hankel integral operators and isometric interpolants on the line abstract the problem of finding an m n continuous isometric matrix valued function with entries from the wiener algebra on the line and with prespecified fourier inverse transform on the half line is studied conditions for existence uniqueness and formulas for the solution of this problem are presented connections are made between the positive factorization indices of certain solutions to this problem and the dimensions of the kernels of hankel operators based on the prespecified data alluded to above the paper uses techniques and results developed in an earlier study of an interpolation problem on the circle the main theorems are in fact continuous analogues of the latter

martingale weak solutions of the stochastic landau lifshitz bloch equation abstract the present paper studies the stochastic landau lifshitz bloch equation which is recommended as the only valid model at temperature around the curie temperature and is especially important for the simulation of heat assisted magnetic recording we study the stochastic landau lifshitz bloch equation in the case that the temperature is raised higher than the curie temperature the global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed

banach algebras generated by an invertible isometry of an l p space abstract we provide a complete description of those banach algebras that are generated by an invertible isometry of an l p space together with its inverse examples include the algebra p f p z of p pseudofunctions on z the commutative c algebra c s 1 and all of its quotients as well as uncountably many exotic banach algebras we associate to each isometry of an l p space a spectral invariant called spectral configuration which contains considerably more information than its spectrum as an operator it is shown that the spectral configuration describes the isometric isomorphism type of the banach algebra that the isometry generates together with its inverse it follows from our analysis that these algebras are semisimple with the exception of p f p z they are all closed under continuous functional calculus and their gelfand transform is an isomorphism as an application of our results we show that banach algebras that act on l 1 spaces are not closed under quotients this answers the case p 1 of a question asked by le merdy 20 years ago

ideals with bounded approximate identities in fourier algebras abstract we make use of the operator space structure of the fourier algebra a g of an amenable locally compact group to prove that if h is any closed subgroup of g then the ideal i h consisting of all functions in a g vanishing on h has a bounded approximate identity this result allows us to completely characterize the ideals of a g with bounded approximate identities we also show that for several classes of locally compact groups including all nilpotent groups i h has an approximate identity with norm bounded by 2 the best possible norm bound

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

operator space structure and amenability for figa talamanca herz algebras column and row operator spaces which we denote by col and row respectively over arbitrary banach spaces were introduced by the first named author for hilbert spaces these definitions coincide with the usual ones given a locally compact group g and p p 1 with 1p 1p 1 we use the operator space structure on cb col lp g to equip the figa talamanca herz algebra ap g with an operator space structure turning it into a quantized banach algebra moreover we show that for p q 2 or 2 q p and amenable g the canonical inclusion aq g ap g is completely bounded with cb norm at most kg2 where kg is grothendieck s constant as an application we show that g is amenable if and only if ap g is operator amenable for all and equivalently for one p 1 this extends a theorem by ruan

completely isometric representations of _ and let g be a locally compact group it is shown that there exists a natural completely isometric representation of the completely bounded fourier multiplier algebra m cb a g which is dual to the representation of the measure algebra m g on b l 2 g the image algebras of m g and m cb a g in cb b l 2 g are intrinsically characterized and some commutant theorems are proved it is also shown that for any amenable group g there is a natural completely isometric representation of ucb g on b l 2 g which can be regarded as a duality result of neufang s completely isometric representation theorem for luc g

completely bounded homomorphisms of the fourier algebras abstract for locally compact groups g and h let a g denote the fourier algebra of g and b h the fourier stieltjes algebra of h any continuous piecewise affine map y h g where y is an element of the open coset ring induces a completely bounded homomorphism a g b h by setting u u on y and u 0 off of y we show that if g is amenable then any completely bounded homomorphism a g b h is of this form and this theorem fails if g contains a discrete nonabelian free group our result generalises results of cohen amer j math 82 1960 213 226 host bull soc math france 1986 114 and of the first author j funct anal 2004 213 we also obtain a description of all the idempotents in the fourier stieltjes algebras which are contractive or positive definite

