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The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such an operator is represented as a sum of a countable set of matrix differential operators of first order acting in $ L_2(\mathbb {R}) $ L 2 (R). Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: (1) in the case of coefficients stabilizing at infinity; and (2) in the case of periodic coefficients. [ABSTRACT FROM AUTHOR]

Given a densely defined and gapped symmetric operator with infinite deficiency index, it is shown how self-adjoint extensions admitting arbitrarily prescribed portions of the gap as essential spectrum are identified and constructed within a general extension scheme. The emergence of new spectrum in the gap by self-adjoint extension is a problem with a long history and recent deep understanding, and yet it remains topical in several recent applications. Whereas it is already an established fact that, in case of infinite deficiency index, any kind of spectrum inside the gap can be generated by a suitable self-adjoint extension, the present discussion has the virtue of showing the clean and simple operator-theoretic mechanism of emergence of such extensions. [ABSTRACT FROM AUTHOR]

A bounded linear operator T on a complex Banach space X is superconvex-cyclic, if for some vector x in the space, C. c o (o r b (T , x)) is dense. Here, co(orb(T, x)) means the convex hull of { T n x : n ≥ 0 }. In this paper, we give necessary conditions for the superconvex-cyclicity of an operator in terms of the point spectrum of its adjoint. Also, we prove that positive multiples of superconvex-cyclic operators are superconvex-cyclic. However, this concept is not necessarily transferred to positive integer powers of an operator. Moreover, we completely characterize superconvex-cyclic matrices whose spectra contain no real number or contain a positive real number. [ABSTRACT FROM AUTHOR]

In this study, we give another generalization of second order backward difference operator ∇ 2 by introducing its quantum analog ∇ q 2. The operator ∇ q 2 represents the third band infinite matrix. We construct its domains c 0 (∇ q 2) and c (∇ q 2) in the spaces c 0 and c of null and convergent sequences, respectively, and establish that the domains c 0 (∇ q 2) and c (∇ q 2) are Banach spaces linearly isomorphic to c 0 and c , respectively, and obtain their Schauder bases and α -, β - and γ -duals. We devote the last section to determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the operator ∇ q 2 over the Banach space c 0 of null sequences. [ABSTRACT FROM AUTHOR]

In this note, we prove a characterization of reflexive Banach spaces with unconditional basis and study the set theoretical complexity of certain sets associated to linear bounded operators acting on reflexive separable Banach spaces. [ABSTRACT FROM AUTHOR]

We show that near periodic rivet chains that connect two Kirchhoff plates and are modeled by point Sobolev transmission conditions, Rayleigh waves arise, propagate along the chains, and decay exponentially in the orthogonal direction. Under additional geometric conditions we discover the standing (periodic) waves that carry no energy. [ABSTRACT FROM AUTHOR]

Consider a tuple (Y 1 , ... , Y d) of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each Y i. We address the question what one can say about the sizes of the eigenspaces for any non-commutative polynomial P (Y 1 , ... , Y d) in those operators? We show that for each polynomial P there are unavoidable eigenspaces, which occur in P (Y 1 , ... , Y d) for any (Y 1 , ... , Y d) with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over the free skew field and via rank calculations - and in analytic terms - where it is given by freely independent random variables with prescribed atoms in their distributions. This allows us to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but more generally also to non-commutative rational functions. Since many random matrix models become asymptotically free in the large N limit, our results allow us to calculate the location and size of atoms in the asymptotic eigenvalue distribution of polynomials and rational functions in randomly rotated matrices. Applications to the Atiyah property and to the calculation of the spectrum of universal covers of graphs are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider an extended coined Szegedy model and discuss the existence of the point spectrum of induced quantum walks in terms of recurrence properties of the underlying birth and death process. We obtain that if the underlying random walk is not null recurrent, then the point spectrum exists in the induced quantum walks. As an application, we provide a simple computational way of the dispersion relation of the edge state part for the topological phase model driven by quantum walk using the recurrence properties of underlying birth and death process. [ABSTRACT FROM AUTHOR]

We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation. Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0, the C0-semigroup is not compact, not eventually compact, even not quasi-compact. Moreover, we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution. In addition, we obtain the spectral radius and essential spectral radius of the C0-semigroup. Lastly, we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum. [ABSTRACT FROM AUTHOR]

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schrödinger operators. [ABSTRACT FROM AUTHOR]

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 Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra. [ABSTRACT FROM AUTHOR]

