Spectral Properties Of Operators Research Articles

Spectral properties of operators in the problem on normal oscillations of a mixture of viscous compressible fluids

In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

Wild dynamics and asymptotically separated sets

Let X X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T T on X X such that A T ≔ { x ∈ X : ‖ T n x ‖ → ∞ } A_T≔\{x\in X:~ \|T^nx\|\to \infty \} is not dense and has nonempty interior, whenever X X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X T:X\rightarrow X for which the set A T A_T and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators.

Products of Positive Operators

On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class {mathcal {L}^{+,2}} of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in {mathcal {L}^{+,2}} are developed, and membership in {mathcal {L}^{+,2}} among special classes, including algebraic and compact operators, is examined.

Local Spectral Theory for R and S Satisfying RnSRn = Rj

In this paper, we analyze local spectral properties of operators R,S and RS which satisfy the operator equations RnSRn=Rj and SnRSn=Sj for same integers j≥n≥0. We also continue to study the relationship between the local spectral properties of an operator R and the local spectral properties of S. Thus, we investigate the transmission of some local spectral properties from R to S and we illustrate our results with an example. The theory is exemplified in some cases.

Spectral Analysis of Operator Polynomials and Second-Order Differential Operators

Studying spectral properties of operator polynomials is reduced to studying the corresponding spectral properties of operators defined by operator matrices. The results are used to investigate second-order differential operators by associating them with the corresponding first-order differential operators and using their properties related to invertibility.

Research on Spectral Properties of Operators with Distant Perturbations (Overview)

Research on Spectral Properties of Operators with Distant Perturbations (Overview)

Representations and Corresponding Operators Induced by Hecke Algebras

In this paper, by establishing free-probabilistic models on the Hecke algebras $$\mathcal {H}(G_{p})$$ , we construct canonical free probability spaces $$(\mathcal {H}(G_{p}), \psi _{p})$$ , where $$G_{p} = GL_{2}(\mathbb {Q} _{p})$$ , for primes $$p$$ . Dependent upon such free-probabilistic structures, we study corresponding representations of $$\mathcal {H}(G_{p})$$ , and consider spectral properties of operators realized under representations.

Spectral analysis of differential operators with unbounded operator coefficients in weighted spaces of functions

The spectral properties of operators constructed from a family of evolution operators are studied in Banach spaces of vector functions defined by a weight function of pre-exponential growth. Necessary and sufficient conditions for the invertibility of such operators are obtained in terms of exponential dichotomy. We prove a spectrum mapping theorem for semigroups of Howland difference operators generated by the operator under study.

Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions

We study spectral properties of Hamiltonians $\rH_{X,\gB,q}$ with $\delta'$-point interactions on a discrete set $X={x_k}_{k=1}^\infty\subset\R_+$. %at the centers $x_n$ on the positive half line in terms of energy forms. Using the form approach, we establish analogs of some classical results on operators $\rH_q=-d^2/dx^2+q$ with locally integrable potentials $q\in L^1_{\loc}(\R_+)$. In particular, we establish analogues of the Glazman-Povzner-Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with $\delta$-interactions, spectral properties of operators $\rH_{X,\gB,q}$ are closely connected with those of $\rH_{X,q}^N=\oplus_{k}\rH_{q,k}^N$, where $\rH_{q,k}^N$ is the Neumann realization of $-d^2/dx^2+q$ in $L^2(x_{k-1},x_k)$.

Estimates for the convergence rate of spectral decompositions of even-order differential operators

This paper studies spectral properties of ordinary differential operators of even order 2n, n > 1, with nonsmooth coefficients in the differential operation, in particular, with nonsmooth coefficient multiplying the (2n – 1)th derivative. We use the Il’in’s approach [1] to studying spectral properties of operators irre� spective of the particular form of the boundary condi� tions. We investigate the dependence of estimates for the rate of local convergence of spectral decomposi� tions on the distance from an interior compact set K to the boundary ∂G in the case where the normalized Fourier coefficients αkfk have asymptotics O () , where ν = const > 0 and λk is the spectral parameter (see Theorem 1) and in the case where the coefficients have asymptotics O (l n –β λ k ), β = const (see Theo� rem 2). We begin with a brief review of related results. Paper [2] studies the rate of convergence of biorthog� onal expansions of functions in systems of root func�

On Selfadjoint Homogeneous Operators

We study unbounded selfadjoint operators that are unitarily equivalent to their affine transformation. We investigate transformation properties of an operator-valued M-function associated with given affine-invariant operator as well as spectral properties of operators that occur in that investigation.