quotients of banach algebras acting on lp spaces we show that the class of banach algebras that can be isometrically represented on an lp space for p 2 is not closed under quotients this answers a question asked by le merdy 20 years ago our methods are heavily reliant on our earlier study of banach algebras generated by invertible isometries of lp spaces

state space formulae for the factorization of all pass matrix functions abstract we consider the factorization of an all pass matrix function e s using proper stable minimum phase factors stale space formulae for the stable minimum phase factors of e s are derived when they exist as well as state space formulae for the wiener hopf factors this is achieved using the state space characterization of all pass matrix functions given by glover 1984 in conjunction with the results of green and anderson 1987

a stochastic partial differential equation with values in a manifold abstract we investigate a certain stochastic partial differential equation which is defined on the unit interval with periodic boundary condition and takes values in a manifold such equation has particularly two different applications namely it determines the evolution law of an interacting constrained system of continuum distributed over the unit circle while it defines a diffusive motion of loops on a manifold we establish the existence and uniqueness results and then show the smoothness property of the solutions some examples are given in the final section

spectral factor reduction by phase matching the continuous time single input single output case a procedure for reducing the order of a random process spectral factor without altering its stable minimum phase property is presented it is based upon the balanced or the hankel norm approximation of a certain all pass function characterizing the phase of the process spectral factor the motivation for this particular methodology for model order reduction stems from the canonical correlation analysis of akaike 1975 further bode s gain phase relation and nehari s extension used in conjunction provide a functional analytic justification of the reduction procedure the net result of this procedure is a stable minimum phase reduced order spectral factor that matches the original one fairly well the paper draws heavily on the early discrete time paper of jonckheere and helton 1984 in the paper we treat the continuous time case and extend the results of jonckheere and helton in several directions

power spectrum reduction by optimal hankel norm approximation of the phase of the outer spectral factor a new approach to stochastic processes power spectrum reduction based on akaike s canonical correlation between the process past and future is developed as a first result the correlation between the canonical past and the canonical future is given its precise functional analytic interpretation as the hankel operator generated lay the all pass function characterizing the phase of the process outer spectral factor the canonical correlation coefficients turn out to he nothing else but the singular values of this hankel operator next the all pass function characterizing the phase of the outer spectral factor is reduced using the optimal hankel norm procedure of adamjan arov and krein the reduction of the all pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through this involves bode s relations between the gain and the phase of an outer function nehari s extension and the winding number of the all pass phase function which is itself shown to be related to the definition of the past and the future finally the net result is a reduced order outer spectral factor that matches the high order spectral factor fairly well

large deviations and transitions between equilibria for stochastic landau lifshitz gilbert equation we study a stochastic landau lifshitz equation on a bounded interval and with finite dimensional noise we first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property next we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method an essential ingredient of the proof is the compactness or weak to strong continuity of the solution map for a deterministic landau lifschitz equation when considered as a transformation of external fields we then apply this large deviations principle to show that small noise can cause magnetisation reversal we also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise the problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories

on p approximation properties for p operator spaces abstract this paper has a two fold purpose let 1 p we first introduce the p operator space injective tensor product and study various properties related to this tensor product including the p operator space approximation property for p operator spaces on l p spaces we then apply these properties to the study of the pseudofunction algebra pf p g the pseudomeasure algebra pm p g and the figa talamanca herz algebra a p g of a locally compact group g we show that if g is a discrete group then most of approximation properties for the reduced group c algebra c g the group von neumann algebra vn g and the fourier algebra a g related to amenability weak amenability and approximation property of g have the natural p analogues for pf p g pm p g and a p g respectively the p completely bounded multiplier algebra m cb a p g plays an important role in this work

