Julia operators and linear systems (nonuniqueness of linear system)

Complementation theory in Krein spaces can be extended for any self-adjoint transformation. There is a close relation between Julia operators and linear systems. The theory of Julia operators can be used to construct distinct Krein spaces which are the state spaces of extended canonical linear systems with given transfer function.     

Spectral properties for perturbations of unitary operators

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,     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

Spectral properties of continuous refinement operators

This paper studies the spectrum of continuous refinement operators and relates their spectral properties with the solutions of the corresponding continuous refinement equations.

     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.     

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H_∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.     

Spectral properties of disjointly strictly singular operators

Spectral properties of strictly singular and disjointly strictly singular operators on Banach lattices are studied. We show that even in the case of positive operators, the whole spectral theory of strictly singular operators cannot be extended to disjointly strictly singular operators. However, several spectral properties of disjointly strictly singular operators are given.     

Spectral and oscillatory properties of a linear pencil of fourth-order differential operators

The paper deals with the spectral and oscillatory properties of a linear operator pencilA ? λB, where the coefficient A corresponds to the differential expression (py″)″ and the coefficient B corresponds to the differential expression ?y″ + cry. In particular, it is shown that all negative eigenvalues of the pencil are simple and, under some additional conditions, the number of zeros of the corresponding eigenfunctions is related to the serial number of the corresponding eigenvalue.     

Spectral synthesis of diagonal operators and representing systems for the space of entire functions

In this paper, we study continuous linear operators on spaces of functions analytic on disks in the complex plane having as eigenvectors the monomials zn whose associated eigenvalues λn are distinct. In particular, we show that under mild conditions, such a diagonal operator has non-spectral invariant subspaces (that is, closed invariant subspaces which are not the closed linear span of collections of monomials) if and only if every entire function of a particular growth rate is representable as a generalized Dirichlet series .     

Spectral properties of a class of reflectionless Schrödinger operators

We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct examples of reflectionless Schrödinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain in C, which exhibit a singular component.     

Spectral properties of operators in linear transport theory

Diagonalizability of operators occurring in linear transport theory is discussed from a general point of view. In particular, the operator A^–1T, where T is the multiplication operator in L² (–1, 1) and A is given by a formula of the type A f = f – _o ⁿ a_j < f, p_j > p_j , is investigated. Diagonalization of this operator which is connected with one-group neutron transport is carried out in the general case that the coefficients a_j are arbitrary complex numbers. Also, a peculiarity of multi-group theory, where the operator involved has a multiple continuous spectrum, is pointed out. A correct interpretation of the main result of that theory is provided.