Analysis of spectral properties of operators for linearized steady-state equations of a viscous compressible heat-conducting fluid

The spectrum of operators for linearized steady-state equations of a viscous, compressible, heat-conducting fluid is studied. Using the factorization of operators and applying the theory of pseudo-differential operators, we prove that the operator is sectorial and its spectrum is discrete everywhere except for some segment on the real line. This segment is the essential spectrum of the operator and it lies in the left half-plane.

Spectral Analysis of Operator in M/M/1 Queueing Model Described by Ordinary Differential Equations

In this paper,the spectral properties of operator A determinedby M/M/1 queueing model described by ordinary differential equations are studied in l~1.By complete spectral analysis, we show that the spectra of A have an elliptic shape,and all the interior points of the ellipse are the eigenvalues of A.0 is the only eigenvalue on the boundary,so it can not be separated from other spectra.The results show that it is impossible to see the steady-state of queueing system described by ordinary differential equations in the limited time.

Spectral properties of operators with polynomial invariants in real finite-dimensional spaces

We consider linear operators lying in the orthogonal group of a quadratic form and study those spectral properties of such operators that can be expressed in terms of the signature of this form. We show that in the typical case these transformations are symplectic. Some of the results can be extended to the general case when the operator admits a homogeneous form of degree ≥3.

On Symmetrizable Operators on Hilbert Spaces

This paper, motivated by transport theory, deals with spectral properties of operators G on a complex Hilbert space H such that SG is self-adjoint where S is a nonnegative operator: We give several lower bounds of the spectral radius of G and determine the latter in some cases. We derive the whole spectrum for power compact G by means of Lagrange multiplier theory. We find out spectral connections between G and SG. We give a (spectral) stability estimate for symmetrizable operators in terms of the spectral radius of the perturbation.

Linear kinetic equation: long-time behavior of one-particle distribution function

We construct asymptotic (long-time) solution of the linear Boltzmann equation using the time-dependent perturbation theory generalized to non-Hermitian operators. We prove that for times much larger than the relaxation time τ0, t ≫τ0, one-particle distribution function separates into spatio-temporal and velocity dependent parts, and provide the explicit expression for the long-time solution of the linear Boltzmann equation. Our analysis does not assume that relative density gradients \(n^{-1}(\partial / \partial \mathaccent"017E{r}) n\) are small. It relates the hydrodynamic form of the one-particle distribution function to spectral properties of operators involved in linear Boltzmann equation.

Spectral Properties of the Periodic Magnetic Schr�dinger Operator in the High-Energy Region. Two-Dimensional Case

The goal is to investigate spectral properties of the operator H=(−i∇ +a(x))2+a0(x) in the two-dimensional situation, a(x), a0(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.

Spectral properties of operators of the theory of harmonic potential

We classify the points of the spectrum of the operatorsB andB * of the theory of harmonic potential on a smooth closed surfaceS ⊂ ℝ3. These operators give the direct value onS of the normal derivative of the simple layer potential and the double layer potential. We show that zero can belong to the point spectrum of both operators inL 2 (S). We prove that the half-interval [−2π, 2π) is densely filled by spectrum points of the operators for a varying surface; this is a generalization of the classical result of Plemelj. We obtain a series of new spectral properties of the operatorsB andB * on ellipsoidal surfaces.

Spectral properties of operators for the systems of hydrodynamics of a rotating liquid and nonuniqueness of the limit amplitude

In addition to the systems describing rotating liquid hydrodynamics traditionally applied to studying the motion of a top filled with a liquid, some recent papers treated systems with the Coriolis term as applied to various problems connected with the atmosphere and ocean dynamics [9, i0], to studying different wave motions in a liquid [ii] such as gravitational waves, inner waves resulting from the interaction of the Coriolis force with the pressure gradient, inertia waves (including the case of unbounded media), and acoustic waves in a compressible liquid. In all these problems the rotation of the Earth is taken into account; that is why the Coriolis term appears in the systems, which endows the solutions with specific features -there appears small waves changing in time in a sophisticated manner.

Etude spectrale d'une famille d'op�rateurs non-sym�triques intervenant dans la th�orie des champs de reggeons

In this paper, we study a few spectral properties of a non-symmetrical operator arising in the Gribov theory. The first and second section are devoted to Bargmann's representation and the study of general spectral properties of the operator: For λ'>0 and λ'2^μλ' + λ2. We prove that the solutions of the equation u'(t) + Hλ>}μfλu(ί) = 0 are expandable in series of the eigenvectors of Hλ>tlXtλ for ί>θ/' ' In the last section, we show that the smallest eigenvalue σ(α) of the operator Hλ\μ,λ,a is analytic in α, and thus admits an expansion: σ(α) = σo + ασ1 + α2σ2+ . . ., where σ0 is the smallest eigenvalue of the operator Hλ'ίfltλ>0.