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

balanced stochastic realizations abstract the balanced stochastic realization is introduced as a balanced solution to the continuous time positive real equations and the dual positive real equations the structure of the associated balanced spectral factors is derived the main result being a product decomposition of the spectral factors the properties of the stochastic model reduction technique of balanced stochastic truncation are considered in particular it is shown that balanced stochastic truncation preserves the right half plane zeros of the spectral factors leading to a transformation approach to the approximation of non minimum phase stochastic systems

existence of global solutions and invariant measures for stochastic differential equations driven by poisson type noise with non lipschitz coefficients the purpose of this paper is twofold firstly we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi martingales secondly we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent

unit canonical correlations between future and past on considere des processus arma stationnaires x t de rang complet et le cas ou il y a des fonctions lineaires du futur x t t 0 et du passe x t t 0 qui ont une correlation unite on montre que le nombre de paires de ce type lineairement independantes est le nombre de zeros du determinant des fonctions de transfert sur le cercle unite

perturbation of semi linear evolution equations under weak assumptions at initial time abstract we prove perturbation results for abstract semi linear evolution equations in a banach space the main feature is that only very weak assumptions are needed at initial time this allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi linear parabolic and hyperbolic boundary value problems with various boundary conditions the theory also implies the well known theory on parameter dependent equations

column and row operator spaces over qslp spaces and their use in abstract harmonic analysis the notions of column and row operator space were extended by a lambert from hilbert spaces to general banach spaces in this paper we use column and row spaces over quotients of subspaces of general lp spaces to equip several banach algebras occurring naturally in abstract harmonic analysis with canonical yet not obvious operator space structures that turn them into completely bounded banach algebras we use these operator space structures to gain new insights on those algebras

state space approach to approximation by phase matching a state space realization of the stable part of the phase function of a proper spectral factor is provided the phase matching problem jonckheere and helton 1985 harshavardhana and jonckheere 1985 can now be solved completely in the state space framework using existing numerically reliable procedures further a connection is made between the stochastic balancing approach and the phase approximation procedure for the stochastic realization approximation

stochastic geometric wave equations with values in compact riemannian homogeneous spaces let m be a compact riemannian homogeneous space e g a euclidean sphere we prove existence of a global weak solution of the stochastic wave equation dt tu dk 1dxk xku fu du gu du w in any dimension d 1 where f and g are continuous multilinear maps and w is a spatially homogeneous wiener process on rd with finite spectral measure a nonstandard method of constructing weak solutions of spdes that does not rely on martingale representation theorem is employed

unitary interpolants factorization indices and infinite hankel block matrices abstract the existence uniqueness and construction of unitary n n matrix valued functions j j j in wiener like algebras on the circle with prescribed matrix fourier coefficients j j for j 0 are studied in particular if j then such an exists with j if and only if 0 1 where v denotes the infinite block hankel matrix j k v j k 0 1 acting in the sequence space l n 2 one of the main results is that the nonnegative factorization indices of every such are uniquely determined by the given data in terms of the dimensions of the kernels of i v v whereas the negative factorization indices are arbitrary it is also shown that there is a unique such if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for in terms of certain schmidt pairs of 0 are given the results depend upon a fine analysis of the structure of the kernels of i v v and of the one step extension problem of adamjan arov and krein funct anal appl 2 1968 1 18 isometric interpolants for the nonsquare case are also considered

factorization of all pass matrix functions we consider the factorization of an all pass matrix function e s into proper stable minimum phase factors this is achieved by applying the state space wiener hopf factorization theory of bart et al 1983 to the state space characterization of all pass matrix functions given by glover 1984 important connections are made between the properties of e s as a nehari extension its partial wiener hopf factorization indices and its hankel singular values

the global attractor for the 2d navier stokes flow on some unbounded domains we extend previous results on the existence of the global attractor for the 2d navier stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth the existence of the global attractor is obtained on arbitrary open sets such that the poincare inequality holds and for forces in the natural dual space v the proof is based on the energy equation and the concept of asymptotic compactness an estimate of the hausdorff and fractal dimensions of the attractor is also given

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

amenability and weak amenability of the fourier algebra let g be a locally compact group we show that its fourier algebra a g is amenable if and only if g has an abelian subgroup of finite index and that its fourier stieltjes algebra b g is amenable if and only if g has a compact abelian subgroup of finite index we then show that a g is weakly amenable if the component of the identity of g is abelian and we prove some partial results towards the converse