the space and scale dependencies of the self similar structure of turbulence two new techniques for analysing the space and scale dependencies of the self similar structure of turbulence are introduced both methods are based on the wavelet transform but use different wavelets first the concept of a turbulence eddy is defined in terms of the mexican hat wavelet transform of the velocity signal in such a way that the eddy has a size and a location in space a new scaling exponent the eddy capacity d text e 0 leq d text e leq 1 is defined in terms of the zero crossings of this wavelet transform d text e is a measure of the space fillingness of eddies and of their scaling in real space whereas the hausdorff dimension d text h prime and the kolmogorov capacity d text k prime are scaling exponents that are sensitive to the scaling and space fillingness in wavenumber space d text e differs from both d text k prime and d text h prime in that it is particularly sensitive to phase correlations for any random phase signal d text e 1 whereas d text h prime and d text k prime depend on the energy spectrum for well defined spiral signals and for signals with a fractal intermittency in space d text e d text k prime secondly we propose a practical test for discriminating between h fractals and k fractals i e between a dense and a sparse distribution of singularities this method is based on successive averages of wavelet transforms and the morlet wavelet is used the technique is shown to be robust with respect to large amounts of phase noise both methods are applied to a one dimensional 1d turbulence velocity signal of very high reynolds number re lambda 2720 it is found that d text e 1 for the inertial scales larger than the taylor microscale lambda thus indicating some degree of phase scrambling of the 1d turbulence velocity signal at the larger inertial scales there is some indication of a spiral type structure in this signal at scales below lambda finally we measure the spatial fluctuation of wavelet energy comparing an experimental turbulence signal with random phase and spiral like signals the magnitude of spatial fluctuations is set by the scrambled phase part of the turbulence but the increase of energy fluctuation at small scales may be due to locally self similar small scale structures

isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

models for n tuples of noncommuting operators abstract this paper proves versions of the rota model theorem the de branges rovnyak model theorem and the coisometric extension theorem for n tuples of not necessarily commuting operators this generalizes the work of a e frazho j funct anal 48 1982 1 11 for pairs of operators the methods involve applying the single operator results to matrices of operators

complements to models for noncommuting operators abstract this paper develops a dilation theory for two noncommutative operators many of the results and techniques in dilation theory for one operator are extended to this setting this is done by extending the schaffer construction for one operator to the fock space setting a new wold decomposition for two isometrics with orthogonal range is obtained this wold decomposition and the schaffer construction is used to extend the sz nagy foias lifting theorem and the characteristic function theory to the noncommutative setting the characteristic function is operator valued and defined in a fock space

invariant subspaces and hyper reflexivity for free semigroup algebras in this paper we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras this enables us to prove that they are reflexive and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces such algebras are called hyper reflexive this property is very strong but it has been established in only a very few cases moreover the prototypes of this class of algebras are the natural candidate for a non commutative analytic toeplitz algebra on n variables the case we make for this analogy is very compelling in particular in this paper the key to the invariant subspace analysis is a good analogue of the beurling theorem for invariant subspaces of the unilateral shift this leads to a notion of inner outer factorization in these algebras in a sequel to this paper 13 we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in cn a free semigroup algebra is the weak operator topology closed algebra generated by a set s1 sn of isometries with pairwise orthogonal ranges these conditions are described algebraically by s i sj iji for 1 i j n f

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

a new method for constructing invariant subspaces the new idea that is used in this article for producing non trivial closed invariant subspaces of bounded linear operators on reflexive banach spaces is the use of fixed points of set valued functions the advantage of this new method is that it is reasonable to expect that the famous method of lomonosov for producing invariant subspaces using fixed points of functions can be viewed as a special case of the use of fixed points of set valued functions further uses of this new idea and open questions are suggested at the end of the article

absolute continuity for commuting row contractions abstract absolutely continuous commuting row contractions admit a weak continuous functional calculus building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the drury arveson space we give a complete characterization of these commuting row contractions in measure theoretic terms we also establish that completely non unitary row contractions are necessarily absolutely continuous in direct parallel with the case of a single contraction finally we consider refinements of this question for row contractions that are annihilated by a given ideal

superposition principles associated with the moutard transformation an integrable discretization of a 2 1 dimensional sine gordon system superposition principles both linear and nonlinear associated with the moutard transformation are found on suitable reinterpretation these constitute an integrable discrete nonlinear system and its associated linear system further it is shown that in a particular form this system is an integrable discretization of a 2 1 dimensional sine gordon system solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the moutard transformation included in these solutions are discrete analogues of the kink solutions of the continuous system

