The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Although various operators in the space of functions of bounded variation have been studied by quite a few authors, no simple necessary and sufficient conditions guaranteeing compactness of linear integral operators acting in such spaces have been known. The aim of the paper is to fully characterize the class of kernels which generate compact linear integral operators in the BV-space. Using this characterization we show that certain weakly singular and convolution operators (such as the Abel and Volterra operators), when considered as transformations of \(BV[a,b]\), are compact. We also provide a detailed comparison of those new necessary and sufficient conditions with various other conditions connected with compactness of linear (integral) operators in the space of functions of bounded variation which already exist in the literature.

On linear integral operators with nonnegative kernels

This article generalizes some spectral inequalities for non-negative matrices (see [2] and [3]) to compact integral operators with non-negative kernels defined on Banach function spaces. The spectral radius of a sum of such operators is estimated under certain conditions and a generalization of this inequality is given. An inequality for the spectral radius of a compact integral operator with the kernel equal to a weighted geometric mean of non-negative kernels is also proved.

On the Schur Test forL2-Boundedness of Positive Integral Operators with a Wiener–Hopf Example

33 - SVD for linear inverse problems

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained.

We are concerned with the numerical solution of the eigenvalue problem $T\varphi = \lambda \varphi ,\varphi \ne 0$, where T is a linear compact integral operator in a Banach space. We present in a unifying manner most of practical numerical methods, i.e. projections and approximate quadratures methods, with a special interest in convergence rates and a posteriors error bounds.The linear operator T and its approximation $T_n$ are defined in the same space. Perturbation theory is then a suitable framework for our problem. It enables us to produce an algorithm which computes $\lambda $ and $\varphi $ by iterations, starting from the eigenelements of $T_n$. The computations require only matrix calculations of a fixed order, which can be kept small relative to the resulting accuracy.The application of the algorithm for projection methods is presented with an emphasis on the Galerkin method and its Sloan variant. The efficiency of the algorithm is illustrated for integral operators, approximated by the projecti...

Let T be a bounded linear operator on a Banach space X. A subset of the spectrum of T which is invariant under certain compact perturbation of T is studied. It consists of the spectrum of T with finite-dimensional poles deleted. In the case of a bounded operator, it coincides with the essential spectrum as defined by F. E. Browder [1]. It is characterized as a set considered by Caradus [2]. A formula of the spectral radius type is proved. Furthermore, a spectral mapping theorem is valid. The notation is that of Taylor [5]. Let R(T) denote the range of T and N(T) the nullspace of T, i.e., N(T) = {x: Tx=0}. The dimension of N(T), n(T), is called the nullity of T and the codimension of R(T), d(T), the defect of T. Suppose for some integer k, N(Th) =N(Tk+l); then the ascent, a(T), is defined as the smallest value of k for which this is true. The smallest integer for which R(TK) =R(TK+1) is called the descent of T and is denoted by a(T). For the operator X-T, n(X-T) is abbreviated to n(A), etc. B(X) will denote the bounded linear operators, C(X) the compact linear operators. ALB means AB = BA = 0. Let [T] fEB (X)/C(X); then ao([T]) denotes the spectrum of [T] as an element of that Banach algebra. For a linear operator T, letP(T)= {CEC(X):T-C?C} and Q(T)= {DEC(X): DT= TD }. The object of this paper is to study the sets

Given an operator T bounded on a weighted Lp space, the factorization technique of Rubio de Francia forces strong conditions on the weight.This algorithm is extended to two weight problems, and is shown to yield not just necessary but sufficient conditions in a wide range of settings.A nonnegative weight w . Ap for some p > 1 if for all intervals (or cubes) I, mlwiI)dx(wJr~m'"f 'sc' where C is a universal constant independent of I. We will use the notation w(I) for fj w(x) dx and I(w) for w(I)/\I\.Muckenhoupt studied these weights in detail, establishing, among other facts, that the Hardy-Littlewood maximal operator / -> /* is bounded on Lp(wdx) if and only if w G Ap [5].A weight w Ai provided I(w) < Cessinf w for all intervals I, or equivalently, w*(x) = supxeII(w) < Cw(x) a.e.Peter Jones [4] showed that w Ap if and only if there exist weights u and v Ai with w = uv1~p.Recently, Jose L. Rubio de Francia applied Maurey's theory of factorization of operators to such weighted norm inequalities and obtained a very elegant proof of the Jones' factorization theorem [7].This technique, which I have dubbed the Rubio de Francia Algorithm, or the RdFA, yields factorization theorems for many types f operators.Guido Weiss [10] observed that in the unweighted case, when the operator is an integral operator with nonnegative kernel, the condition forced by the RdFA corresponds to the hypotheses in Schur's lemma.So, the RdFA produces necessary and sufficient conditions for the boundedness of an operator.In 1, we will apply the RdFA to the two-weight problem for such integral operators, obtaining necessary and sufficient conditions.In 2, we will look at a number of applications.1.The RdFA.Let X denote an appropriate measure space, 7?" for n > 1 or the unit circle in the complex plane.Let p and A be nonnegative weight functions.Lp(p) will denote LP(X,p(x)dx).THEOREM (THE RDFA).Let Q(x,y) > 0 and let T be the operator Tf(x) = Q(x,y)f(y)dy.Let p > 1, with l/p + 1/q = 1.Then T: Lp(p) -> LP(X) is a

In this work, we present methods for constructing representations of polynomial covariance type commutation relations AB=BF(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AB=BF(A)$$\\end{document} by linear integral operators in Banach spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.

In recent articles the first author and H. Gonska [e.g., see G. Anastassiou, C. Cottin, and H. Gonska, Global smoothness of approximating functions, Analysis, 11, 43–57 (1991); G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] studied global smoothness preservation by some univariate and multivariate linear operators over compact domains and ℝ n , n ≥ 1. In particular, they studied a very general positive linear integral type operator [e.g., see G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] over ℝ n that was introduced through a convolution-like integration of another general positive linear operator with a scaling-type function. In this article the authors, among others, extend and generalize [G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)]. Also certain new similar but more general integral operators are introduced and studied. These operators arise in a natural way, and for all these sufficient conditions are given for shift invariance, preservation of higher-order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preservation, and preservation of continuous probabilistic distribution functions. Several examples of very general specialized operators, old and new, are given that satisfy all the above properties.

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

Variational principles for finding real and complex nonzero eigenvalues, and associated eigenvectors, of a linear compact operator K on a Hilbert space are developed and analyzed. When K is self-adjoint, certain unconstrained variational problems are described for finding the positive, respectively, negative, eigenvalues of K and the corresponding eigenvectors. These principles are extended to generalized eigenproblems and to nonlinear compact operators. For nonself-adjoint linear operators, a minimization problem for certain positive real eigenvalues is described. All the positive real eigenvalues may be described as critical points of a Lagrangian functional. These characterizations are then extended to describe complex eigenvalues and eigenvectors of nonself-adjoint, compact linear operators.

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behavior of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composite operator decay faster than those of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

Linear non-compact operators are difficult to study because they do not exist in the finite-dimensional world. Recently, Hofmann and Mathé [Electron. Trans. Numer. Anal., 57 (2022), pp. 1–16] studied the singular values of the compact composition of the non-compact Hausdorff moment operator and the compact integral operator and found credible arguments, but no strict proof, that those singular values fall only slightly faster than those of the integral operator alone. However, the fact that numerically the singular values of the combined operator fall exponentially fast was not mentioned. In this note, we supply the missing numerical results and provide an explanation why the two seemingly contradictory results may both be true.