On graph measures in banach spaces and description of essential spectra of multidimensional transport equation

In this paper, we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness. Moreover, we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces. These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L₁ spaces.     

A note on the essential pseudospectra and application

In this research article, we work with the notion of essential pseudospectra of closed, densely defined linear operators in the Banach space. We start by giving the definition and we investigate the characterization, the stability and some properties of these essential pseudospectra.     

Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators

In this paper, we prove that, if the product A=A₁?An is a Fredholm operator where the ascent and descent of A are finite, then Aj is a Fredholm operator of index zero for all j, 1?j?n, where A₁,…,An be a symmetric family of bounded operators. Next, we investigate a useful stability result for the Rako?evi?/Schmoeger essential spectra. Moreover, we show that some components of the Fredholm domains of bounded linear operators on a Banach space remain invariant under additive perturbations belonging to broad classes of operators A such as γ(Am)<1 where γ(⋅) is a measure of noncompactness. We also discuss the impact of these results on the behavior of the Rako?evi?/Schmoeger essential spectra. Further, we apply these latter results to investigate the Rako?evi?/Schmoeger essential spectra for singular neutron transport equations in bounded geometries.     

Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

A note on structured pseudospectra

In this note, we study the notion of structured pseudospectra. We prove that for Toeplitz, circulant, Hankel and symmetric structures, the structured pseudospectrum equals the unstructured pseudospectrum. We show that this is false for Hermitian and skew-Hermitian structures. We generalize the result to pseudospectra of matrix polynomials. Indeed, we prove that the structured pseudospectrum equals the unstructured pseudospectrum for matrix polynomials with Toeplitz, circulant, Hankel and symmetric structures. We conclude by giving a formula for structured pseudospectra of real matrix polynomials. The particular type of perturbations used for these pseudospectra arise in control theory.     

Measure of weak noncompactness, some new properties in Fredholm theory, characterization of the Schechter essential spectrum and application to transport operators

Measures of weak noncompactness have been successfully applied in topology, functional analysis and operator theory. In this paper, we consider one axiomatic approach to this notion which includes the most important classical definitions. We work with the notion of the measures of weak noncompactness in order to establish some results concerning the class of semi-Fredholm and Fredholm operators. Further, we apply the obtained results to prove, under some conditions on the perturbed operator, the invariance of the Schechter essential spectrum on Banach spaces. These results are exploited to investigate the Schechter essential spectrum of a multidimensional neutron transport operator.     

Approximated structured pseudospectra

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank‐one or projected rank‐one perturbations of the given matrix is proposed. The choice of rank‐one or projected rank‐one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank‐one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra.     

Controllability for a class of second-order evolution differential inclusions without compactness

In this paper, we discuss the existence and controllability for a class of second-order evolution differential inclusions without compactness in Banach spaces. By applying the technique of weak topology and Glicksberg–Ky Fan fixed point theorem, we prove our main results without the hypotheses of compactness on the operator generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Further, we extend our study to existence and controllability of second-order evolution differential inclusions with nonlocal conditions and impulses. Finally, an example is given for the illustration of the obtained theoretical results.     

A fast algorithm for computing the pseudospectra of Toeplitz matrices

The concept of pseudospectra was introduced by Trefethen during the 1990s and became a popular tool to explain the behavior of non-normal matrices. In this paper, we propose a fast algorithm for computing the pseudospectra of Toeplitz matrices by using fast Toeplitz QR factorization. Numerical experiments are given to illustrate the efficiency of the new algorithm.     

Convolution Operators on Expanding Polyhedra: Limits of the Norms of Inverse Operators and Pseudospectra

We consider matrix convolution operators with integrable kernels on expanding polyhedra. We study their connection with convolution operators on the cones at the vertices of polyhedra. We prove that the norm of the inverse operator on a polyhedron tends to the maximum of the norms of the inverse operators on the cones, and the pseudospectrum tends to the union of the corresponding pseudospectra. The study bases on the local method adapted to this kind of problems.     

Some New Results Related to Compact Matrix Operators in the Class((l_p)T,l_∞)

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈((lp)T,l∞), where T is an arbitrary triangle, 1 ≤ p ≤∞, to be a compact operator. In the past,only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces.     

Solvability of mixed type operator equations

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.     

Pseudospectra and delay differential equations

In this paper, we present a new method for computing the pseudospectra of delay differential equations (DDEs) with fixed finite delay. This provides information on the sensitivity of eigenvalues under arbitrary perturbations of a given size, and hence insight into how stability may change under variation of parameters. We also investigate how differently weighted perturbations applied to the individual matrices of the delayed eigenvalue problem affect the pseudospectra. Furthermore, we compute pseudospectra of the infinitesimal generator of the DDE, from which a lower bound on the maximum transient growth can be inferred. To illustrate our method, we consider a DDE modelling a semiconductor laser subject to external feedback.     

Approximate controllability of a fractional stochastic partial integro-differential systems via noncompact operators

Objective: in this article, we discuss the approximate controllability problems of a new class of fractional impulsive stochastic partial integro-differential systems in separable Hilbert spaces. Methods: by applying the fractional calculus, the measure of noncompactness, properties of fractional resolvent operators and fixed point theorems. Results: we prove our main results without the hypotheses of compactness on the operator generated by the linear part of systems. Instead we suppose that the nonlinear term only satisfies a weakly compactness condition. Conclusion: the approximate controllability for the control systems with noncompact operators is established. Finally, an example is given for the illustration of the obtained theoretical results.     

几类非紧性测度之间的比较性质

首先列举了几类不同意义下的非紧性测度,并对这几种非紧性测度作了比较,得到了两个重要的比较性质.     

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied. This work was partially funded by the Natural Sciences and Engineering Research Council of Canada, and by the MITACS Network of Centres of Excellence.     

The characterization of compact operators on spaces of strongly summable and bounded sequences

We use the characterizations of the classes of all infinite matrices that map the spaces of sequences which are strongly summable or bounded by the Cesàro method of order 1 into the spaces of null or convergent sequences given by Ba?ar, Malkowsky and Altay [Matrix transformations on the matrix domains of triangles in the spaces of strongly C₁-summable and bounded sequences, Publ. Math. Debrecen 73 (1-2) (2008), 193-213] and the Hausdorff measure of noncompactness to characterize the classes of all compact operators between those spaces.     

On some measures of noncompactness in the space of continuous functions

We consider some quantities in the space of functions continuous on a bounded interval, which are related to monotonicity of functions. Based on those quantities we construct a few measures of noncompactness in the mentioned function space. Several properties of those measures are established; among others it is shown that they are regular or “partly” regular measures and equivalent to the Hausdorff measure of noncompactness.     

Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means

In the present paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence spaces of generalized means. Further, by applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact.     

Strongly Generated Banach Spaces and Measures of Noncompactness

To generalize the Hausdorff measure of noncompactness to other classes of bounded sets (like e. g. conditionally weakly compact or Asplund sets), we introduce Grothendieck classes. We deduce integral inequalities for quantities (called Grothendieck measures) related to these classes. As a by-product, we can answer a question concerning the measure of noncompactness for linear T : X → Y introduced in [14], and generalize a theorem about weak solutions of differential equations in Banach spaces.