1 hyperreflexivity and complete hyperreflexivity the subspaces and subalgebras of b h which are hyperreflexive with constant 1 are completely classified it is shown that there are 1 hyperreflexive subspaces for which the complete hyperreflexivity constant is strictly greater than 1 the constants for ct b h are analyzed in detail

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

scalings of scalar structure functions in a velocity field with coherent vortical structures in planar turbulence modeled as an isotropic and homogeneous collection of two dimensional noninteracting compact vortices the structure functions s p r of a statistically stationary passive scalar field have the following scaling behavior in the limit where the peclet number pe s n r const in r l pe 1 3 for l pe 1 3 r l s p r r l pe 1 3 6 1 d for l pe 1 2 r l pe 1 3 where l is a large scale and d is the fractal codimension of the spiral scalar structures generated by the vortices d 2 3 note that l pe 1 2 is the scalar taylor microscale that stems naturally from our analytical treatment of the advection diffusion equation the essential ingredients of our theory are the locality of interscale transfer and lundgren s time average assumption a phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalization of this phenomenology to spatially smooth chaotic flows yields k ln k 1 generalized power spectra for the advected scalar fields

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

the algebraic structure of non commutative analytic toeplitz algebras the non commutative analytic toeplitz algebra is the wot closed algebra generated by the left regular representation of the free semigroup on n generators we develop a detailed picture of the algebraic structure of this algebra in particular we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n ball the k dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of h over the unit disk in 6 17 18 20 a good case is made that the appropriate analogue for the analytic toeplitz algebra in n non commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators the papers cited obtain a compelling analogue of beurling s theorem and inner outer factorization in this paper we add further evidence the main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of cn the kernel is the subgroup of quasi inner automorphisms which are trivial modulo the wot closed commutator ideal additional evidence of analytic properties comes from the structure of k dimensional completely contractive representations which have a structure very similar to the fibration of the maximal ideal space of h over the unit disk an important tool in our analysis is a detailed structure theory for wot closed right ideals curiously left ideals remain more obscure the non commutative analytic toeplitz algebra ln is determined by the left regular representation of the free semigroup fn on n generators z1 zn which acts on 2 fn by w v wv for v w in fn in particular the algebra ln is the unital wot closed algebra generated by the isometries li zi for 1 i n this algebra and its norm closed version the noncommutative disk algebra were introduced by popescu 19 in an abstract sense in connection with a non commutative von neumann inequality and 1991 mathematics subject classification 47d25 march 9 1997 october 9 1997 final draft first author partially supported by an nserc grant and a killam research fellowship second author partially supported by an nsf grant

potential theory and analytic properties of a cantor set the authors consider the electrostatic potential due to a uniform distribution of charge on a cantor set mellin transforms are used to find expansions for the potential at a distance r from the end of the charge distribution it is found that the potential is a power law multiplied by a function periodic in ln r together with a power series in r a recursion relation for the moments of the distribution is used to reveal a similar structure in these moments finally fourier transform techniques are used to find explicit representations for the distribution and its moments

free pluriharmonic majorants and commutant lifting in this paper we initiate the study of sub pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball b h n 1 x1 xn b h n x1x1 xnxn 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h several classical results from complex analysis have analogues in this noncommutative multivariable setting we present basic properties for sub pluriharmonic curves characterize the class of sub pluriharmonic curves that admit free pluriharmonic majorants and find in this case the least free pluriharmonic majorants we show that for any free holomorphic function on b h n 1 the map 0 1 c r1 rn r rr1 rrn rr1 rrn is a sub pluriharmonic curve in the cuntz toeplitz algebra generated by the right creation operators r1 rn on the full fock space with n generators we prove that is in the noncommutative hardy space hball2 if and only if has a free pluriharmonic majorant in this case we find herglotz riesz and poisson type representations for the least pluriharmonic majorant of moreover we obtain a characterization of the unit ball of hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of when is in the unit ball of hball2 in the second part of this paper we introduce a generalized noncommutative commutant lifting gncl problem which extends to our noncommutative multivariable setting several lifting problems including the classical sz nagy foias commutant lifting problem and the extensions obtained by treil volberg foias frazho kaashoek and biswas foias frazho as well as their multivariable noncommutative versions we solve the gncl problem and using the results regarding sub pluriharmonic curves and free pluriharmonic majorants on noncommutative balls we provide a complete description of all solutions in particular we obtain a concrete schur type description of all solutions in the noncommutative commutant lifting theorem

