(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.     

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.     

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnans formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.     

Fredholm Indices of Band-Dominated Operators

The Fredholmness of a band-dominated operator on is closely related with the invertibility of its limit operators: the operator is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded. The goal of the present note is to show how the Fredholm index of a Fredholm band-dominated operator can be determined in terms of its limit operators.     

Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

To an evolution family on the half-line of bounded operators on a Banach space X we associate operators I_X and I_Z related to the integral equation and a closed subspace Z of X. We characterize the exponential dichotomy of by the exponential dichotomy and the quasi-exponential dichotomy of the operators X we associate operators I_X and I_Z, respectively.     

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.     

On Marcinkiewicz Integral Operators with Rough Kernels

This paper is devoted to the study on the L^p-mapping properties of Marcinkiewicz integral operators with rough kernels along “polynomial curves” on The boundedness of the Marcinkiewicz integrals for some fixed 1 < p < ∞ are obtained under some size conditions, which essentially improve or extend some well-known results.     

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.     

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.     

Norms of Moore-Penrose Inverses of Fredholm Toeplitz Operators

In this paper we estimate the norm of the Moore-Penrose inverse T(a)⁺ of a Fredholm Toeplitz operator T(a) with a matrix-valued symbol a∈L_{N × N}^∞ defined on the complex unit circle. In particular, we show that in the ”generic case” the strict inequality ||T(a)⁺|| > ||a⁻¹||_∞ holds. Moreover, we discuss the asymptotic behavior of ||T(t^ra)⁺|| for . The results are illustrated by numerical experiments.     

A Fredholm Determinant Formula for Section Determinants of Bounded Operators

We show that the section determinant of e^A can be expressed, under certain conditions, by the Fredholm determinant of an integral operator. The kernel function of this integral operator is computed explicitly in terms of the operator A. As a simple consequence we derive a Weierstrass type product expansion for the section determinant.     

On a Class of Integral Operators

In this paper we consider the space where dv _s is the Gaussian probability measure. We give necessary and sufficient conditions for the boundedness of some classes of integral operators on these spaces. These operators are generalizations of the classical Bergman projection operator induced by kernel function of Fock spaces over .      

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.     

Inverse Problems for Sturm–Liouville Operators on Noncompact Trees

We study Sturm–Liouville differential operators on noncompact graphs without cycles (i.e., on trees) with standard matching conditions in internal vertices. First we establish properties of the spectral characteristics and then we investigate the inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness theorem and propose a procedure for constructing the solution using the method of spectral mappings. Received: February 13, 2007.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

CBMO Estimates for Commutators of Multilinear Fractional Integral Operators on Herz Spaces

The CBMO estimates for commutators of fractional integral and Multilinear fractional integral operators with rough kernel are established. This work was completed with the support of Hunan Provincial Natural Science Foundation of China 06A0074.     

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     

Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.