be a cyclic subspace for mz restricted to m

two representations of a conditioned superprocess we consider a class of measure valued markov processes constructed by taking a superprocess over some underlying markov process and conditioning it to stay alive forever we obtain two representations of such a process the first representation is in terms of an immortal particle that moves around according to the underlying markov process and throws off pieces of mass which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess as a consequence of this representation we show that the tail field of the conditioned superprocess is trivial if the tail field of the underlying process is trivial the second representation is analogous to one obtained by legall in the unconditioned case it represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process this representation is useful for studying the transience and recurrence properties of the closed support process

nonlinear parabolic p d e and additive functionals of superdiffusions suppose that e is an arbitrary domain in r d l is a second order elliptic differential operator in s r x e and s e is the extremal part of the martin boundary for the corresponding diffusion let 1 2 we investigate a boundary value problem u r lu u in s u v on s e u 0 on x e involving two measures and for the existence of a solution we give sufficient conditions in terms of a martin capacity and necessary conditions in terms of hitting probabilities for an l superdiffusion x if a solution exists then it can be expressed by an explicit formula through an additive functional a of x an l superdiffusion is a branching measure valued process a natural linear additive nla functional a of x is determined uniquely by its potential h defined by the formula p a 0 h r x dr dx for all m the determining set of a every potential h is an exit rule for and it has a unique decomposition into extremal exit rules if and are measures which appear in this decomposition then can be replaced by an integral equation u r x p r x t dy u t y ds h r x where p r x t dy is the transition function of we prove that h is the potential of a nla functional if and only if has a solution u moreover u r x log p r x e a 0 by applying these results to homogeneous functionals of time homogeneous superdiffusions we get a stronger version of theorems proved in an earlier publication the foundation for our present investigation is laid by a general theory developed in the accompanying paper

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

solvability of the hamiltonians related to exceptional root spaces rational case solvability of the rational quantum integrable systems related to exceptional root spaces g2 f4 is re examined and for e6 7 8 is established in the framework of a unified approach it is shown that hamiltonians take algebraic form being written in certain weyl invariant variables it is demonstrated that for each hamiltonian the finite dimensional invariant subspaces are made from polynomials and they form an infinite flag a notion of minimal flag is introduced and minimal flag for each hamiltonian is found corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants the hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite dimensional but finitely generated lie algebra of differential operators taken in a finite dimensional representation