factorization theorems and the structure of operators on hilbert space soit 7b h un espace de hilbert complexe et soit 7b l 7b h l algebre des operateurs lineaires bornes agissant sur 7b h on demontre des theoremes de factorisation on etudie la structure des operateurs sur 7b h

free holomorphic functions on the unit ball of b h n abstract in this paper we continue the study of free holomorphic functions on the noncommutative ball b h n 1 x 1 x n b h n x 1 x 1 x n x n 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h and n 1 2 or n several classical results from complex analysis have free analogues in our noncommutative setting we prove a maximum principle a naimark type representation theorem and a vitali convergence theorem for free holomorphic functions with operator valued coefficients we introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by schwarz and harnack the borel caratheodory theorem is extended to our noncommutative setting using a noncommutative generalization of schwarz s lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball b h n 1 we obtain an analogue of julia s lemma for free holomorphic functions f b h n 1 b h m 1 we also obtain pick julia theorems for free holomorphic functions with operator valued coefficients we provide a noncommutative generalization of a classical inequality due to lindelof which turns out to be sharper then the noncommutative von neumann inequality finally we introduce a pseudohyperbolic metric on b h n 1 which is invariant under the action of the free holomorphic automorphism group of b h n 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on b n the open unit ball of c n in this setting we obtain a schwarz pick type lemma we also provide commutative versions of these results for operator valued multipliers of the drury arveson space

wavelet representations and fock space on positive matrices abstract we show that every biorthogonal wavelet determines a representation by operators on hilbert space satisfying simple identities which captures the established relationship between orthogonal wavelets and cuntz algebra representations in that special case each of these representations is shown to have tractable finite dimensional co invariant doubly cyclic subspaces further motivated by these representations we introduce a general fock space hilbert space construction which yields creation operators containing the cuntz toeplitz isometries as a special case

operators affiliated to the free shift on the free hardy space abstract the smirnov class for the classical hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators this definition extends naturally to the commutative and non commutative multi variable settings of the drury arveson space and the full fock space over c d identifying the fock space with the free multi variable hardy space of non commutative or free holomorphic functions in a non commutative open unit ball in several matrix variables we prove that any closed densely defined operator affiliated to the right free multiplier algebra of the full fock space acts as right multiplication by a non commutative function in the right free smirnov class and analogously replacing right with left

fast chemical reaction and multiple scale concentration fields in singular vortices two species involved in a simple fast reaction tend to become segregated in patches composed of a single of these reactants these patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs fed by diffusion stirred by advection this boundary and the concentration fields within the patches may tend to present multiple scale characteristics based on this segregated state this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it when concentrations presenting initial self similar fluctuations are advected by a singular vortex first the two sources of singularity i e the self similar initial conditions and the singular vortex are considered isolatedly on the one hand self similar initial conditions are imposed to a diffusion reaction system for one and two dimensional cases on the other hand an imposed singular vortex advects initially on off concentration fields in combination with diffusion and reaction this problem is addressed analytically by characterizing the boundary by a box counting dimension and the concentration fields by a h older exponent and numerically by direct numerical simulations of the advection diffusion reaction equations second the way the two sources hang together shows that depending on the self similar properties of the initial concentration fields the vortex promotes the chemical activity close to its inner smoothed out core or close to the outer region where the boundary starts to spiral for all the considered situations the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws

duality convexity and peak interpolation in the drury arveson space abstract we consider the closed algebra a d generated by the polynomial multipliers on the drury arveson space we identify a d as a direct sum of the preduals of the full multiplier algebra and of a commutative von neumann algebra and establish analogues of many classical results concerning the dual space of the ball algebra these developments are deeply intertwined with the problem of peak interpolation for multipliers and we generalize a theorem of bishop carleson rudin to this setting by means of choquet type integral representations as a byproduct we shed some light on the nature of the extreme points of the unit ball of a d

hyper reflexivity and the factorization of linear functionals abstract it is shown that hyper reflexivity of a space of linear operators on a hilbert space follows from a factorization property of linear functionals continuous in the weak operator topology this provides new examples of hyper reflexive algebras and new proofs for the hyper reflexivity of the noncommutative disk algebras

