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We investigate the point spectrum associated with travelling wave solutions in a Keller–Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise. • Travelling wave solutions are transiently unstable for a range of parameter values. • The point eigenvalue of order two at the origin persists for small diffusion. • The constant consumption model is spectrally stable in the weighted function space. • The spectral gap vanishes for linear consumption. • Eigenfunctions for the origin are contained in the weighted function space. [ABSTRACT FROM AUTHOR]

The spectrum of the Dirac operator is investigated on a Riemannian manifold one of whose end is asymptotically flat in some sense. It is proved that there is no non-zero eigenvalue and the essential spectrum is equal to R . [ABSTRACT FROM AUTHOR]

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if X is an infinite-dimensional complex Banach space then every operator T ∈ L ( X ) admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well. [ABSTRACT FROM AUTHOR]

The paper is concerned with essential spectra of singular linear Hamiltonian systems of arbitrary order with arbitrary equal defect indices. Several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding Hamiltonian system, and a sufficient and necessary condition for the essential spectral points is obtained for Hamiltonian systems of even-order with one singular endpoint. An advantage of these results is that one can determine the essential spectral points of Hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations. In addition, two illustrative examples are provided to show how to get some information about the essential spectrum by our results. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko’s lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36(2) (2004) 643–658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59–84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices. [Copyright &y& Elsevier]

Abstract: In this article we calculate the asymptotic behaviour of the point spectrum for some special self-adjoint unbounded Jacobi operators J acting in the Hilbert space . For given sequences of positive numbers and real the Jacobi operator is given by , where and are diagonal operators, S is the shift operator and the operator J acts on the maximal domain. We consider a few types of the sequences and and present three different approaches to the problem of the asymptotics of eigenvalues of various classes of J''s. In the first approach to asymptotic behaviour of eigenvalues we use a method called successive diagonalization, the second approach is based on analytical models that can be found for some special J''s and the third method is based on an abstract theorem of Rozenbljum. [Copyright &y& Elsevier]

Abstract: The absense of the point spectrum of the Dirac operator is investigated. By the method of an integral identity, it is shown that some non-compact complete manifold with a pole has no -eigenspinors for non-zero eigenvalues. [Copyright &y& Elsevier]

Selfadjoint operators on Hilbert spaces have been extended to more general scopes such as certain real Banach spaces and complex Banach spaces endowed with a continuous Hermitian bilinear form. Here we propose a definition of selfadjoint operator that works for both real and complex Banach spaces and that naturally extends the classical concept of selfadjointness for operators on Hilbert spaces. We strongly rely on selectors of the duality mapping, obtaining a very natural extension in the case of smooth Banach spaces. We provide nontrivial examples of selfadjoint operators on nonHilbert smooth Banach spaces and on nonHilbert general Banach spaces. Finally, we extend the classical theory on eigenvalues and supporting vectors of selfadjoint positive operators on Hilbert spaces to the scope of reflexive, strictly convex and smooth Banach spaces. [ABSTRACT FROM AUTHOR]