Spectral Sets and the Drazin Inverse with Applications to Second Order Differential Equations (original) (raw)

A generalized Drazin inverse

Glasgow Mathematical Journal, 1996

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

The inverse spectral problem

We prove that given any compact subset of the complex plane containing zero, there exists a Hankel operator having this set as its spectrum.

The σg-Drazin Inverse and the Generalized Mbekhta Decomposition

Integral Equations and Operator Theory, 2007

In this paper we define and study an extension of the g-Drazin for elements of a Banach algebra and for bounded linear operators based on an isolated spectral set rather than on an isolated spectral point. We investigate salient properties of the new inverse and its continuity, and illustrate its usefulness with an application to differential equations. Generalized Mbekhta subspaces are introduced and the corresponding extended Mbekhta decomposition gives a characterization of circularly isolated spectral sets.

Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators

We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order. The uniqueness is achieved from a discrete set of data, namely, the values at the points −n 2 (n ∈ N) of (a physically appropriate generalization of) the Weyl-Titchmarsh m-function m(λ) for the problem. As a corollary, we establish a uniqueness result for a physically motivated inverse problem inspired by Berry and Dennis ('Boundary-conditionvarying circle billiards and gratings: the Dirichlet singularity',

Differential equations in the spectral parameter

Communications in Mathematical Physics, 1986

We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that some family of eigenfunctions φ satisfies a differential equation in the spectral parameter k of the form B(k, d k)φ = Θ(x)φ. For each such V{x) we determine the algebra of all possible operators B and the corresponding functions Θ(x). Table of Contents 0. Introduction 177 1. (adL) m + 1 ((9) = 0 180 2. K(oo) is Finite 187 3. The Rational KdV Potentials 192 4. The Even Case 203 5. The Even Potentials Work Too 213 6. F(oo) = oo is the Airy Case 218 7. Some Illustrative Examples 222

SPECTRAL THEORY AND ITS APPLICATIONS Book of Abstracts

Existence of a saddle-point in differential game, 2019

It is known that many problems of the economy are brought to the solution of differential games. In this thesis, we consider a differential game involving two players. Let some process be described by the system of differential equations

B-Fredholm Spectra of Drazin Invertible Operators and Applications

Axioms

In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and the transfer between some of the spectral properties defined through B-Fredholm spectra of this class of operators. Among other results, we investigate the transfer of generalized a-Weyl’s theorem from T to their Drazin inverse S, if it exists.

Inverse spectral problems and closed exponential systems

Annals of Mathematics, 2005

Consider the inverse eigenvalue problem of the Schrödinger operator defined on a finite interval. We give optimal and almost optimal conditions for a set of eigenvalues to determine the Schrödinger operator. These conditions are simple closedness properties of the exponential system corresponding to the known eigenvalues. The statements contain nearly all former results of this topic. We give also conditions for recovering the Weyl-Titchmarsh m-function from its values m(λ n).

On two-spectra inverse problems

Proceedings of the American Mathematical Society, 2020

We consider a two-spectra inverse problem for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this problem.