On the exponential dichotomy of linear difference equations

We consider a system of linear difference equationsx ⁿ⁺¹ =A (n)xⁿ in anm-dimensional real or complex spaceVsum with detA(n) = 0 for some or alln εZ. We study the exponential dichotomy of this system and prove that if the sequence {A(n)} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {A (n)} is almost periodic and the system has exponential dichotomy on the finite interval {k, ...,k +T},k εZ, with sufficiently largeT, then the system is exponentially dichotomous onZ.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

On exponential dichotomy, Bohl-Perron type theorems and stability of difference equations

For a delay difference equation

     

Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup

We find necessary and sufficient conditions on a Banach spaceX in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak type (1, 1) or strong type (p, p) in the range 1<p<∞. In this setting, we consider the Riesz transforms and the Littlewood-Paleyg-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure. In all cases, we show that the condition onX is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property. In doing so, we also find some new equivalences even for the harmonic case. The first and third authors were partially supported by CONICET (Argentina) and Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral. The second author was partially supported by the European Commission via the TMR network “Harmonic Analysis”.     

On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

In this paper, for each given $ we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from into We approach the poblem by proving some vector valued inequalities. As a byproduct we obtain the strong type (1,1) for the “global” part of the centered maximal function. Received May 18, 1999 / Revised December 9, 1999 Published online July 20, 2000     

Uniform exponential dichotomy of parabolic operators with rapidly oscillating coefficients

The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients.     

On the exponential dichotomy of pulse evolution systems

The equivalence of regularity and exponential dichotomy is established for linear pulse differential equations with unbounded operators in a Banach space. The separatrix manifolds of a linear pulse system exponentially dichotomous on a semiaxis are studied in a finite-dimensional space. The conditions of weak regularity of this system are given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 418–424, April, 1994.This work was supported by Ukrainian State Committee for Science and Technology (the Foundation for Fundamental Studies).     

Maximal operators related to the Ornstein-Uhlenbeck semigroup with complex time parameter

Let γ be the Gaussian measure in Rd and Ht, t>0, the corresponding Ornstein-Uhlenbeck semigroup, whose infinitesimal generator is . For each p with 1<p<∞, let Ep⊂C be the closure of the region of holomorphy of the map t?Ht taking values in the space of bounded operators on Lp(γ). We examine the maximal operator . The known results about concern mainly the case p<2. We prove that for p>2 this operator is of weak type but not of strong type (p,p) for γ. However, if a neighbourhood of the origin is deleted from Ep in the definition of , the resulting operator is shown to be of strong type.     

On the proof of characterizations of the exponential dichotomy

We describe explicitly the generator of the evolutionary semigroup associated with the evolutionary operator generated by the linear differential equation . From this we give a short proof of some known characterizations of the exponential dichotomy of the above mentioned equation.

     

On local spectral properties of complex symmetric operators

In this paper we study properties of complex symmetric operators. In particular, we prove that every complex symmetric operator having property (β) or (δ) is decomposable. Moreover, we show that complex symmetric operator T has Dunford?s property (C) and it satisfies Weyl?s theorem if and only if its adjoint does.     

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)^l +V inR ⁿ, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.     

On local spectral properties of Hamilton type operators

In this paper, we introduce the class of Hamilton type operators and study various properties of this class. We show that every Hamilton type operator with property (β) or (δ) is decomposable. In addition, we prove that a Hamilton type operator T satisfies property (β), Dunford’s property (C) and Weyl’s theorem if and only if its adjoint does.     

The concept of spectral dichotomy for linear difference equations II

This paper is a continuation of a previous one (J. Math. Anal. Appl. 185 (1994), 275–287) in which the concept of spectral dichotomy has been introduced. This new notion of dichotomy has proved to be useful since it allows to apply the well known theory of linear operators to study dynamic properties of nonautonomous linear difference equations. In the present paper we extend our result on the equivalence of the spectral dichotomy and the well known exponential dichotomy to the class of linear differenc equations whose right-hand sides are not necessarily invertible. We furthermore investigate equations on the set of positive integers for which we establish necessary and sufficient conditions for exponential and unifrom stability.     

The exponential dichotomy of solutions to systems of linear difference equations with almost periodic coefficients

By the direct Lyapunov method we prove a sufficient condition for the exponential dichotomy with a weakened (in comparison with the case of arbitrary coefficients) requirement to the difference derivative of the Lyapunov function along the system trajectory. We give an illustrative example.     

On spectral properties of some classes of linear operators

Voronezh State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 2, pp. 61–64, April–June, 1995.     

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ={Φ(m,n)}_(m,n)∈Δ the subspace . Supposing that X₁ is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on X₁, we obtain that the admissibility of the pair with p∈(1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.     

On the exponential dichotomy of linear almost periodic pulse systems

For a linear almost periodic pulse system, we prove that the exponential dichotomy on a semiaxis implies the exponential dichotomy on the entire axis. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 136–142, January, 1998.