essentially normal homogeneous quotient modules on the polydisc abstract in the present paper we give a complete characterization of all essentially normal homogeneous quotient modules of the hardy space on the unit polydisc it turns out that our approach also applies to quotients of weighted bergman modules

translation representations for automorphic solutions of the wave equation in non euclidean spaces i this paper deals with the spectral theory of the laplace beltrami operator acting on automorphic functions in n dimensional hyperbolic space hn we study discrete subgroups which have a fundamental polyhedron f with a finite number of sides and infinite volume concerning these we have shown previously that the spectrum of contains at most a finite number of point eigenvalues in 1 2 n 1 2 0 and none less than 1 2 n 1 2 here we prove that the spectrum of is absolutely continuous and of infinite multiplicity in 1 2 n 1 2 our approach uses the non euclidean wave equation introduced by faddeev and pavlov energy ef is defined as ut ut u lu where the bracket is the l2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric energy is conserved under the group of operator u t relating initial data to data at time t we construct two isometric representations of the space of automorphic data by l2 r n which transmute the action of u t into translation these representations are given explicitly in terms of integrals of the data over horospheres in part ii we shall show the completeness of these representations utt lu 0 l 1 2 n 1 2

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

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

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

exponential lower bounds for polytopes in combinatorial optimization we solve a 20 year old problem posed by yannakakis and prove that no polynomial size linear program lp exists whose associated polytope projects to the traveling salesman polytope even if the lp is not required to be symmetric moreover we prove that this holds also for the cut polytope and the stable set polytope these results were discovered through a new connection that we make between one way quantum communication protocols and semidefinite programming reformulations of lps

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

beurling type quotient modules over the bidisk and boundary representations this paper mainly concerns beurling type quotient modules of h2 d2 over the bidisk by establishing a theorem of function theory over the bidisk it is shown that a beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most 1 1 furthermore we apply this result to the study of boundary representations of toeplitz algebras over quotient modules it is proved that the identity representation of c sz sw is a boundary representation of b sz sw in all nontrivial cases this extends a result of arveson to toeplitz algebras on beurling type quotient modules over the bidisk cf w arveson subalgebras of c algebras acta math 123 1969 141 224 w arveson subalgebras of c algebras ii acta math 128 1972 271 308 the paper also establishes k homology defined by beurling type quotient modules over the bidisk

restrictions of h p functions in the polydisk manuscript received 19 may 1987 this work was partially supported by an n s f grant 2 added in proof as r rochberg and n salinas have pointed out to me this actually goes back to n aronszajn trans amer math soc 68 1950 p 361 where it was proved that the product kl xl yl k x y2 of reproducing kernels for spaces fl f corresponds to the tensor product space f f while the product kl x y k x y corresponds to the restriction of f f to the diagonal americant journial of mathematics 110 1988 1119 1152 1119

eigenvalue asymptotics of the neumann laplacian of regions and manifolds with cusps abstract we study the eigenvalue asymptotics of a neumann laplacian n in unbounded regions of r2 with cusps at infinity a typical example is x y r 2 x 1 y x 2 and prove that n e n n e h v e 2 vol where hv is the canonical one dimensional schrodinger operator associated to the problem we establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a dirichlet laplacian d for a class of cusp type regions of infinite volume

the reducibility of compressed shifts on beurling type quotient modules over the bidisk abstract in this paper we study the compressed shift operator s z 1 on the beurling type quotient module k of hardy space h 2 d 2 over the bidisk firstly we give a necessary and sufficient condition such that s z 1 has nontrivial pure isometry reducing subspace as an application we show that s z 1 has agler reducing subspaces if and only if is the product of two one variable inner functions secondly for a rational inner function with degree n 1 we show that s z 1 is reducible on k if and only if s z 1 has agler reducing subspaces furthermore we study the case when the rational inner functions have degree n 2 and this case is quite different from that the degree of is n 1

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

approximation limits of linear programs beyond hierarchies we develop a framework for proving approximation limits of polynomial size linear programs lps from lower bounds on the nonnegative ranks of suitably defined matrices this framework yields unconditional impossibility results that are applicable to any lp as opposed to only programs generated by hierarchies using our framework we prove that o n1 2 approximations for clique require lps of size 2n this lower bound applies to lps using a certain encoding of clique as a linear optimization problem moreover we establish a similar result for approximations of semidefinite programs by lps our main technical ingredient is a quantitative improvement of razborov s 38 rectangle corruption lemma for the high error regime which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix

