Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

Lyapunov-type Inequalities for Differential Equations

Let us consider the linear boundary value problem

The Strong Variant of a Barbashin Theorem on Stability of Solutions for Non-autonomous Differential Equations in Banach Spaces

Among others we shall prove that an exponentially bounded evolution family U = {U(t, s)} _t≥s≥0 of bounded linear operators acting on a Banach space X is uniformly exponentially stable if and only if there exists q [1, ∞) such that

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

Elements of the Theory of Linear Volterra Operators in Banach Spaces

The new definition of Volterra operator introduced in [5] allows specification of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations.     

Positive Schatten(-Herz) Class Toeplitz Operators on Pluriharmonic Bergman Spaces

We study characterizations of arbitrary positive Toeplitz operators of Schatten (or Schatten-Herz) type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman space. This work was supported by a Hanshin University Research Grant.     

Riesz Systems and Controllability of Heat Equations with Memory

The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system. Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems. In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target.

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This paper fits into the research program of the GNAMPA-INDAM .     

On Vector Integral Inequalities

The first author introduced an integration theory of vector functions with respect to an operator-valued measure in complete bornological locally convex vector spaces. In this paper some important results behind this Dobrakov-type integration technique in non-metrizable spaces are given. Received: December 10, 2007., Accepted: May 6, 2008.     

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).     

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.      

Boundedness of Commutators on Hardy-Type Spaces

In this paper, the boundedness of the commutator generated by strongly singular Calderón-Zygmund operator and a Lipschitz function is discussed on the classical Hardy spaces and the Herz-type Hardy spaces. The authors also get the boundedness of the strongly singular Calderón-Zygmund operator itself and the commutator generated by strongly singular Calderón-Zygmund operator and a BMO function on the Herz-type Hardy spaces.     

On Positive Linear Volterra-Stieltjes Differential Systems

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new. The first author is supported by the Alexander von Humboldt Foundation.     

The Problem of the Centre for Cubic Systems with Two Parallel Invariant Straight Lines and One Invariant Conic

For cubic differential systems with two parallel invariant straight lines and at least one invariant conic it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first three Liapunov quantities L _j , j = 1, 2, 3 vanish.      

The Aumann-Gould Integral

We introduce and study a multi-valued integral of Aumann-type where the selectors are Gould integrals of real functions with respect to finitely additive vector-valued measures. We also discuss the relationship between this kind of integral and the M-Gould set-valued integral with respect to a finitely additive multimeasure M. Received: November 17, 2006, Revised: September 17, 2007 and January, 8, 2008, Accepted: March 13, 2008.     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

Theorem of Completeness for a Dirac-Type Operator with Generalized λ-Depending Boundary Conditions

A completeness theorem is proved involving a system of integro-differential equations with some λ-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established. The research was supported by the Academy of Finland (project 129092).     

Value distribution of the difference operator

We investigate the value distribution of difference operators for entire functions, and establish a difference analogue of the Brück conjecture for entire functions of order less than 1. This work was supported by the NNSF of China (No. 10671109).     

Fock Spaces Connected with Spherical Mean Operator and Associated Operators

We define and study the Fock space associated with the spherical mean operator. Next, we establish some results for the Segal-Bergmann transform for this space. Lastly, we prove some properties concerning Toeplitz operators on this space. Received: May 11, 2007. Revised: May 20, 2008. Accepted: May 23, 2008.