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

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

efficient algorithms for universal portfolios a constant rebalanced portfolio is an investment strategy that keeps the same distribution of wealth among a set of stocks from day to day there has been much work on cover s universal algorithm which is competitive with the best constant rebalanced portfolio determined in hindsight cover 1991 helmbold et al 1998 blum and kalai 1999 foster and vohra 1999 vovk 1998 cover and ordentlich 1996a cover 1996c while this algorithm has good performance guarantees all known implementations are exponential in the number of stocks restricting the number of stocks used in experiments helmbold et al 1998 cover and ordentlich 1996a ordentlich and cover 1996b cover 1996c blum and kalai 1999 we present an efficient implementation of the universal algorithm that is based on non uniform random walks that are rapidly mixing applegate and kannan 1991 lovasz and simonovits 1992 frieze and kannan 1999 this same implementation also works for non financial applications of the universal algorithm such as data compression cover 1996c and language modeling chen et al 1999

w gauge structures and their anomalies an algebraic approach starting from flat two dimensional gauge potentials we propose the notion of w gauge structure in terms of a nilpotent becchi rouet stora brs differential algebra the decomposition of the underlying lie algebra with respect to an sl 2 subalgebra is crucial for the discussion of conformal covariance in particular the appearance of a projective connection different sl 2 embeddings lead to different w gauge structures we present a general soldering procedure which allows one to express zero curvature conditions for the w currents in terms of conformally covariant differential operators acting on the w gauge fields and to obtain at the same time the complete nilpotent brs differential algebra generated by w currents gauge fields and the ghost fields corresponding to w diffeomorphisms as illustrations we treat the cases of sl 2 itself and the two different sl 2 embeddings in sl 3 viz the w 1 3 and w 2 3 gauge structures in some detail in these cases we determine algebraically w anomalie

switching strategies for sequential decision problems with multiplicative loss with application to portfolios a wide variety of problems in signal processing can be formulated such that decisions are made by sequentially taking convex combinations of vector valued observations and these convex combinations are then multiplicatively compounded over time a ldquouniversalrdquo approach to such problems might attempt to sequentially achieve the performance of the best fixed convex combination as might be achievable noncausally by observing all of the outcomes in advance by permitting different piecewise fixed strategies within contiguous regions of time the best algorithm in this broader class would be able to switch between different fixed strategies to optimize performance to the changing behavior of each individual sequence of outcomes without knowledge of the data length or the number of switches necessary the algorithms developed in this paper can achieve the performance of the best piecewise fixed strategy that can choose both the partitioning of the sequence of outcomes in time as well as the best strategy within each time segment we compete with an exponential number of such partitions using only complexity linear in the data length and demonstrate that the regret with respect to the best such algorithm is at most o ln n in the exponent where n is the data length finally we extend these results to include finite collections of candidate algorithms rather than convex combinations and further investigate the use of an arbitrary side information sequence

holomorphic isometries of mathbb b m into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals we study general properties of images of holomorphic isometric embeddings of complex unit balls mathbb b m into irreducible bounded symmetric domains varomega of rank at least 2 in particular we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections varlambda of the compact dual x c of varomega the question naturally arises as to which linear sections z varlambda cap varomega are actually images of holomorphic isometries of complex unit balls we study the latter question in the case of bounded symmetric domains varomega of type iv alias lie balls i e bounded symmetric domains dual to hyperquadrics we completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains varomega especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by mok proc am math soc 144 4515 4525 2016 from varieties of minimal rational tangents and that moreover any holomorphic isometric embedding f mathbb b m rightarrow varomega extends to a holomorphic isometric embedding f mathbb b n 1 rightarrow varomega dim varomega n the case of lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type iv domain are expected to be more rigid

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

hilbert series of nearly holomorphic sections on generalized flag manifolds abstract let x g p be a complex flag manifold and e x be a g homogeneous holomorphic vector bundle fix a u invariant kahler metric on x with u g maximal compact we study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in e coincides with the space of u finite smooth sections in e the degree of nearly holomorphic sections defines a u invariant filtration on this space using sheaf cohomology we determine in suitable cases the corresponding hilbert series it turns out that this is given in terms of lusztig s q analog of kostant s weight multiplicity formula and hence gives a new representation theoretic interpretation of these formulas

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

fast universalization of investment strategies a universalization of a parameterized investment strategy is an online algorithm whose average daily performance approaches that of the strategy operating with the optimal parameters determined offline in hindsight we present a general framework for universalizing investment strategies and discuss conditions under which investment strategies are universalizable we present examples of common investment strategies that fit into our framework the examples include both trading strategies that decide positions in individual stocks and portfolio strategies that allocate wealth among multiple stocks this work extends in a natural way cover s universal portfolio work we also discuss the runtime efficiency of universalization algorithms while a straightforward implementation of our algorithms runs in time exponential in the number of parameters we show that the efficient universal portfolio computation technique of kalai and vempala proceedings of the 41st annual ieee symposium on foundations of computer science redondo beach ca 2000 pp 486 491 involving the sampling of log concave functions can be generalized to other classes of investment strategies thus yielding provably good approximation algorithms in our framework

