Spectral Theory Of Linear Operators Research Articles

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1<|x|<r2}(N≥2) with the boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:[r1,r2]×R×R+×R→R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ1 of the Laplace operator −▵ with boundary condition u|∂Ω=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,ξ,ζ,η) to be a superlinear growth on ξ,ζ,η as |(ξ,ζ,η)|→∞. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.

Positive solutions for fractional differential equation at resonance under integral boundary conditions

Abstract By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.

Gaps between Operator Norm and Spectral and Numerical Radii of the Tensor Product of Operators

One of the fundamental problem of the Spectral Theory of Linear Operators is to determine of the geometric place of the spectrum of the given operator and calculate the spectral and numerical radii of this operator. Other important problem in this theory is to explained the situation the spectral (numerical) radius is equal or not to operator norm. The only known way to calculate the spectral radius to date is the classical Gelfand formula, which often presents great technical challenges. Also, there is not yet a method of calculating the numerical radius for an operator. It should be noted that the finding the numerical range and numerical radius means maximally localizing the spectrum of an operator. The main purpose of this paper is to determine the relations gaps between operator norm and spectral and numerical radii of the tensor product operators associated with the compatible gaps of coordinate operators.

Semigroup method for a deteriorating system with multiple vacations of a repairman

By using the method of functional analysis, especially $$C_0$$ -semigroup theory and spectral theory of linear operators we prove the well-posedness and the existence of a unique positive dynamic solution of a deteriorating system with multiple vacations of a repairman. Moreover, we show spectral properties of the system operator which imply that the system does not have a steady-state solution.

FINITENESS OF THE NUMBER OF CRITICAL VALUES OF THE HARTREE-FOCK ENERGY FUNCTIONAL LESS THAN A CONSTANT SMALLER THAN THE FIRST ENERGY THRESHOLD

We study the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation which is the corresponding Euler-Lagrange equation is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with the Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fr\'echet second derivative of the functional at the limit point.

Singular value automata and approximate minimization

AbstractThe present paper uses spectral theory of linear operators to construct approximatelyminimal realizations of weighted languages. Our new contributions are: (i) a new algorithm for the singular value decomposition (SVD) decomposition of finite-rank infinite Hankel matrices based on their representation in terms of weighted automata, (ii) a new canonical form for weighted automata arising from the SVD of its corresponding Hankelmatrix, and (iii) an algorithmto construct approximateminimizations of given weighted automata by truncating the canonical form.We give bounds on the quality of our approximation.

Sufficient Conditions for Stability of Periodic Linear Impulsive Delay Systems

For a linear periodic system with impulsive action and delay, new approaches to the study of stability were proposed on the basis of the methods of spectral theory of linear operators, direct Lyapunov method, and N.G. Chetaev method for construction of the Lyapunov functions for the periodic linear systems, as well as the perturbation method for construction of the Lyapunov functions. These methods underlie the sufficient conditions for asymptotic stability of the linear periodic systems with impulsive action and delay. We gave some illustrative examples of studying stability of such systems under different assumptions about the dynamic properties of the continuous and discrete components of the impulsive system.

On the iterates of Jackson type operator GS,N in Hilbert space

In this paper we study the limit of the iterates of Jackson type operator. Our results continue the works of Badea [2] and Nagler et al. [9, 10]. The proofs are based on spectral theory of linear operators and are performed at first for Hilbert space and then are extended for some Banach spaces.

A SPECTRAL THEORY OF LINEAR OPERATORS ON RIGGED HILBERT SPACES UNDER ANALYTICITY CONDITIONS II: APPLICATIONS TO SCHRÖDINGER OPERATORS

A spectral theory of linear operators on a rigged Hilbert space is applied to Schrodinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach t o resonances (generalized eigenvalues) for both classes of potentials without using a ny spectral deformation tech- niques. Generalized eigenvalues for one dimensional Schrodinger operators (ordinary differential operators) are investigated in detail. A certain h olomorphic functionD(λ) is constructed so thatD(λ) = 0 if and only ifλ is a generalized eigenvalue. It is proved that D(λ) is equivalent to the analytic continuation of the Evans fun ction. In particular, a new formulation of the Evans function and its analytic continuation is given.

On the Iterates of Jackson Type Operator $${G_{s,n}}$$ G s , n

In this paper, we study the limit of the iterates of a large class of linear bounded operators preserving constants. We obtain in addition the limit of the iterates of algebraic version of the trigonometric Jackson integrals. The proofs are based on spectral theory of linear operators.

On Comparison of Dynamics of Dissipative and Finite-Time Systems Using Koopman Operator Methods

Abstract: Comparison of different dynamical systems or a dynamical system with data is one of the core issues in both dynamical systems and control theory. The fruitful approaches are particularly hard to come by for systems and data that show nonlinear behavior and non-Gaussian noise characteristics. We describe a theory that utilizes spectral theory of linear operators (composition, or Koopman) to provide the methodology, that was originally formulated for measure-preserving systems. Here we extend it to capture dissipative and finite-time dynamics. The approach combines a version of ergodic partition theory with Hardy space theory in observable space (rather than in time).