a reflexivity theorem for weakly closed subspaces of operators it was proved in 4 that the ultraweakly closed algebras generated by certain contractions on hilbert space have a remarkable property this property in conjunction with the fact that these algebras are isomorphic to h was used in 3 to show that such ultraweakly closed algebras are reflexive in the present paper we prove an analogous result that does not require isomorphism with h and applies even to linear spaces of operators our result contains the reflexivity theorems of 3 2 and 9 as particular cases let g denote the algebra of linear bounded operators acting on the hilbert spale x and let xf denote a linear subspace of then ff is endowed with the weak and ultraweak topologies that it inherits from y 3 cf 6 15 for two arbitrary vectors x y e h we can define the ultra weakly continuous functional x y on kby x c y a ax y a e x where k stands for the scalar product in j definition 1 let n be a natural number n 1 the subspace 4 has property bn respectively an if for every positive number e there exists a positive number 8 6 n such that for every system f i 1 i j n of weakly respectively ultraweakly continuous functionals on and every system xi yj 1 i j n of vectors in x satisfying the inequalities irk1j xi yj 11 8 there exist vectors x yj 1 i n in3 such that oi x yj i i va n and x x li e iiy y i j n since every weakly continuous functional on xf is also ultraweakly continuous property bn is weaker than an added in proof it was pointed out by c apostol that b7 and an are in fact equivalent this fact is not used below we recall now from 8 that a linear subspacea1 of y i is said to be reflexive if it contains every operator t e y i with the property that tx e x for every received by the editors january 31 1984 1980 mathemnatics sub ect classification primary 47c05 the author was partially supported by a grant from the national science foundation 1985 american mathematical society 0002 9947 85 1 00 25 per page 139 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 39 22 utc all use subject to http about jstor org terms

on the structure of contraction operators with applications to invariant subspaces on ameliore certaines conditions suffisantes pour un membre de la classe a d 0 on obtient de nouvelles conditions suffisantes pour qu une contraction sur un espace de hilbert dont le spectre contient le cercle unite ait des sous espaces invariants non triviaux

surface theory in discrete projective differential geometry i a canonical frame and an integrable discrete demoulin system we present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved we propose a canonical frame in terms of which the associated projective gauss weingarten and gauss mainardi codazzi equations adopt compact forms based on a scaling symmetry which injects a parameter into the linear gauss weingarten equations we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of lie quadrics and their envelopes in the case of discrete demoulin surfaces we derive a backlund transformation for the underlying discrete demoulin system and show how the latter may be formulated as a two component generalization of the integrable discrete tzitzeica equation which has originally been derived in a different context at the geometric level this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry

models for noncommuting operators abstract this paper develops a model theory for a pair of noncommuting operators using backward shift operators on a fock space rota s theorem is generalized i e it is shown that any two bounded operators on a hilbert space are simultaneously similar to part of a pair of backward shift operators on a fock space these shift operators and the fock space framework are also used to develop a dilation theory for two noncommuting operators

unitary isomorphism of fock spaces of bosons and fermions arising from a representation of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation according to branching laws associated with these descriptions a certain representation of the cuntz algebra o2 induces fock representations hb and hf of bosons and fermions simultaneously from this a unitary operator u from hb to hf is obtained we show the explicit formula of the action of u on the standard basis of hb it is shown that u preserves the particle number of hb and hf

noncommutative transforms and free pluriharmonic functions abstract in this paper we study free pluriharmonic functions on noncommutative balls b h n 0 and their boundary behavior these functions have the form f x 1 x n k 1 k b x a 0 i k 1 k a x a b c where the convergence of the series is in the operator norm topology for any x 1 x n b h n and b h denotes the algebra of all bounded linear operators on a hilbert space h the main tools used in this study are certain noncommutative transforms which are introduced in the present paper and which generalize the classical transforms of berezin poisson fantappie herglotz and cayley several classical results from complex analysis have free analogues in our noncommutative multivariable setting

finitely correlated representations of product systems of c correspondences over nk abstract we study isometric representations of product systems of correspondences over the semigroup n k which are minimal dilations of finite dimensional fully coisometric representations we show the existence of a unique minimal cyclic coinvariant subspace for all such representations the compression of the representation to this subspace is shown to be a complete unitary invariant for a certain class of graph algebras the nonself adjoint wot closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace this class includes free semigroup algebras this result extends to a class of higher rank graph algebras which includes higher rank graphs with a single vertex