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 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

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

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

bounded geometry for kleinian groups we show that a kleinian surface group or hyperbolic 3 manifold with a cusp preserving homotopy equivalence to a surface has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants bounded geometry is a positive lower bound on the lengths of closed geodesics when the surface is a once punctured torus the coefficients coincide with the continued fraction coefficients associated to the ending laminations

on a class of nearly holomorphic automorphic forms with a b 1 e c and 0 n e z the point that attracts our attention is its nonholomorphic nature it is noteworthy that f is holomorphic if k 2 and also such nonholomorphy doesn t appear in the hilbert modular case therefore one may naturally ask whether this is an isolated oddity in our previous papers 10 and 13 we showed that it was not so in fact a similar eisenstein series on sp n q or on su n n of a particular weight can have the form

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

the plancherel formula for spherical functions with a one dimensional k type on a simply connected simple lie group of hermitian type abstract we generalize the harish chandra inversion formula for the spherical transform on a riemannian symmetric space to homogeneous line bundles on a hermitian symmetric space we determine the plancherel measure explicitly

fonctions holomorphes a croissance moderee et vecteurs distributions the space of distribution vectors of a holomorphic discrete series representation of scalar type can be identified to a space of holomorphic functions with moderate growth this is the main result of the paper we give an application of it to the hankel transform and to multivariate laguerre series expansions

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

function theory on cartan domains and the berezin toeplitz symbol calculus cn with dv z normalized lebesgue measure on 9 we let l l2 q dv be the usual space of lebesgue square integrable complex valued functions on 9 by h2 h2 q dv we denote the bergman subspace of l2 consisting of holomorphic functions the orthogonal projection operator from l2 onto h2 is denoted by p forf in l q the space of essentially bounded measurable functions on 9 we will consider the multiplication operator

ranks of mapping tori via the curve complex is generated by nielsen transformations this isequivalent to requiring that our two generating sets for g are related by asequence of the following moves 1 if a6 bare generators replace awith ab 2 if a6 bare generators switch their places in the ordering 3 if ais a generator replace it with a

hua operators and poisson transform for bounded symmetric domains let be a bounded symmetric domain of non tube type in c n with rank r and s its shilov boundary we consider the poisson transform p s f z for a hyperfunction f on s defined by the poisson kernel p s z u h z z n r h z u n r 2 s z u s s c for all s satisfying certain non integral condition we find a necessary and sufficient condition for the functions in the image of the poisson transform in terms of hua operators when is the type i matrix domain in m n m c n less than or slanted equal to m we prove that an eigenvalue equation for the second order m n n valued hua operator characterizes the image 2006 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

operator valued bessel functions holomorphic discrete series and harmonic representations of u p q abstract this paper presents representation theoretic applications of the theory of operator valued bessel functions which was developed in the first paper of this series using the reduced bessel function k we study the harmonic representation and holomorphic discrete series for the group u p q of biholomorphic automorphisms of the siegel domains of type ii

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

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

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

finite covers of random 3 manifolds a 3 manifold is haken if it contains a topologically essential surface the virtual haken conjecture posits that every irreducible 3 manifold with infinite fundamental group has a finite cover which is haken in this paper we study random 3 manifolds and their finite covers in an attempt to shed light on this difficult question in particular we consider random heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface for this model of random 3 manifold we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds our results contrast with the analogous probabilities for groups coming from random balanced presentations giving quantitative theorems to the effect that 3 manifold groups have many more finite quotients than random groups the next natural question is whether these covers have positive betti number for abelian covers of a fixed type over 3 manifolds of heegaard genus 2 we show that the probability of positive betti number is 0

solving convex programs by random walks minimizing a convex function over a convex set in n dimensional space is a basic general problem with many interesting special cases here we present a simple new algorithm for convex optimization based on sampling by a random walk it extends naturally to minimizing quasi convex functions and to other generalizations

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

on the classification of automorphic lie algebras the problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants this provides a powerful tool of analysis and it opens the road to new applications of automorphic lie algebras beyond the con text of integrable systems in this paper it is shown that sl2 c based automorphic lie algebras associated to the icosahedral group i the octahedral group o the tetrahedral group t and the dihedral group dn are isomorphic the proof is based on techniques from classical invariant theory and makes use of clebsch gordan decomposition and transvectants molienfunctionsandthetrace form thisresultprovidesacompleteclas sification of sl2 c based automorphic lie algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of automorphic lie algebras

