Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

LetU=(U(t, s)) _tsO be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG _O,G _X andI _X on certain spaces ofX-valued continuous functions connected with the integral equation , and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofG _O,G _X andI _X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.This work was done while the first author was visiting the Department of Mathematics of the University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in Tübingen for their kind hospitality and constant encouragement.Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged.     

Two-Wavelet Localization Operators on $$ L^p(\mathbb{R}^n) $$ for the Weyl-Heisenberg Group

We give results on the boundedness and compactness of localization operators with two admissible wavelets on for the Weyl-Heisenberg group.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coe.cient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive Böttchers formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Solvability and Fredholm Properties of Integral Equations on the Half-Line in Weighted Spaces

The solvability of integral equations of the form and the behaviour of the solution x at infinity are investigated. Conditions on k and on a weight function w are obtained which ensure that the integral operator K with kernel k is bounded as an operator on X_w, where X_w denotes the weighted space of those continuous functions defined on the half-line which are O(w(s)) as We also derive conditions on w and k which imply that the spectrum and essential spectrum of K on X_w are the same as on BC[0,). In particular, the results apply when when the integral equation is of Wiener-Hopf type. In this case we show that our results are particularly sharp.     

Regular Functions of Operators on Tensor Products of Hilbert Spaces

A class of linear operators on tensor products of Hilbert spaces is considered. Estimates for the norm of operator-valued functions regular on the spectrum are derived. These results are new even in the finite-dimensional case. By virtue of the obtained estimates, we derive stability conditions for semilinear differential equations. Applications of the mentioned results to integro-differential equations are also discussed.     

A Note on the Spectrum of Invertible p-hyponormal Operators

A bounded linear operator T is clalled p-hyponormal if (T*T)^p ≥ (TT)^p, 0 < p < 1. It is known that for semi-hyponormal operators (p = 1/2), the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. In this paper we prove a somewhat weaker result for invertible p-hyponormal operators for 0 < p < 1/2.     

On the Spectrum of Invertible Semi-hyponormal Operators

It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a different proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model.      

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.     

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      

LU- versus UL-Factorization of Integral Operators with Semi-Separable Kernel

The problem we consider is how to obtain a UL-factorization from a LU-factorization for integral operators with semi-separable kernels in both the time varying and the time invariant cases. We also consider the special situation where the integral operators are self-adjoint.     

Polar Wavelet Transforms and Localization Operators

The notion of a polar wavelet transform is introduced. The underlying non-unimodular Lie group, the associated square-integrable representations and admissible wavelets are studied. The resolution of the identity formula for the polar wavelet transform is then formulated and proved. Localization operators corresponding to the polar wavelet transforms are then defined. It is proved that under suitable conditions on the symbols, the localization operators are, in descending order of complexity, paracommutators, paraproducts and Fourier multipliers. This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators acting on an ordered finite dimensional Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well us in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequalities) or forward affine equations (inequalities).     

On Uniform Exponential Trichotomy of Evolution Operators in Banach Spaces

This paper presents necessary and sufficient conditions for uniform exponential trichotomy of nonlinear evolution operators in Banach spaces. Thus are obtained results which extend well-known results for uniform exponential stability in the linear case.      

On the Range of Elementary Operators

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A₁,A₂,.., A_n) and B = (B₁, B₂,.., B_n) be n-tuples in B(H), we define the elementary operator by In this paper we initiate the study of some properties of the range of such operators.     

Evolution equations governed by the sweeping process

This paper is concerned with variants of the sweeping process introduced by J.J. Moreau in 1971. In Section 4, perturbations of the sweeping process are studied. The equation has the formX(t) -N _C(t) (X(t)) +F(t, X(t)). The dimension is finite andF is a bounded closed convex valued multifunction. WhenC(t) is the complementary of a convex set,F is globally measurable andF(t, ·) is upper semicontinuous, existence is proved (Th. 4.1). The Lipschitz constants of the solutions receive particular attention. This point is also examined for the perturbed version of the classical convex sweeping process in Th. 4.1. In Sections 5 and 6, a second-order sweeping process is considered:X (t) -N _C(X(t)) (X(t)). HereC is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved whenC is dissipative (Th. 5.1) or when allC(x) are contained in a compact setK (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension whenC is continuous.     

Analysis of Non-normal Operators via Aluthge Transformation

Let T be a bounded linear operator on a complex Hilbert space . In this paper, we show that T has Bishops property () if and only if its Aluthge transformation has property (). As applications, we can obtain the following results. Every w-hyponormal operator has property (). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chs problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re(T) = (Re T) or not.     

Image of the Spectral Measure of a Jacobi Field and the Corresponding Operators

By definition, a Jacobi field is a family of commuting selfadjoint three-diagonal operators in the Fock space The operators J(ϕ) are indexed by the vectors of a real Hilbert space H₊. The spectral measure ρ of the field J is defined on the space H₋ of functionals over H₊. The image of the measure ρ under a mapping is a probability measure ρ_K on T₋. We obtain a family J_K of operators whose spectral measure is equal to ρ_K. We also obtain the chaotic decomposition for the space L²(T₋, d_{ρ K}).     

Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations

By using the Schauder fixed point theorem, we establish a result for the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. We also present the application of our result to the special case of second order nonlinear ordinary differential equations as well as to a specific class of second order nonlinear delay differential equations. Moreover, we give a general example which demonstrates the applicability of our result. Received: 10 May 2004