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

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

uniform decay in viscoelasticity for kernels with small non decreasingness zones abstract we consider a viscoelastic problem with a relaxation function which may be strictly increasing in some sub intervals that is we go beyond the zero which was a bound for the derivative of the kernel in the previous works it is proved that we have different types of decay rates including the exponential one provided that the rate and or the non decreasingness zone including the flat zone is small enough in a certain sense

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

asymptotics of viscoelastic materials with nonlinear density and memory effects abstract this paper is concerned with the nonlinear viscoelastic equation t u t t u t t u u 0 s u t s d s f u h suitable to modeling extensional vibrations of thin rods with nonlinear material density t u t u and presence of memory effects this class of equations was studied by many authors but well posedness in the whole admissible range 0 4 and for f growing up to the critical exponent were established only recently the existence of global attractors was proved in presence of an additional viscous or frictional damping in the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor a for 4 and critical nonlinearity f

a symbol class for some schrodinger equations on r n on considere l equation de schrodinger dans r n id t dt h t et on interprete certains resultats de zelditch d une facon plus geometrique on suit les singularites de t a travers l espace x en introduisant une notion d ensemble front d onde qui est metaplectiquement covariant

long time behavior of a quasilinear viscoelastic equation with past history abstract this paper is concerned with a class of quasilinear wave equations with memory u t u t t u u t t t t s u s d s u t f u h 0 which was considered by several authors with 0 since 2001 existing results are mainly devoted to global existence energy decay existence with small data and blow up of solutions however uniqueness seems to be an open problem and existence of attractors was no yet considered the objective of the present paper is to provide some results on the well posedness and long time behavior to this equation in a more general setting which includes past history that is by taking in the memory term

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

the schur algorithm and reproducing kernel hilbert spaces in the ball abstract using reproducing kernel hilbert spaces methods we develop a schur type algorithm for a subclass of the functions analytic and contractive in the ball we also consider the nevanlinna pick interpolation problem in that class

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

decay estimates for some semilinear damped hyperbolic problems let be a bounded open domain in rn g r r a non decreasing continuous function such that g 0 0 and h e lloc1 r l2 under suitable assumptions on g and h the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u lu g u h t x as t the results obtained by use of differential inequalities can be applied to the case of the semilinear wave equation u u delta u g text u t text h text in r times omega text u text 0 on r times partial omega in r u 0 onr for instance if g s c left s right p 1 s d left s right q 1 s with c d 0 and 1 p q n 2 q n 2 then if h in l infty r l 2 omega all solutions are bounded in the energy space for t 0 and if u v are two such solutions the energy norm of u t v t decays like t 1 p 1 as t

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star shaped domain we study the asymptotic stability and give the energy decay rates for solutions of the wave equation with boundary dissipation of the memory type

arbitrary decays in linear viscoelasticity it is by now well known that a necessary condition for the exponential polynomial decay of the energy of a problem arising in viscoelasticity is that the kernel which appears in the memory term itself be of exponential polynomial type by a kernel of exponential polynomial type we mean that the product of this kernel with an exponential polynomial function is summable some researchers have started from this condition to seek other sufficient conditions ensuring exponential or polynomial decay of the energy in this work we generalize these works to an arbitrary decay not necessarily of an exponential or polynomial rate

optimal g closure bounds via stability under lamination optimal bounds for a two dimensional g closure problem are proved they are obtained by combining together bounds derived using fine properties of quasiconformal mappings and a stability property of the g closure which is known as stability under lamination

existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source abstract we consider the system of nonlinear wave equations u t t u t m 1 u t div u 2 u f 1 u v x t 0 t v t t v t r 1 v t div v 2 v f 2 u v x t 0 t with initial and dirichlet boundary conditions under some suitable assumptions on the functions f 1 and f 2 the initial data and the parameters in the equations the theorems of global existence and nonexistence are proved

uniform decay of solutions for a quasilinear system of viscoelastic equations abstract in this paper we consider a system of two coupled quasilinear viscoelastic equations in canonical form with dirichlet boundary condition we use the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure uniform decay with exponential and polynomial rates of the solutions energy which extends some existing results for a single equation to the case of a coupled system

energy decay rates for von karman system with memory and boundary feedback abstract in this paper we study the von karman plate model with long range memory and boundary nonlinear feedback we establish an explicit and general decay rate result using some properties of the convex functions our result is obtained without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function