Let (X, d) be a locally compact separable ultrametric space. Given a measure m on X and a function C defined on the set B of all balls B ⊂ X, we consider the hierarchical Laplacian L = LC. The operator L acts in L²(X, m), is essentially self-adjoint, and has a purely point spectrum. Choosing a family {ε(B)}B∈B of i.i.d. random variables, we define the perturbed function C(B) = C(B)(1 + ε(B)) and the perturbed hierarchical Laplacian L = LC. All outcomes of the perturbed operator L are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process M defined in terms of L-eigenvalues. Under some natural assumptions, M can be approximated by a Poisson point process. Using a result of Arratia, Goldstein, and Gordon based on the Chen--Stein method, we provide total variation convergence rates for the Poisson approximation. We apply our theory to random perturbations of the operator Dα, the p-adic fractional derivative of order α > 0. [ABSTRACT FROM AUTHOR]

Abstract The space of entire functions which are integrable with respect to the Gaussian weight, known also as the Fock space, is one of the preferred functional Hilbert spaces for modeling and experimenting harmonic analysis, quantum mechanics or spectral analysis phenomena. This space of entire functions carries a three parameter family of canonical isometric involutions. We characterize the linear differential operators acting on Fock space which are complex symmetric with respect to these conjugations. In parallel, as a basis of comparison, we discuss the structure of self-adjoint linear differential operators. The computation of the point spectrum of some of these operators is carried out in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x) :=AΔv(x) + + Df (vΔ(x))v(x), x ∈ Rd, d ≥ 2, where vΔ : Rd→Rm is the profile of a rotating wave satisfying vΔ(x)→v∞ ∈ Rm as x →∞, the map f : Rm→Rm is smooth and the matrix A ∈ Rm,m has eigenvalues with positive real parts and commutes with the limit matrix Df (v∞). The matrix S ∈ Rd,d is assumed to be skew-symmetric with eigenvalues (λ1, . . . , λd)=(±iσ1, . . . ,±iσk, 0, . . . , 0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction-diffusion systems. We prove under appropriate conditions that every λ ∈ C satisfying the dispersion relation det(λIm + η2A - Df (v∞) + iIm)=0 for some η ∈ R and n ∈ Zk, σ =(σ1, . . . , σk)T ∈ Rk belongs to the essential spectrum σess(L) in Lp. For values Re λ to the right of the spectral bound for Df (v∞), we show that the operator λI - L is Fredholm of index 0, solve the identification problem for the adjoint operator (λI - L)∗ and formulate the Fredholm alternative. Moreover, we show that the set σ(S) ∪ {λi + λj : λi, λj ∈ σ(S), 1 ≤ i

A linear system of mixed-type differential difference equations is studied. A step derivation method and some conditions on an initial function are used to guarantee existence, uniqueness and smoothness of the solution. Further, a transformation group is defined on a complete countably normed space of initial functions and the spectrum of the infinitesimal generator of this group is studied. The same technique applies to a linear system of retarded differential difference equations. A problem to extend solutions of such a system to the left on the real line is solved. [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]

We will study the linear spectral properties of steady Saffman–Taylor bubbles with surface tension in a Hele-Shaw cell. It was known [21] that the Saffman–Taylor exact solution without surface tension is linearly unstable and the point spectrum is a continuum set. In this paper, we rigorously conclude that the point spectrum of the stability operator of any branch of Saffman–Taylor bubbles is a discrete set when surface tension is not zero. [ABSTRACT FROM AUTHOR]

Aq-analog expression of the determinantal polynomial of a weighted shift operator is introduced and applied to find the numerical radius of the operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundedness of the Libera operator on Dirichlet spaces in terms of the Schur test. Moreover, we get its point spectrum and norm. [ABSTRACT FROM AUTHOR]

The notion of atomic observable was introduced by S.Gudder for effect test spaces in 1997. In this paper an observable is a σ-homomorphism from the Borel algebra on a line to some logic. Roughly, an observable on a logic is atomic, if it is completely determined by its restriction to one-element subsets of its point spectrum. In particular, every discrete observable is atomic. We study some elementary properties of such observables, and discuss a possible notion of functional dependency between them. Algebraically, a dependency is a certain preorder relation on the set of all atomic observables, which induces an order relation on the set of all maximal orthogonal subsets of the logic. Several properties, as well as characteristics in terms of the underlying logic, of these relations are stated. [ABSTRACT FROM AUTHOR]

This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum of the operator D(r, 0, 0, s) on a sequence space bv0, the space of all sequences x such that limn→∞ xn = 0 and ∥ x ∥ = Σ n=0∞ ∣ x n+1 - xn ∣ < ∞. [ABSTRACT FROM AUTHOR]