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

free biholomorphic functions and operator model theory ii let f f1 fn be an n tuple of formal power series in noncommutative indeterminates z1 zn such that f 0 0 and the jacobian detjf 0 0 and let g g1 gn be its inverse with respect to composition we assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball b h n 1 where b h is the algebra of bounded linear operators an a hilbert space h in this paper several results concerning the noncommutative multivariable operator theory on the unit ball b h n 1 are extended to the noncommutative domain bf h x b h n g f x x and f x 1 for an appropriate evaluation x f x we develop an operator model theory and dilation theory for bf h where the associated universal model is an n tuple mz1 mzn of left multiplication operators acting on a hilbert space h2 f of formal power series all the results of this paper have commutative versions

absolutely continuous representations and a kaplansky density theorem for free semigroup algebras we introduce notions of absolutely continuous functionals and representations on the non commutative disk algebra an absolutely continuous functionals are used to help identify the type l part of the free semigroup algebra associated to a extendible representation a extendible representation of an is regular if the absolutely continuous part coincides with the type l part all known examples are regular absolutely continuous functionals are intimately related to maps which intertwine a given extendible representation with the left regular representation a simple application of these ideas extends reflexivity and hyper reflexivity results moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of an is weak dense in the unit ball of the associated free semigroup algebra if and only if is regular we provide some explicit constructions related to the density theorem for specific representations a notion of singular functionals is also defined and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts

structure of free semigroupoid algebras abstract a free semigroupoid algebra is the wot closure of the algebra generated by a tck family of a graph we obtain a structure theory for these algebras analogous to that of free semigroup algebra we clarify the role of absolute continuity and wandering vectors these results are applied to obtain a lebesgue von neumann wold decomposition of tck families along with reflexivity a kaplansky density theorem and classification for free semigroupoid algebras several classes of examples are discussed and developed including self adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence

mixing entrainment and fractal dimensions of surfaces in turbulent flows some basic thoughts are set down on the relation between the fractal dimension of various surfaces in turbulent flows and the practically important processes of mixing between two streams reacting or otherwise separated by a convoluted surface as well as of entrainment of irrotational flow by a turbulent stream an expression based on heuristic arguments is derived for the flux of transportable properties such as mass momentum and energy across surfaces and a prediction made on this basis for the fractal dimension of surfaces in fully turbulent flows is shown to be in essential agreement with measurements it is further shown that this prediction remains robust when corrected for the non uniform effects along the surface a related prediction concerning the dependence of mixing on the reynolds number and the fractal dimension of the surface is substantiated in the developing as well as the fully developed states by independent measurements of both the fractal dimension and the amount of mixing between reactants in a temporally evolving countercurrent shear flow

discrete projective minimal surfaces abstract we propose a natural discretisation scheme for classical projective minimal surfaces we follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects thus we introduce discrete analogues of classical lie quadrics and their envelopes and classify discrete projective minimal surfaces according to the cardinality of the class of envelopes this leads to discrete versions of godeaux rozet demoulin and tzitzeica surfaces the latter class of surfaces requires the introduction of certain discrete line congruences which may also be employed in the classification of discrete projective minimal surfaces the classification scheme is based on the notion of discrete surfaces which are in asymptotic correspondence in this context we set down a discrete analogue of a classical theorem which states that an envelope of the lie quadrics of a surface is in asymptotic correspondence with the surface if and only if the surface is either projective minimal or a q surface accordingly we present a geometric definition of discrete q surfaces and their relatives namely discrete counterparts of classical semi q complex doubly q and doubly complex surfaces

wavelet analysis of fractal boundaries part 2 multifractal analysis this second part deals with the global analysis of the boundary of domains omega rd we develop methods for determining the dimensions of the sets where the local behaviors introduced in part 1 occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

on the structure of contraction operators ii soit 7b h un espace de hilbert complexe separable de dimension infinie et 7b l 7b h l algebre de tous les operateurs lineaires bornes sur 7b h soit d le disque unite ouvert dans c d et n est l ensemble des entiers positifs il existe r 6 tel que chaque contraction t dans 7b l 7b h avec c t soit a un sous espace hyperinvariant non trivial soit appartient a la classe a 1 r

nest representations of directed graph algebras this paper is a comprehensive study of the nest representations for the free semigroupoid algebra mathfrak l g of a countable directed graph g as well as its norm closed counterpart the tensor algebra mathcal t g we prove that the finite dimensional nest representations separate the points in mathfrak l g and a fortiori in mathcal t g the irreducible finite dimensional representations separate the points in mathfrak l g if and only if g is transitive in components which is equivalent to being semisimple also the upper triangular nest representations separate points if and only if for every vertex x in mathcal t g supporting a cycle x also supports at least one loop edge we also study faithful nest representations we prove that mathfrak l g or mathcal t g admits a faithful irreducible representation if and only if g is strongly transitive as a directed graph more generally we obtain a condition on g which is equivalent to the existence of a faithful nest representation we also give a condition that determines the existence of a faithful nest representation for a maximal type mathbb n nest