The spectral method and ergodic theorems for general Markov chains

We study the ergodic properties of Markov chains with an arbitrary state space and prove a geometric ergodic theorem. The method of the proof is new: it may be described as an operator method. Our main result is an ergodic theorem for Harris–Markov chains in the case when the return time to some fixed set has finite expectation. Our conditions for the transition function are more general than those used by Athreya–Ney and Nummelin. Unlike them, we impose restrictions not on the original transition function but on the transition function of an embedded Markov chain constructed from the return times to the fixed set mentioned above. The proof uses the spectral theory of linear operators on a Banach space.

A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions

A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace X of H such that the resolvent (λ−T)−1ϕ of the operator T has an analytic continuation from the lower half plane to the upper half plane as an X′-valued holomorphic function for any ϕ∈X, even when T has a continuous spectrum on R, where X′ is a dual space of X. The rigged Hilbert space consists of three spaces X⊂H⊂X′. A generalized eigenvalue and a generalized eigenfunction in X′ are defined by using the analytic continuation of the resolvent as an operator from X into X′. Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.

Spectral Theory for Linear Operators of Mixed Type and Applications to Nonlinear Dirichlet Problems

For a class of linear partial differential operators L of mixed elliptic-hyperbolic type in divergence form with homogeneous Dirichlet data on the entire boundary of suitable planar domains, we exploit the recent weak well-posedness result of [8] and minimax methods to establish a complete spectral theory in the context of weighted Lebesgue and Sobolev spaces. The results represent the first robust spectral theory for mixed type equations. In particular, we find a basis for a weighted version of the space comprised of weak eigenfunctions which are orthogonal with respect to a natural bilinear form associated to L. The associated eigenvalues {λ k } k∈ℕ are all non-zero, have finite multiplicity and yield a doubly infinite sequence tending to ± ∞. The solvability and spectral theory are then combined with topological methods of nonlinear analysis to establish the first results on existence, existence with uniqueness and bifurcation from (λ k , 0) for associated semilinear Dirichlet problems.

Analytic perturbation of generalized inverses

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. We provide an algorithm for computing the coefficients of the Laurent series of the perturbed Drazin generalized inverse. In particular, the regular part coefficients can be efficiently calculated by recursive formulae. Finally we apply the obtained results to the perturbation analysis of the Moore–Penrose generalized inverse in the real domain.

A Semigroup Approach to the System with Primary and Secondary Failures

We investigate the solution of a repairable parallel system with primary as well as secondary failures. By using the method of functional analysis, especially, the spectral theory of linear operators and the theory ofC0-semigroups, we prove well-posedness of the system and the existence of positive solution of the system. And then we show that the time-dependent solution strongly converges to steady-state solution, thus we obtain the asymptotic stability of the time-dependent solution.

On a size-structured two-phase population model with infinite states-at-birth

In this work, we introduce and analyze a linear size-structured population model with infinite states-at-birth. We model the dynamics of a population in which individuals have two distinct life-stages: an “active” phase when individuals grow, reproduce and die and a second “resting” phase when individuals only grow. Transition between these two phases depends on individuals’ size. First we show that the problem is governed by a positive quasicontractive semigroup on the biologically relevant state space. Then, we investigate, in the framework of the spectral theory of linear operators, the asymptotic behavior of solutions of the model. We prove that the associated semigroup has, under biologically plausible assumptions, the property of asynchronous exponential growth.

Bernstein Polynomials and Operator Theory

Kelisky and Rivlin have proved that the iterates of the Bernstein operator (of fixed order) converge to L, the operator of linear interpolation at the endpoints of the interval [0, 1]. In this paper we provide a large class of (not necessarily positive) linear bounded operators T on C[0, 1] for which the iterates Tn converge towards L in the operator norm. The proof uses methods from the spectral theory of linear operators.

Multiparameter spectral theory and separation of variables

Multiparameter spectral theory generalizes the classical spectral theory of linear operators to n linear operators linked by n spectral parameters. The spectral parameters arise as separation constants when, for example, separation of variables techniques are used to solve boundary-value problems for partial differential equations. In this paper, we discuss some of the fundamental aspects of the theory such as solvability, commutativity and abstract representation theory. Many of these concepts are of importance to modern developments in separation of variables techniques. The ideas are illustrated in application to differential equations and special functions of mathematical physics.

Bounds for Zeros of Entire Functions

Let f be an entire function. Denote by z1(f),z2(f),… the zeros of f with their multiplicities. In the paper, estimates for the sums $$\sum_{k=1}^{j}\frac{1}{|z_{k}(f)|}\quad (j=1,2,\ldots )$$ and for the counting function of the zeros of f are established. If f is of finite order ρ(f), we derive bounds for the series $$S_{p}(f):=\sum_{k=1}^{\infty}\frac{1}{|z_{k}(f)|^{p}}\quad (p\geq \rho(f))\quad \mbox{and}\quad \sum_{k=1}^{\infty}\biggl(\mathop{Im}\frac{1}{z_{k}(f)}\biggr)^{2}\quad (\rho(f)<2),$$ as well as relations between the series $$\sum_{k=1}^{\infty}\frac{1}{z_{k}^{m}(f)}\quad (m\geq \rho(f))$$ and the traces of certain matrices. The contents of the paper is closely connected with the following well-known results: the Hadamard theorem on the convergence exponent of the zeros of an entire function, the Jensen inequality for the counting function, the Cauchy theorem on the comparison of the zeros of polynomials, Ostrowski’s inequalities for the real and imaginary parts of the zeros of polynomials and the Cartwright–Levinson theorem. The suggested approach is based on the recent development of the spectral theory of linear operators.