In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable ζ = Σ∞k=1 2/3k εηk ≡ Δη1…ηk…, where the indices ηk are independent random variables taking the values 0, 1, 2, 3, 4 with probabilities p0k, p1k, p2k, p3k, p4k respectively, pik ≥ 0, Σ4i=0 pik = 1, Σ4i=0 pik = 1 for any k ∊ ℕ, εm = emπi/2 = im (m = 0, 1, 2, 3) are 4th roots of unity, ε4 = 0. We prove that the distribution of ζ is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of ζ to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found. [ABSTRACT FROM AUTHOR]

In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable ... where the indices k are independent random variables taking the values 0,1,2,3,4 with probabilities p0k, p1k, p2k, p3k, p4k respectively, ... are 4th roots of unity, ε4 = 0. We prove that the distribution of ζ is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of ζ to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found. [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]

In this paper we have examined the spectra of the operator D(r, 0, 0, s) on sequence spaces c0 and c. [ABSTRACT FROM AUTHOR]

Let A be a bounded linear operator on a Banach space X. We investigate the conditions of existing rank-one operator B such that I + f(A)B is invertible for every analytic function f on (A). Also, we compare the invariant subspaces of f(A)B and B. This work is motivated by an operator method on the Banach space l² for solving some PDEs, extended to general operator space under some conditions here. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of 2nth-order regular differential operator with eigenparameter-dependent boundary conditions and transmission conditions at an interior discontinuous point. By constructing a new linear operator A associated with the problem, we prove that the operator A is self-adjoint in a suitable Hilbert space H, and the eigenvalues of the problem coincide with those of A. In terms of basic solutions of differential equation, we show that the eigenvalues of this problem coincide with the zeros of the entire function det Φ(1, λ), and obtain that the operator A has only point spectrum. [ABSTRACT FROM AUTHOR]

The spectral properties of the off-diagonal infinite-dimensional Hamiltonian operator are studied in this article. Characterizations of the spectrum, the point spectrum and the residual spectrum are given. A spectral inclusion property of numerical range is proved. [ABSTRACT FROM AUTHOR]

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

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class denoted by l-*- A, of operators satisfying T*| T| T ≥ T*| T*| T, and we prove the basic properties of these operators. Using these results, we also prove that if T or T* ∈ l-*- A, then w( f( T)) = f( w( T)), σ( f( T)) = f( σ( T)) for every f ∈ H( σ( T)), where H( σ( T)) denotes the set of all analytic functions on an open neighborhood of σ( T). [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 paper we investigate perturbations of the regular spectrum of an upper triangular operator matrix such as acting on a Hilbert space . [Copyright &y& Elsevier]

Abstract: 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 (in the sense that a semispectral measure of S vanishes on ). 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). [Copyright &y& Elsevier]

Abstract: Let us consider the time-dependent Schrödinger equation, on the Hilbert space , where is a repulsive periodic time-dependent potential, with period T. We denote by its associated propagator. First, using a multiplier method, we rule out the existence of regular eigenvectors of the Floquet operator . Secondly, strengthening the hypotheses on the potential V, we prove that the spectrum of does not contain any eigenvalues, by means of positive commutator methods. [Copyright &y& Elsevier]

Abstract: We consider operators of convolution with measures μ on locally compact groups. We characterize the spectrum of by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of , respectively. The proofs rely on commutator methods. [Copyright &y& Elsevier]

It is known that relatively bounded compact perturbations of analytic semigroups yield again generators of analytic semigroups. On the other hand, for each non-analytic semigroup there is a relatively bounded rank one perturbation which is no longer a generator. In this note we show that in fact any non-analytic semigroup admits a relatively bounded rank one perturbation with a sequence of eigenvalues zj such that the real parts R(zj) converge to infinity. [ABSTRACT FROM AUTHOR]

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]

Abstract: Let T be an operator on a separable Banach space, and denote by its point spectrum. We answer a question left open in (Israel J. Math. 146 (2005) 93–110) by showing that it is possible that be uncountable, yet . We further investigate the relationship between the growth of sequences such that and the possible size of . Analogous results are also derived for continuous operator semigroups . [Copyright &y& Elsevier]

We find all elastic systems with point spectrum. For almost elastic nonconservative systems we obtain large time asymptotics for the velocities of the balls. This one-dimensional ball model was invoked in order to elucidate some problems of gas kinetics and turbulence but turned out hard to study in its own right. [ABSTRACT FROM AUTHOR]

We investigate the spectral properties of a class of Jacobi matrices resulting from periodic pertubations of Jacobi operators with smooth coefficients. [ABSTRACT FROM AUTHOR]

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]

Our main goal is to analyze the geometric and spectral properties of the Picard modular group with Gaussian integer entries acting on the two-dimensional complex hyperbolic space. [ABSTRACT FROM AUTHOR]