multi toeplitz operators and free pluriharmonic functions abstract we initiate the study of weighted multi toeplitz operators associated with noncommutative regular domains d q m h b h n m n 1 where b h is the algebra of all bounded linear operators on a hilbert space h these operators are acting on the full fock space with n generators and have as symbols free pluriharmonic functions on the interior of the domain d q m h we prove that the set of all weighted multi toeplitz operators coincides with a d q m a d q m w o t where the domain algebra a d q m is the norm closed unital non selfadjoint algebra generated by the universal model w 1 w n of the noncommutative domain d q m h these results are used to study the class of free pluriharmonic functions on d q m h several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting in particular we show that the bounded free pluriharmonic functions are precisely those which are noncommutative berezin transforms of weighted multi toeplitz operators and solve the dirichlet extension problem in this setting using noncommutative cauchy transforms we provide a free analytic functional calculus for n tuples of operators which extends to free pluriharmonic functions our study of weighted multi toeplitz operators on fock spaces is a blend of multi variable operator theory noncommutative function theory operator spaces and harmonic analysis

circles in the spectrum and the geometry of orbits a numerical ranges approach abstract we prove that a bounded linear hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost orthogonality relations this result is obtained via the study of numerical ranges of operator tuples where several new results are also obtained as consequences of our numerical ranges approach we derive in particular wide generalizations of arveson s theorem as well as show that the weak convergence of operator powers implies the uniform convergence of their compressions on an infinite dimensional subspace

kolmogorov s contributions to the physical and geometrical understanding of small scale turbulence and recent developments this paper reviews how kolmogorov postulated for the first time the existence of a steady statistical state for small scale turbulence and its defining parameters of dissipation rate and kinematic viscosity thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes we present theoretical arguments and experimental evidence to indicate when the small scale motions might tend to a universal form paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the inertial range spectrum these may be particular forms of eddy or eigenstructure such as spiral vortices which may not be unique to turbulent flows also they tend to lead to the notable spiral contours of scalars in turbulence whose self similar structure enables the boxcounting technique to be used to measure the capacity dk of the contours themselves or of their intersections with lines dk although the capacity a term invented by kolmogorov and studied thoroughly by kolmogorov tikhomirov is like the exponent 2p of a spectrum in being a measure of the distribution of length scales d being related to 2p in the limit of very high reynolds numbers the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the reynolds number thus kolmogorov tikhomirov provide the basis for a more widely applicable measure of the self similar structure of turbulence finally we also review how kolmogorov s concept of the universal spatial structure of the small scales together with appropriate additional physical hypotheses enables other aspects of turbulence to be understood at these scales in particular the general forms of the temporal statistics such as the highfrequency inertial range spectra in eulerian and lagrangian frames of reference and the perturbations to the small scales caused by non isotropic non gaussian and inhomogeneous large scale motions 1 kolmogorov s papers review and comments a introduction in this review we join with the other contributors to this special publication in celebrating some of kolmogorov s great contributions to fluid mechanics and mathematics and showing in some small way how his genius has inspired further

analytic contractions nontangential limits and the index of invariant subspaces let h be a hilbert space of analytic functions on the open unit disc d such that the operator m of multiplication with the identity function defines a contraction operator in terms of the reproducing kernel for h we will characterize the largest set delta h subset of partial derivative d such that for each f g is an element of h g not equal 0 the meromorphic function f g has nontangential limits a e on delta h we will see that the question of whether or not delta h has linear lebesgue measure 0 is related to questions concerning the invariant subspace structure of m zeta we further associate with h a second set sigma h subset of partial derivative d which is defined in terms of the norm on h for example sigma h has the property that vertical bar zeta n f vertical bar vertical bar 0 for all f is an element of h if and only if sigma h has linear lebesgue measure 0 it turns out that h s h a e by which we mean that delta h backslash sigma h has linear lebesgue measure 0 we will study conditions that imply that delta h sigma h a e as one corollary to our results we will show that if dim h zeta h 1 and if there is a c 0 such that for all f is an element of h and all lambda is an element of d we have parallel to 1 lambda over bar zeta zeta lambda f parallel to c parallel to f then delta h sigma h a e and the following four conditions are equivalent 1 parallel to zeta n f parallel to negated right arrow 0 for some f is an element of h 2 parallel to zeta n f parallel to negated right arrow 0 for all f is an element of h f not equal 0 3 h has nonzero lebesgue measure 4 every nonzero invariant subspace m of m zeta has index 1 i e satisfies dim m zeta m 1 less