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

clark theory in the drury arveson space abstract we extend the basic elements of clark s theory of rank one perturbations of backward shifts to row contractive operators associated to de branges rovnyak type spaces h b contrastively contained in the drury arveson space on the unit ball in c d the aleksandrov clark measures on the circle are replaced by a family of states on a certain noncommutative operator system and the backward shift is replaced by a canonical solution to the gleason problem in h b in addition we introduce the notion of a quasi extreme multiplier of the drury arveson space and use it to characterize those h b spaces that are invariant under multiplication by the coordinate functions

a well posedness result for nonlinear viscoelastic equations with memory abstract we establish an existence uniqueness and continuous dependence result for the weak solutions to the nonlinear viscoelastic equation with hereditary memory on a bounded three dimensional domain t u t t u t t u t u u 0 s u t s d s f u h with dirichlet boundary conditions in particular the parameter belongs to the interval 0 4 the value 4 being critical for the sobolev embeddings while f can reach the critical polynomial order 5

general decay rate estimates for viscoelastic dissipative systems abstract the linear viscoelastic equation is considered we prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity the proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates

energy decay rate for a von karman system with a boundary nonlinear delay term abstract in this paper we show the energy decay rate for a von karman system with a boundary nonlinear delay term this work is devoted to investigate the influence of kernel function g and the effect of the boundary nonlinear term 1 u t t m 1 u t t a boundary nonlinear time delay term 2 u t t m 1 u t t and prove energy decay rates of solutions when g do not necessarily decay exponentially and the boundary condition has a time delay

noncommutative reproducing kernel hilbert spaces abstract the theory of positive kernels and associated reproducing kernel hilbert spaces especially in the setting of holomorphic functions has been an important tool for the last several decades in a number of areas of complex analysis and operator theory an interesting generalization of holomorphic functions namely free noncommutative functions e g functions of square matrix arguments of arbitrary size satisfying additional natural compatibility conditions is now an active area of research with motivation and applications from a variety of areas e g noncommutative functional calculus free probability and optimization theory in linear systems engineering the purpose of this article is to develop a theory of positive kernels and associated reproducing kernel hilbert spaces for the setting of free noncommutative function theory

the metaplectic representation weyl operators and spectral theory abstract let x be the grassmannian of lagrangian subspaces of r 2 n and x the bundle of negative half forms we construct a canonical imbedding s r n even c which intertwines the metaplectic representation of m p n on s r n with the induced representation of m p n on c this imbedding converts the algebra of weyl operators into an algebra of pseudodifferential operators and enables us to prove theorems about the spectral properties of weyl operators by reducing them to standard facts about pseudodifferential operators for instance we are able to prove a weyl theorem on the asymptotic growth of eigenvalues with an optimal error estimate for such operators and an analogue of the helton clustering theorem and the chazarain duistermaat guillemin trace formula

the douglas property for multiplier algebras of operators abstract for a collection of reproducing kernels k which includes those for the hardy space of the polydisk and the ball and for the bergman space k is a complete pick kernel if and only if the multiplier algebra of h 2 k has the douglas property consequences for solving the operator equation a x y are examined

non commutative clark measures for the free and abelian toeplitz algebras abstract we construct a non commutative aleksandrov clark measure for any element in the operator valued free schur class the closed unit ball of the free toeplitz algebra of vector valued full fock space over c d here the free analytic toeplitz algebra is the unital weak operator topology wot closed algebra generated by the component operators of the free shift the row isometry of left creation operators this defines a bijection between the free operator valued schur class and completely positive maps non commutative ac measures on the operator system of the free disk algebra the norm closed algebra generated by the free shift identifying drury arveson space with symmetric fock space we determine the relationship between the non commutative ac measures for elements of the operator valued commutative schur class the closed unit ball of the wot closed toeplitz algebra generated by the arveson shift and the ac measures of their free liftings to the free schur class

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

global existence and uniform decay for a nonlinear viscoelastic equation with damping abstract in this paper we investigate a nonlinear viscoelastic equation with linear damping global existence of weak solutions and the uniform decay estimates for the energy have been established

global and blow up solutions for a system of nonlinear hyperbolic equations with dissipative terms abstract in this paper we study a nonlinear hyperbolic system with strong damping firstly by use of the successive approximation method and a series of classical estimates we prove the local existence and uniqueness of weak solution secondly via some inequalities applying the potential method and the concave method we discuss the asymptotic and blow up behavior of weak solution with different conditions

general decay and blow up of solution for a quasilinear viscoelastic problem with nonlinear source abstract in this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous dirichlet boundary condition we prove that for certain class of relaxation functions and certain initial data in the stable set the decay rate of the solution energy is similar to that of the relaxation function this result improves earlier ones obtained by messaoudi and tatar s a messaoudi n e tatar global existence and uniform stability of solutions for a quasilinear viscoelastic problem math methods appl sci 30 2007 665 680 in which only the exponential and polynomial decay rates are considered conversely for certain initial data in the unstable set there are solutions that blow up in finite time the last result is new since it allows a larger class of initial energy which may take positive values