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

properties of beurling type submodules via agler decompositions abstract in this paper we study operator theoretic properties of the compressed shift operators s z 1 and s z 2 on complements of submodules of the hardy space over the bidisk h 2 d 2 specifically we study beurling type submodules namely submodules of the form h 2 d 2 for inner using properties of agler decompositions of to deduce properties of s z 1 and s z 2 on model spaces h 2 d 2 h 2 d 2 results include characterizations in terms of of when a commutator s z j s z j has rank n and when subspaces associated to agler decompositions are reducing for s z 1 and s z 2 we include several open questions

scattering theory for the wave equation on a hyperbolic manifold this talk is a survey of work done jointly mainly with peter lax 3 4 5 but also with bettina wilkott and alex woo 10 over the past ten years it deals with the spectrum of the perturbed and unperturbed laplace beltrami operator acting on automorphic functions on an n dimensional hyperbolic space ihn the associated discrete subgroup of motions is assumed to have the finite geometric property but is otherwise unrestricted this means that the fundamental domain when derived by the polygonal method has a finite number of sides its volume may be finite or infinite and it can have cusps of arbitrary rank with the obvious identifications the fundamental domain can be treated as a manifold m

a parallel repetition theorem we show that a parallel repetition of any two prover one round proof system mip 2 1 decreases the probability of error at an exponential rate no constructive bound was previously known the constant in the exponent in our analysis depends only on the original probability of error and on the total number of possible answers of the two provers the dependency on the total number of possible answers is logarithmic which was recently proved to be almost the best possible u feige and o verbitsky proc 11th annual ieee conference on computational complexity ieee computer society press los alamitos ca 1996 pp 70 76

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

the spectrum of kleinian manifolds abstract we obtain the plancherel theorem for l 2 g where g is a classical simple lie group of real rank one and g is convex cocompact discrete subgroup and deduce its consequences for the spectrum of locally invariant differential operators on bundles over kleinian manifolds as the main tool we develop a geometric version of scattering theory which in particular contains the meromorphic continuation of the eisenstein series for this situation the central role played by invariant distribution sections supported on the limit set is emphasized

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

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

spectral analysis of second order elliptic operators on noncompact manifolds on demontre l estimation de mourre et les bornes qui y sont reliees pour une classe d operateurs elliptiques qui inclut des laplaciens pour des quotients cofinis et coinfinis d espaces hyperboliques

inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in euclidean space we propose a new approach to solve the inverse boundary value problems in the euclidean space the idea consists of embedding the problem into hyperbolic manifolds and using their spectral properties as a by product one can discuss the reconstruction of local conformal deformation of the metric of hyperbolic manifolds from the spectral data at infinity we also propose a new spectral data observed from the cusp neighborhood at infinity

scattering for schrodinger operators in a class of domains with non compact boundaries abstract schrodinger operators in a class of domains with asymptotic cones are considered a generalized fourier transform representing the absolutely continuous part of the schrodinger operator as multiplication by 2 in the asymptotic cone is constructed wave operators relating the free laplacian to schrodinger operators are computed using the generalized fourier transform the wave operators relating schrodinger operators acting in domains with the same asymptotic cone are computed and shown to be complete

the noncommutative choquet boundary let s be an operator system a self adjoint linear sub space of a unital c algebra a such that 1 s and a c s is generated by s a boundary representation for s is an irreducible rep resentationof c s on a hilbert space with the property that s has a unique completely positive extension to c s the set s of all unitary equivalence classes of boundary representations is the non commutative counterpart of the choquet boundary of a function system s c x that separates points of x it is known that the closure of the choquet boundary of a function system s is the y boundary of x relative to s the corresponding noncommutative problem of whether every operator system has suf ficiently many boundary representations was formulated in 1969 but has remained unsolved despite progress on related issues in particu lar it was unknown if s 6 for generic s in this paper we show that every separable operator system has sufficiently many boundary representations our methods use separability in an essential way

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

incoming and outgoing waves on etudie des espaces de solutions de l equation d onde u tt lu 0 ou lf f xx qf qui apparait dans la theorie de la diffusion de lax phillips

the laplace operator on a hyperbolic manifold i spectral and scattering theory using techniques of stationary scattering theory for the schrodinger equation we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the laplace beltrami operator on manifolds mn arising as the quotient of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries we consider manifolds mn of infinite volume in subsequent papers we will use the techniques developed here to analytically continue eisenstein series for a large class of discrete groups including some groups with parabolic elements

a mourre estimate and related bounds for hyperbolic manifolds with cusps of non maximal rank we consider the laplace operator on quotients of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non maximal rank using methods developed by the first two authors we prove a mourre estimate and commutator estimates on the laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates these estimates will be used elsewhere to study the scattering matrix and eisenstein series and their meromorphic continuations and should be useful in studying trace formulas for these discrete groups