hardy algebras w correspondences and interpolation theory given a von neumann algebra m and a w correspondence e over m we construct an algebra h e that we call the hardy algebra of e when m open image in new window e h e is the classical hardy space h open image in new window of bounded analytic functions on the unit disc when m open image in new window and e open image in new window h e is the free semigroup algebra studied by popescu davidson and pitts and many others we show that given any faithful normal representation of m on a hilbert space h there is a natural correspondence e over the commutant m called the dual of e and that h e can be realized in terms of b h valued functions on the open unit ball open image in new window e in the space of adjoints of elements in e we prove analogues of the nevanlinna pick theorem in this setting and discover other aspects of the value distribution theory for elements in h e we also analyze the boundary behavior of elements in h e and obtain generalizations of the sz nagy foias functional calculus and the functional calculus of popescu for c n c row contractions the correspondence e has a dual that is naturally isomorphic to e and the commutants of certain so called induced representations of h e can be viewed as induced representations of h e for these induced representations a double commutant theorem is proved

interpolation problems in several variables abstract we obtain a noncommutative nudel man type interpolation theorem in several variables and a commutative analogue forh b n the hardy space of bounded analytic functions in the open ball of c n using this result we obtain an operatorial analogue in several variables of the rosenblum rovnyak restriction theorem as well as an extension to the noncommutative analytic toeplitz algebraf we provide new extensions of the caratheodory fejer and nevanlinna pick interpolation theorems forf

application of an incidence theorem for conics cauchy problem and integrability of the dckp equation we demonstrate that pascal s classical theorem for hexagons inscribed in conics allows one to define in a compact manner maps which are governed algebraically by the integrable discrete ckp equation a theorem for conics on oriented triangulated surfaces is used to construct a well posed cauchy problem for these dckp maps moreover the same theorem is exploited to construct in a purely geometric manner a backlund transformation for dckp maps thus the integrability of dckp maps and their underlying nonlinear soliton equation is shown to be encoded in an incidence theorem of projective geometry

notes on invariant subspaces the main purpose of this article is to give an approach to the recent invariant subspace theorem of brown chevreau and pearcy every contraction on a hubert space whose spectrum contains the unit circle has nontrivial invariant subspaces our proof incorporates several of the recent ideas tying together function theory and operator theory 1 i n t r o d u c t i o n the jordan structure theorem for finite matrices has been known now for over one hundred years and its usefulness can hardly be overstated it says that every square matrix a over the complex numbers c is similar to another matrix b i e b xax for some invertible matrix x which is a direct sum of jordan cells that is b can be written in block form bx 0 0 b o 0 bk and each bt has the form a 0 0

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

compressions graphs and hyperreflexivity abstract in this note we extend the ideas in 11 to prove reflexivity and especially hyperreflexivity results that pertain to compressions rather than summands these results in turn are relevant for some remarkable classes of operators

some counterexamples in nonselfadjoint algebras in this note a construction is given of a special operator t on hilbert space a free variable in this construction is a weakly closed subspace ty of operators by appropriate choice of j we give counterexamples to several well known

invariant subspaces and operator model theory on noncommutative varieties let mathcal q subset mathbb c left be an arbitrary set of polynomials in noncommutative indeterminates such that q 0 0 for all q in mathcal q the noncommutative variety begin aligned mathcal v f mathcal q m mathcal h left x x 1 ldots x n in mathbf d f m mathcal h q x 0 text for all q in mathcal q right end aligned where mathbf d f m mathcal h is a noncommutative regular domain in b mathcal h n and b mathcal h is the algebra of bounded linear operators on a hilbert space mathcal h admits a universal model b m b 1 m ldots b n m such that q b m 0 q in mathcal q acting on a model space which is a subspace of the full fock space with n generators in this paper we obtain a beurling type characterization of the joint invariant subspaces under the operators b 1 m ldots b n m in terms of partially isometric multi analytic operators acting on model spaces more generaly a beurling lax halmos type representation is obtained and used to parameterize the wandering subspaces of the joint invariant subspaces under b 1 m otimes i mathcal e ldots b n m otimes i mathcal e and to characterize when they are generating for the corresponding invariant subspaces similar results are obtained for any pure n tuple x 1 ldots x n in the noncommutative variety mathcal v f mathcal q m mathcal h we characterize the elements in the noncommutative variety mathcal v f mathcal q m mathcal h which admit characteristic functions develop an operator model theory for the completely non coisometric elements and prove that the characteristic function is a complete unitary invariant for this class of elements this extends the classical sz nagy foias functional model for completely non unitary contractions based on characteristic functions our results apply in particular when mathcal q consists of the noncommutative polynomials z iz j z jz i i j 1 ldots n in this case the model space is a symmetric weighted fock space which is identified with a reproducing kernel hilbert space of holomorphic functions on a reinhardt domain in mathbb c n and the universal model is the n tuple m lambda 1 ldots m lambda n of multipliers by the coordinate functions

