Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

For a sectorial operator A with spectrum (A) that acts in a complex Banach space B, we prove that the condition (A) i R = Ø is sufficient for the differential equation where is a small positive parameter, to have a unique bounded solution x for an arbitrary bounded function f: R B that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as 0+ to the unique bounded solution of the differential equation x(t) = Ax(t) + f(t).     

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

Increasing Monotone Operators in Banach Space

An operator mapping a separable reflexive Banach space X into the dual space X is called increasing if as . Necessary and sufficient conditions for the superposition operators to be increasing are obtained. The relationship between the increasing and coercive properties of monotone partial differential operators is studied. Additional conditions are imposed that imply the existence of a solution for the equation with an increasing operator A.     

On the Neumann-Poincaré operator

Let be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L _C ² () the space of all complex-valued functions on which are square integrable w.r. to the arc-length on . Let L ²() stand for the space of all real-valued functions in L _C ² () and put Since the Cauchy singular operator is bounded on L _C ² (), the Neumann-Poincaré operator C ₁ sending each h L ²() into , is bounded on L ²(). We show that the inclusion characterizes the circle in the class of all AD-regular Jordan curves .     

On a certain Maximum Principle for Positive Operators in an Ordered Normed Space

The equation for a positive linear continuous operator is considered in an ordered normed space , where the cone is assumed to be closed and having a nonempty interior. Then the dual cone of K possesses a base . Generalizing the well known maximum principle for positive matrices an operator A is said to satisfy the maximum principle, if for any there exists a positive linear continuous functional which is both, maximal on the element Ax, i.e. , and positive on the element x, i.e. 0$$ " align="middle" border="0"> . This property is studied and characterized both analytically by some extreme point condition and geometrically by means of the behaviour under A of the faces of the cone K. It is shown that the conditions which have been obtained for finite dimensional spaces in earlier relevant papers are special cases of conditions presented in this paper. The maximum pinciple is proved for simple operators in the spaces and c.     

On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space which can roughly be described as follows: (1) If is an open subset of and all spectral subspaces for A corresponding to compact subsets of have finite rank of negativity, the same is true for a selfadjoint operator B in for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood of such that the restriction of A to a spectral subspace for A corresponding to is a nonnegative operator in is preserved under relative perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.     

Invariance theory for infinite dimensional linear control systems

In this paper we characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system , whereA is the infinitesimal generator of aC ₀-semigroup of operators and whereB is a bounded linear operator mapping a Hilbert space intoX. WhenS D(A^*) is dense inS , it is shown that a necessary (but insufficient) condition for holdability is (1): . A stronger condition than (1) is shown to be sufficient for a type of approximate holdability. In the finite dimensional setting, (1) reduces to (A, B)-invariance, which is known to be equivalent to the existence of a (bounded) linear feedback control law which achieves holdability inS. We prove that this equivalence holds in infinite dimensions as well, whenA is bounded and the linear spacesS, B andS+ B are closed.In the unbounded case, our results are illustrated by the shift semigroup and by the heat equation on an infinite rod with distributed controls. In the bounded case, our example is an integro-differential control system.Research sponsored by the National Research Council of Canada under Grant A7271.Research sponsored by the National Research Council of Canada under Grant A4641.     

Accretive perturbations and error estimates for the Trotter product formula

We study the operator-norm error bound estimate for the exponential Trotter product formula in the case of accretive perturbations. LetA be a semibounded from below self-adjoint operator in a separable Hilbert space. LetB be a closed maximal accretive operator such that, together withB ^*, they are Kato-small with respect toA with relative bounds less than one. We show that in this case the operator-norm error bound estimate for the exponential Trotter product formula is the same as for the self-adjointB [12]:

A Functional Interpretation of Cycloid Processes: Application to the Study of Asymptotic Behaviour

We show that the classic Chapman–Kolmogorov equations of certain Markovian transition semigroups on finite state spaces have a formal analogy, of a homologic nature, in terms of cycloids ₁, ..., _B, and positive numbers w₁, ..., w_B. The collection _k ,w _k completely determines a Markov process {_n}, called a cycloid process, admitting an invariant probability distribution, and decomposes its distribution Prob(_n = , _{n + 1} = ) into a linear expression. The latter is further used in the study of the asymptotic behaviour of the cycloid process.     

