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

spectrum and analytical indices of the c algebra of wiener hopf operators we study multivariate generalisations of the classical wiener hopf algebra which is the c algebra generated by the wiener hopf operators given by convolutions restricted to convex cones by the work of muhly and renault this c algebra is known to be isomorphic to the reduced c algebra of a certain restricted action groupoid given by the action of euclidean space on a certain compactification using groupoid methods we construct composition series for the wiener hopf c algebra by a detailed study of this compactification we compute the spectrum and express homomorphisms in k theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms namely these symbols maps turn out to be given by an analytical family index of a continuous family of fredholm operators in a subsequent paper we also obtain a topological expression of these indices

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

an additional gibbs state for the cubic schrodinger equation on the circle an invariant gibbs state for the nonlinear schrodinger equation on the circle was constructed by bourgain 2 3 and mckean 6 out of the basic hamiltonian using a trigonometric cutoff the cubic nonlinear schrodinger equation is a completely integrable system having an infinite number of additional integrals of motion in this paper we construct the second invariant gibbs state from one of these additional integrals for the cubic nls on the circle this additional gibbs state is singular with respect to the gibbs state previously constructed from the basic hamiltonian our approach employs the ablowitz ladik system a completely integrable discretization of the cubic schrodinger equation 2001 john wiley sons inc

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

symplectic structures and volume elements in the function space for the cubic schrodinger equation we consider various trace formulas for the cubic schrodinger equa tion in the space of infinitely smooth functions subject to periodic boundary conditions the formulas relate conventional integrals of motion to the periods of some abelian differentials holomorphic one forms on the spectral curve we show that the periods of abelian differentials are global coordinates on the moduli space of spectral curves the exterior derivatives of the holomorphic one forms are the basic and higher symplectic structures on the phase space we write ex plicitly these symplectic structures in qp coordinates we compute the ratio of two symplectic volume elements in the infinite genus limit

a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation abstract we give a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation nls from many body quantum theory the starting point of our proof is 11 on the time independent problem and 16 on the corresponding problem on a finite lattice an important new obstacle in our analysis is the need to work with a cutoff in the number of particles which breaks the gaussian structure of the free quantum field and prevents the use of the wick theorem we overcome it by means of complex analytic methods our methods apply to the nonlocal nls with bounded convolution potential in the periodic setting we also consider the local nls arising from short range interactions in the many body setting to that end we need the dispersion of the nls in the form of periodic strichartz estimates in x s b spaces

matrix cartan superdomains super toeplitz operators and quantization we present a general theory of non perturbative quantization of a class of hermitian symmetric supermanifolds the quantization scheme is based on the notion of a super toeplitz operator on a suitable z2 graded hilbert spaces of super holomorphic functions the quantized supermanifold arises as the c algebra generated by all such operators we prove that our quantization framework reproduces the invariant super poisson structure on the classical supermanifold as planck s constant tends to zero

invariant tori for the cubic szego equation we continue the study of the following hamiltonian equation on the hardy space of the circle i partial tu pi u 2u where denotes the szego projector this equation can be seen as a toy model for totally non dispersive evolution equations in a previous work we proved that this equation admits a lax pair and that it is completely integrable in this paper we construct the action angle variables which reduces the explicit resolution of the equation to a diagonalisation problem as a consequence we solve an inverse spectral problem for hankel operators moreover we establish the stability of the corresponding invariant tori furthermore from the explicit formulae we deduce the classification of orbitally stable and unstable traveling waves

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

toeplitz algebras and rotational automorphisms associated to polydiscs introduction given a bounded symmetric domain d in cm with d being normalized to be circular and to contain the origin upmeier 22 showed that there is a natural way to associate a c algebra to it this c algebra called the toeplitz c algebra of d and denoted 35 is a solvable c algebra with a composition series jk o k n l of length n where n is the rank of the symmetric domain 22 theorem 3 12 we observe that there is a natural group action on 3 by the compact group g of all biholomorphic automorphisms of d which lie in the connected component of the identity and which fix the origin proposition 1 1 every element 0 in g preserves the solvable series jk o k n l of 3 d we thereby obtain a composition series jk x0 z for the crossed product c algebra 3 xo z in this paper we show by introducing the notion of stratification of composition series see section 2 that the series jk x0 z is exactly the maximal radical series of 3 x0 z section 3 and thus obtain an isomorphism invariant for the latter c algebra when is the polydisc d we determine the k theory of the series jk xo z and their subquotients section 4 techniques from k theory using strong morita equivalence yield a complete classification of the c algebras 3 xo z where d is the n dimensional polydisc up to isomorphism for all 0 in g tn

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

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

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

finite state unification automata and relational languages abstract we define the new notion of a finite state unification automaton a device for finite state recognition of relational languages by means of unification transitions words in such a language are formed by composing base relations and have the general form r i 1 x j 1 x k 1 r i n x j n x k n for some n generation of such languages by regarding horn clauses as grammers has been considered before but to the best of our knowledge recognizing such languages by suitably designed automata is a new approach the main result presented is a pumping lemma forming a necessary condition for finite state recognizability some example results about such automata are given

finite memory automata abstract a model of computation dealing with infinite alphabets is proposed this model is based on replacing the equality test by substitution it appears to be a natural generalization of the classical rabin scott finite state automata and possesses many of their closure and decision properties also when restricted to finite alphabets the model is equivalent to finite state automata

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

boundary orbit strata and faces of invariant cones and complex ol shanskiĭ semigroups let d g k be an irreducible hermitian symmetric domain then g is contained in a complexification gc and there exists a closed complex subsemigroup g gc characterised by fact that all holomorphic discrete series representations of g extend holomorphically to the socalled minimal ol shanskiuo semigroup parallel to the classical theory of boundary strata for the symmetric

reinhardt domains boundary geometry and toeplitz c algebras abstract in this paper we have found explicitly how the boundary geometry of a reinhardt domain in c 2 determines the structure of its toeplitz c algebra more precisely our main result is an explicit simple algorithm to describe the structure of the toeplitz c algebra of a reinhardt domain d in c 2 satisfying some mild boundary condition in terms of rotation c algebras based on the degree of contact of the boundary curve of c the logarithmic domain of d at each point of intersection with the linear faces of the convex hull of c and the slopes of these faces a consequence of this result is that the toeplitz c algebras t d and t d of d and its pseudoconvex hull d are rarely isomorphic but are always of the same type i e either both are of type i or both are not of type i in other words the isomorphism class of t d is usually changed under taking pseudoconvex hull but the type of t d is not affected