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

wavelet analysis of fractal boundaries part 1 local exponents let be a domain of rd in part 1 of this paper we introduce new tools in order to analyse the local behavior of the boundary of classifica tions based on geometric accessibility conditions are introduced and compared they are related to analytic criteria based either on local lp regularity of the characteristic function 1 or on its wavelet coefficients part 2 deals with the global analysis of the boundary of we develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

minimality of the data in wavelet filters abstract orthogonal wavelets or wavelet frames for l2 r are associated with quadrature mirror filters qmf a set of complex numbers which relate the dyadic scaling of functions on r to the z translates in this paper we show that generically the data in the qmf systems of wavelets are minimal in the sense that the data cannot be nontrivially reduced the minimality property is given a geometric formulation in the hilbert space l2 z and it is then shown that minimality corresponds to irreducibility of a wavelet representation of the algebra o 2 and so our result is that this family of representations of o 2 on the hilbert space l2 z is irreducible for a generic set of values of the parameters which label the wavelet representations

singular inner functions and biinvariant subspaces for dissymetric weighted shifts abstract we consider dissymetric weights on z i e nonincreasing unbounded weights z 1 such that n 1 for n 0 and such that lim n n 1 n 1 where n sup p z n p p a dissymetric weight is said to be quasianalytic if n log n n 2 denote by the unit circle and by d the open unit disc set l 2 f l 2 f n z f n 2 2 n 1 2 we identify the usual hardy space h 2 d to the set f l 2 f n 0 n s f e it e it f e it the usual shift on l 2 if u is a singular inner function denote by e u the closure of span s n u n z in l 2 we show that if is a dissymetric weight such that log n n n then e u is a proper subspace of l 2 for every singular inner function u also if is any dissymetric weight then there exists some singular inner function u such that e u is a proper subspace of l 2 so that s always has nontrivial biinvariant subspaces no nontrivial biinvariant subspaces of s were known so far in the quasianalytic case

l1 factorization forc00 contractions with isometric functional calculus abstract let t be an absolutely continuous contraction acting on a hilbert space h for x y h define x t y l 1 t by its fourier coefficients x t y n t n x y if n n x n t w l 1 h 1 0 w t x n l 1 h 1 0 0 w h implies that lim n x n t w l 1 0 w h using known factorization techniques we exhibit a borel set t such that for any f l 1 t there exist x y h such that f x t y t in the case where t a c 00 this leads to a simple proof of the fact that for every f l 1 t there exists x y h such that f x t y in this case we also show using dilation theory in the unit disk that every strictly positive lower semicontinuous function l 1 t can be written in the form x t x examples show that this is the best possible result for the class a c 00

atomic representations of rank 2 graph algebras abstract we provide a detailed analysis of atomic representations of rank 2 graphs on a single vertex they are completely classified up to unitary equivalence and decomposed into a direct sum or direct integral of irreducible atomic representations the building blocks are described as the minimal dilations of defect free representations modelled on finite groups of rank 2

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

reflexivity of non commutative hardy algebras abstract let h e be a non commutative hardy algebra associated with a w correspondence e these algebras were introduced in 2004 by muhly and solel and generalize the classical hardy algebra of the unit disc h d as a special case one obtains also the algebra f d of popescu which is h c d in our setting in this paper we view the algebra h e as acting on a hilbert space via an induced representation we write it h e and we study the reflexivity of h e this question was studied by arias and popescu in the context of the algebra f d and by other authors in several other special cases as it will be clear from our work the extension to the case of a general w correspondence e over a general w algebra m requires new techniques and approach we obtain some partial results in the general case and we turn to the case of a correspondence over a factor under some additional assumptions on the representation m b h we show that h e is reflexive then we apply these results to analytic crossed products h m and obtain their reflexivity for any automorphism aut m whenever m is a factor finally we show also the reflexivity of the compression of the hardy algebra to a suitable coinvariant subspace m which may be thought of as a generalized symmetric fock space