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

invariant differential operators on hermitian symmetric spaces our object of study is the ring of invariant differential operators on a hermitian symmetric space g k of classical and noncompact type here as usual g is a connected noncompact semisimple lie group with finite center and k is a maximal compact subgroup of g it is well known that the ring denoted by g k is isomorphic to a polynomial ring of 1 variables 1 being the rank of g k this is true even in the nonhermitian case if 1 1 it is generated by the laplace beltrami operator z which is essentially self adjoint moreover iz2 is nonnegative in the general case an easily posable problem is to find an explicitly defined set of generators we can go one step further by focusing our attention on the nonnegativity which necessarily limits the range of eigenvalues under a suitable integrability condition on functions it is thus natural to ask whether there exist some canonically defined operators which generate 9 g k and have the property of nonnegativity the main purpose of the present paper is to give an affirmative answer to this question in fact our answer applies to a somewhat more general type of rings of differential operators which includes 9 g k as a special case to be explicit we take an irreducible representation p k gl v with a complex vector space v of finite dimension and consider the set c p of all v valued cx functions f on g such that f xk 1 p k f x for every k e k we then denote by p the ring of left invariant difrrential operators on g which map c p into itself now the complexification g of the lie algebra of g has abelian subalgebras p and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on g k at the origin for any complex vector space w let sr w denote the ring of all complex valued homogeneous polynomial functions on w of degree r and let s w er 0sr w through the adjoint representation of g on g k acts naturally on sr it is a known fact due to hua and schmid that each irreducible constituent of this representation of k has multiplicity one now our principal

singular invariant trilinear forms and covariant bi differential operators under the conformal group abstract the residues of the meromorphic family of conformally invariant trilinear forms on the sphere constructed in clerc and orsted in press 2 are computed their expression involves conformally covariant differential and bidifferential operators for the latter new formulae are obtained

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

covariant q differential operators and unitary highest weight representations for u q su n n we investigate a one parameter family of quantum harish chandra modules of uqsl2n this family is an analog of the holomorphic discrete series of representations of the group su n n for the quantum group uqsun n we introduce a q analog of the wave operator a determinant type differential operator and prove certain covariance property of its powers this result is applied to the study of some quotients of the above mentioned quantum harish chandra modules we also prove an analog of a known result by j faraut and a koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series

the lie theory of the rankin cohen brackets and allied bi differential operators abstract the lie theoretic nature of the rankin cohen brackets is here uncovered these bilinear operations which among other purposes were devised to produce a holomorphic automorphic form from any pair of such forms are instances of sl 2 r equivariant holomorphic bi differential operators on the upper half plane all of the latter are here characterized and explicitly obtained by establishing their one to one correspondence with singular vectors in the tensor product of two sl 2 c verma modules the rankin cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one dimensional the picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two dimensional explicit formulae for basis vectors in both situations are obtained and universal lie algebraic objects subsuming all of them are exhibited a few applications of these results and lie theoretic approach are then considered first a generalization of the latter yields rankin cohen type brackets for hilbert modular forms then some rankin cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of sl 2 r with another such representation occurring in the tensor product decomposition finally the sought for precise relationship between the rankin cohen brackets and gordan s transvection processes of the nineteenth century invariant theory is unveiled

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

spectrally similar incommensurable 3 manifolds reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable building toward a negative answer to this question we construct examples of hyperbolic 3 manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable more precisely for every n 0 we construct a pair of incommensurable hyperbolic 3 manifolds nn and nn whose volume is approximately n and whose length spectra agree up to length n both nn and nn are built by gluing two standard submanifolds along a complicated pseudo anosov map ensuring that these manifolds have a very thick collar about an essential surface the two gluing maps differ by a hyper elliptic involution along this surface our proof also involves a new commensurability criterion based on pairs of pants

on differential operators attached to certain representations of classical groups classification et construction de sous espaces irreductibles de sr integrales contenant les applications polynomiales l operateur differentiel d operateurs differentiels de type arithmetique applications aux series de eisenstein

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

universal portfolios with side information we present a sequential investment algorithm the spl mu weighted universal portfolio with side information which achieves to first order in the exponent the same wealth as the best side information dependent investment strategy the best state constant rebalanced portfolio determined in hindsight from observed market and side information outcomes this is an individual sequence result which shows the difference between the exponential growth wealth of the best state constant rebalanced portfolio and the universal portfolio with side information is uniformly less than d 2n log n 1 k n log 2 for every stock market and side information sequence and for all time n here d k m 1 is the number of degrees of freedom in the state constant rebalanced portfolio with k states of side information and m stocks the proof of this result establishes a close connection between universal investment and universal data compression