The Generalized Holditch Theorem for the Homothetic Motions on the Planar Kinematics

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E be a 1-parameter closed planar Euclidean motion with the rotation number and the period T. Under the motion E/E, let two points A = (0, 0), B = (a + b, 0) E trace the curves k _A, k _B E and let F _A, F _B be their orbit areas, respectively. If F _X is the orbit area of the orbit curve k of the point X = (a, 0) which is collinear with points A and B then

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

Invariance of asymptotic stability of perturbed linear systems on Hilbert space

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form are considered. The operatorA is assumed to be the infinitesimal generator of aC ₀-semigroup of contractionsT(t),t0, in a Hilbert spaceX;B is a bounded linear operator from another Hilbert spaceU toX; and {P(r),r } is a family of bounded or unbounded perturbations ofA inX, where is an arbitrary set, not necessarily carrying any topology. Sufficient conditions are presented that guarantee controllability and stabilizability of the perturbed system, given that the unperturbed system has similar properties. In particular, it is shown that, for certain classes of perturbations, weak and strong stabilizability properties are preserved for the same state feedback operator.This work was supported in part by the Natural Science and Engineering Research Council of Canada under Grant No. A7109.     

Characterization of Smoothness of Multivariate Refinable Functions and Convergence of Cascade Algorithms of Nonhomogeneous Refinement Equations

This paper concerns multivariate homogeneous refinement equations of the form

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

On the approximation to solutions of operator equations by the least squares method

We consider the equation Au = f, where A is a linear operator with compact inverse A ^–1 in a separable Hilbert space . For the approximate solution u _n of this equation by the least squares method in a coordinate system {e _k } _k that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A ( (B) = (A)), we give a priori estimates for the asymptotic behavior of the expressions r _n = u _n – u and R _n = Au _n – f as n . A relationship between the order of smallness of these expressions and the degree of smoothness of u with respect to the operator B is established.__________Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 85–90, 2005Original Russian Text Copyright © by M. L. GorbachukSupported by CRDF and Ukrainian Government Joint Grant UM1-2567-OD03.Translated by V. M. Volosov     

On Duality in Spaces of Solutions to Elliptic Systems

Suppose that D is a bounded domain in (n2) with connected real-analytic boundary, A is an elliptic system with real-analytic coefficients in a neighborhood of the closure of D, and sol(A,D) is the space of solutions to the system Au=0 in D furnished with the standard Frechet–Schwartz topology. Then the dual of sol(A,D) represents the space sol(A, ) of solutions to the system Au=0 in a neighborhood of furnished with the standard inductive limit topology over some decreasing net of neighborhoods of . The corresponding pairing is generated by the inner product in the Lebesgue space L ²(D).     

Tensor ideals in the category of tilting modules

We study the tensor category of tilting modules over a quantum groupU _q with divided powers. The setX ₊ of dominant weights is a union of closed alcoves numbered by the elementswW ^f of a certain subset of affine Weyl groupW. G. Lusztig and N. Xi defined a partition ofW ^f into canonical right cells and the right order _R on the set of cells. For a cellAW ^f we consider a full subcategory formed by direct sums of tilting modulesQ() with highest weights . We prove that is a tensor ideal in , generalizing H. Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then . The proof is an application of a recent result by W. Soergel who has computed the characters of tilting modules.This material is based upon work supported by the U.S. Civilian Research and Development Foundation under Award No. RM1-265.     

Sur le problème des marges

Summary LetX be a set,A an algebra of subsets ofX, m andM two mappings fromA to . Then there exists a finitely additive measure onA such thatmM if and only if for all the sequences (A ₁, ...,A _p ) and (B ₁, ...,B _q ) inA such that the inequality is satisfied. This simple condition permits us to deduce and generalize many previous results relating to the marginal problem.     

A stability theorem for a class of linear evolution systems

Dynamical systems of the form ,u(0)=u ₀, where,B is a densely defined linear operator mapping its domainD (B ) into — the infinitesimal generator of a semigroup of operatorsT (t, B) of classC ₀ — are investigated, such that for each solutionu to , whereP is the spectral eigenprojection onto the null space ofB.It is shown that under some general hypotheses concerning spectral properties ofB the above stability condition is equivalent with the following situation: There exist (i) a normal generating coneK such thatT(t;B)KK fort0 and (ii) a strictly positive element in the dual coneK such that , whereB denotes the dual ofB. Condition (ii) implies the so called total concentration time preservation, i. e. .