Generalized Fredholm Operators Research Articles

On a algebraic characterization of the generalized Fredholm operators

On a algebraic characterization of the generalized Fredholm operators

Index for generalized Fredholm operators and generalized perturbation theory

Abstract In this paper, we develop some generalized Fredholm perturbation results for bounded linear operators acting on non-reflexive Banach spaces satisfying certain conditions. Furthermore, we investigate the stability of the generalized Schechter S-essential spectrum by means of the so-called generalized Fredholm index. This concept is introduced as a semigroup homomorphism satisfying certain properties. Some generalized Fredholm results for block operator matrices acting on a non-reflexive product of two Banach spaces are also developed.

Generalized Fredholm operator equation and its solution based on intermediate coordinate-momentum representation**Work supported by the specialized research fund for the doctorial progress of higher education in China.

Based on the intermediate coordinate-momentum representations, we construct the appropriate generalized Fredholm operator equation and then find its solution. The application of this equation for deriving Hermite-polynomial operator formulae is demonstrated. The connection of the generalized Wigner operator to the form of the Gaussian bivariate distribution of random variables in statistics is shown, which is useful for studying the tomogram of quantum states.

Finite-dimensional perturbations of linear operators and some applications to boundary integral equations

Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such a perturbing operator to the original one reduces the eigen-space dimension and can, particularly, lead to an unconditionally and uniquely solvable perturbed equation. For the second kind Fredholm operators, the perturbing operators are analyzed such that the spectrum points for an original and the perturbed operators coincide except a spectrum point considered, which can be removed for the perturbed operator. A relation between resolvents of original and perturbed operators is obtained. Effective procedures are described for calculation of the undetermined constants in the right-hand side of an operator equation for the case when these constants must be chosen to satisfy the solvability conditions not written explicitly. Implementation of the methods is illustrated on a boundary integral equation of elasticity.

Generalizations of Semi-Fredholm Operators

On nonseparable Hilbert spaces there are multiple sets of operators that are analogous to the semi-Fredholm operators on a separable space. We develop the properties of these sets and relate those properties to some recent research. We conclude with a theorem that indicates precisely how far one can go from a given generalized semi-Fredholm operator (or generalized Fredholm operator) and retain the property of generalized semi-Fredholmness (or generalized Fredholmness).

Generalizations of semi-Fredholm operators

On nonseparable Hilbert spaces there are multiple sets of operators that are analogous to the semi-Fredholm operators on a separable space. We develop the properties of these sets and relate those properties to some recent research. We conclude with a theorem that indicates precisely how far one can go from a given generalized semi-Fredholm operator (or generalized Fredholm operator) and retain the property of generalized semi-Fredholmness (or generalized Fredholmness).

Generalized Fredholm operators

0. Introduction. The classical Fredholm theory in Banach spaces studies normally solvable operators with null space or conull space in F, the ideal of all finite dimensional Banach spaces. The aim of this paper is to study normally solvable operators with null space or conull space in an arbitrary space ideal A. We use the operator ideal Op(A) to replace the operator ideal Fi := Op (F) of the classical theory in a natural way. Operators invertible modulo Op (A) are studied as well. Since F c A and Fi ~Op (A), Fredholm operators are particular A-Fredholm operators. Yang [10], [11] developed Fredholm theory relative to the ideal R of all reflexive Banach spaces with a functorial approach which does not allow to develop the theory in an arbitrary space ideal; theorems of [9], [ 10] and [ 11] are particular cases of some results presented here.

The generalized Fredholm operators with finite indices

The generalized Fredholm operators with finite indices

Generalized Fredholm Operators and the Boundary of the Maximal Group of Invertible Operators

Generalized Fredholm Operators and the Boundary of the Maximal Group of Invertible Operators

Generalized Fredholm operators and the boundary of the maximal group of invertible operators

Let V denote an infinite dimensional Banach space over the complex field and let G [ V ] G[V] denote the subset of bounded operators on V with the property that the null space has a closed complement and the range is closed, where the null space and range are proper subspaces of V. Necessary and sufficient conditions for T ∈ G [ V ] T \in G[V] to be in the boundary, B \mathcal {B} , of the maximal group, M \mathcal {M} , of invertible operators are determined. As a result, B ∩ G [ V ] \mathcal {B} \cap G[V] is the set of products of operators in M \mathcal {M} and operators in P \mathcal {P} , where P \mathcal {P} is the set of projections other than the identity operator and null operator.

Perturbation theory for generalized Fredholm operators. II

If T is a bounded linear operator with generalized inverse S, i.e. T S T = T TST = T , we obtain conditions so that T − U T - U has a generalized inverse. When T is a Fredholm operator, the conditions become simply the requirement that I − U S I - US (or equivalently I − S U I - SU ) is a Fredholm operator. This result includes the classical perturbation theorems where U is required to have a small norm or to be compact.

The Generalized Fredholm Operators

Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if $T(X)$ is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces $\leftrightarrow$ finite-dimensional spaces, weakly compact operators $\leftrightarrow$ compact operators, generalized Fredholm operators $\leftrightarrow$ Fredholm operators, Tauberian operators with closed range $\leftrightarrow$ semi-Fredholm operators.

The generalized Fredholm operators

Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if T ( X ) T(X) is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces ↔ \leftrightarrow finite-dimensional spaces, weakly compact operators ↔ \leftrightarrow compact operators, generalized Fredholm operators ↔ \leftrightarrow Fredholm operators, Tauberian operators with closed range ↔ \leftrightarrow semi-Fredholm operators.

Perturbation theory for generalized Fredholm operators

Perturbation theory for generalized Fredholm operators

Determinant system for composite of generalized Fredholm operators

Determinant system for composite of generalized Fredholm operators

Formulae of Fredholm type for solutions of linear equations with generalized Fredholm operator

Formulae of Fredholm type for solutions of linear equations with generalized Fredholm operator

The determinant theory of generalized Fredholm operators

The determinant theory of generalized Fredholm operators