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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.

Usually backward shift is neither chaotic nor hypercyclic. We will show that on the space A(Omega) of real analytic functions on a connected set Omega subset of R with 0 is an element of Omega, the backward shift operator is chaotic and sequentially hypercyclic. We give criteria for chaos and for many other dynamical properties for weighted backward shifts on A(Omega). For special classes of them we give full characterizations. We describe the point spectrum and eigenspaces of weighted backward shifts on A(Omega) as above. National Center of Science (Poland) ; TÜBİTAK

Consider a unitary operator U0 acting on a complex separable Hilbert space H. In this paper we study spectral properties for perturbations of U0 of the type,Uβ=U0eiKβ, with K a compact self-adjoint operator acting on H and β a real parameter. We apply the commutator theory developed for unitary operators in Astaburuaga et al. (2006) [1] to prove the absence of singular continuous spectrum for Uβ. Moreover, we study the eigenvalue problem for Uβ when the unperturbed operator U0 does not have any. A typical example of this situation corresponds to the case when U0 is purely absolutely continuous. Conditions on the eigenvalues of K are given to produce eigenvalues for Uβ for both cases finite and infinite rank of K, and we give an example where the results can be applied.

The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from \(1\) to \(N\), for a Jacobi matrix \(J\) by the eigenvalues of the finite submatrix \(J_n\) of order \(pn \times pn\), where \(N = \max \{k \in \mathbb{N}: k \leq rpn\}\) and \(r \in (0,1)\) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of \(J\) in the case \(p=3\).

The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in l 2 ( N ) 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.

Let Ω= [a, b] × [c, d] and T1, T2 be partial integral operators in \( C \)(Ω): (T1f)(x, y) = \( \mathop \smallint \limits_a^b \)k1(x, s, y)f(s, y)ds, (T2f)(x, y) = \( \mathop \smallint \limits_c^d \)k2(x, ts, y)f(t, y)dt where k1 and k2 are continuous functions on [a, b] × Ω and Ω × [c, d], respectively. In this paper, concepts of determinants and minors of operators E−τT1, τ ∈ ℂ and E−τT2, τ ∈ ℂ are introduced as continuous functions on [a, b] and [c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T1, T2, and T = T1 + T2 are proved.

14 pages; Journal of Functional Analysis; International audience; We consider operators $H_\mu$ of convolution with measures $\mu$ on locally compact groups. We characterize the spectrum of $H_\mu$ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of $H_\mu$, respectively. The proofs rely on commutator methods.

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 l^2=l^2(N). For given sequences of positive numbers @l"n and real q"n the Jacobi operator is given by J=SW+WS^*+Q, where Q=diag(q"n) and W=diag(@l"n) 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 {q"n} and {@l"n} 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.

Let T∈B(H) be a bounded linear operator on a complex Hilbert space H. Let λ0 be an isolated point of σ(T) and letE=12πi∫|λ−λ0|=r(λ−T)−1dλbe the Riesz idempotent for λ0. In this paper, we prove that if T is a p-quasihyponormal operator with 0

We characterize distributional chaos for linear operators on Fréchet spaces in terms of a computable condition (DCC), and also as the existence of distributionally irregular vectors. A sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented, which is very general and can be applied to many classes of operators. Distributional chaos is also analyzed in connection with frequent hypercyclicity, and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results. © 2013 Elsevier Inc., The fourth author was supported in part by MEC and FEDER, Project MTM2010-14909.

We consider the spectrum of the Stokes operator with and without rotation effect for the whole space and exterior domains in $L^q$-spaces. Based on similar results for the Dirichlet-Laplacian on $mathbf{R}^n$, $n \geq 2$, we prove in the whole space case that the spectrum as a set in $\mathbf{c}$ does not change with $q \in (1,\infty)$, but it changes its type from the residual to the continuous or to the point spectrum with $q$. The results for exterior domains are less complete, but the spectrum of the Stokes operator with rotation mainly is an essential one, consisting of infinitely many equidistant parallel half-lines in the left complex half-plane. The tools are strongly based on Fourier analysis in the whole space case and on stability properties of the essential spectrum for exterior domains.

We give a complete analysis of the spectrum of the unique self-adjoint extension of the sub-Laplacian on the one-dimensional Heisenberg group.

Let be the free product of q + 1 copies of Zn+1 and let denote its Cayley graph (with respect to aj, 1 ≤ j ≤ q + 1). We may think of G as a group acting on the “homogeneous space” , This point of view is inspired by the case of SL2(R) acting on the hyperbolic disk and is developed in [FT-P] [I-P] [FT-S] [S] (but see also [C]).Since G is a group we may investigate some classical topics: the full (reductive) C* algebra, its dual space, the regular Von Neumann algebra and so on. See [B] [P] [L] [V] and also [H]. These approaches give results pointing up the analogy between harmonic analysis on these groups and harmonic analysis on more classical objects.

In this paper we investigate perturbations of the left essential spectrum, right essential spectrum, essential spectrum and the regular spectrum of an upper triangular operator matrix such as M sub C - [A sub 1 sub 1 C sub 1 sub 2 O sub 2 sub 1 B sub 2 sub 2] acting on a Hilbert space H + K.

Let be a connected bounded open set in R N , N 2, with lipschitzian boundary. The best constant in the Poincar e type inequality: j uj 2 C()k grad(u)k 2 1 ;8u2 L 2 ()=R is the inverse of the smallest spectral value of the the bounded self-adjoint linear operator T = div( ) 1 grad in L 2 ()=R ([4]). In this paper we show that, in the case of an elliptical domain of R 2 , the point spectrum of this operator is the set p(T )= fn; e n; 1; n2 N*g, where 1 is an eigenvalue

We will investigate the point spectrum on the imaginary axis of the linearized Boltzmann operator with an external-force potential in an unbounded domain. The boundary condition considered is the perfectly reflective boundary condition. We suppose that the boundary surface is sufficiently smooth, but is not cylindrical. In this case, the point spectrum on the imaginary axis is only equal to f0g.

A generalized version of the Glauber–Klauder basic formula of quantum optics is shown to be valid for any cyclic subnormal operator S whose adjoint has a rich point spectrum σp(S∗) (in the sense that a semispectral measure of S vanishes on C∖σp(S∗)∗). It is exhibited that such operators always have analytic models. The point spectrum of the adjoint of a subnormal operator which satisfies a generalized version of the Glauber–Klauder formula is proved to be rich (in the above sense).

Let us consider the time-dependent Schrodinger equation, i φ t = − Δ φ + V ( x , t ) φ , on the Hilbert space L 2 ( R n ) , where V ( x , t ) is a repulsive periodic time-dependent potential, with period T. We denote by ( U ( t , s ) ) ( t , s ) ∈ R × R its associated propagator. First, using a multiplier method, we rule out the existence of regular eigenvectors of the Floquet operator U ( T , 0 ) . Secondly, strengthening the hypotheses on the potential V, we prove that the spectrum of U ( T , 0 ) does not contain any eigenvalues, by means of positive commutator methods.