Linear Operators In Banach Spaces Research Articles

Null Controllability of Nonlinear Mixed Integro-differential Systems

The present paper investigates the existence of mild solutions and provides sufficient conditions for the null controllability of nonlinear mixed integrodifferential systems with unbounded linear operators in Banach spaces. It is observed that, if the linear system is controllable, then the nonlinear equivalence is also controllable by subjecting certain smooth conditions on the perturbation function. The results are obtained using semi group of linear operators, fractional powers of operators and the Schauder fixed point theorem.

Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.

On the generalized spectrum of bounded linear operators in Banach spaces

<abstract><p>Utilizing the stability characterizations of generalized inverses, we investigate the generalized resolvent of linear operators in Banach spaces. We first prove that the local analyticity of the generalized resolvent is equivalent to the continuity and the local boundedness of generalized inverse functions. We also prove that several properties of the classical spectrum remain true in the case of the generalized one. Finally, we elaborate on the reason why we use the generalized inverse rather than the Moore-Penrose inverse or the group inverse to define the generalized resolvent.</p></abstract>

A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations

The aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the extensions of linear operators in Banach spaces and establish conditions for the decomposition of the integro-differential operator into two lower-order integro-differential operators. We also create solvability criteria and derive the unique solution in closed form. Two example problems for an ordinary and a partial intergro-differential equation respectively are solved.

Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces.

Decomposition of abstract linear operators in Banach spaces

In this paper we investigate a class of integro-differential equations οn a Banach space with nonlocal and initial boundary conditions in terms of an abstract operator equation
 B1 x=Ax-S0 F(A x)-G0 Φ(Ax)=f ,x∈D(B1) (1) 
 where A, A are linear abstract operators, S0, G0 are vectors and Φ, F the functional vectors. The operator B1 under study has a decomposition of the form B1=B0 B with B and B0 being different abstract linear operators of special forms. Methods. Extensions of operators on Banach spaces are used. The Decomposition Method proposed here is essentially different from other Decomposition Methods in the relevant literature. Results. Our main research result is the existence and uniqueness of solution of and its representation in the closed form. The necessary and sufficient conditions for the correctness of the operator are intermediate, secondary results. Overall, a direct method analytically solving problem (1) is proposed, in an algorithmic procedure that is reproducible in any program of symbolic calculations. The stages of the solution method are illustrated by three examples. Computer algebra system Mathematica is employed to demonstrate the solution outcomes.

Acute and Stable Perturbations of the Drazin Inverse of Bounded Linear Operators in Banach Spaces

Let be the set of bounded linear operators on a Banach space X, and a liner operator be Drazin invertible. A linear operator is said to be a stable perturbation of A if B is Drazin invertible and is invertible, where I is the identity operator on X, and are the spectral projectors of A and B respectively. We call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius Under the similar condition that a matrix B is a stable perturbation of a matrix A, an explicit formula for the Drazin inverse BD is derived by Xu, Song and Wei (The stable perturbation of the Drazin inverse of the square matrices, SIAM J. Matrix Anal. Appl., 31(3) (2010), pp. 1507-1520). This formula is generalized to the infinite-dimensional case, and some new formulas for spectral radius of is generalized from the finite-dimensional case to the Banach space.

Regular factorizations and group inverses of linear operators in Banach spaces

The main topic of this paper is the group invertibility and expressions of group inverses of linear operators in Banach spaces. We first give some properties of regular factorizations and then derive some new characterizations of the group invertibility of linear operators. Based on these results, we use any one of generalized inverses to give some concise expressions of the group inverse.

Drazin inverse: representation, approximation, continuity and illustrations

In this paper, we present some characteristics and expressions of the Drazin inverse for matrices and bounded linear operators in Banach spaces. We give a survey of some of results on the continuity of the Moore-Penrose and Drazin inverse, direct technics for computing the Drazin inverse are discussed, they are based on Euler-Knopp Method and characterized in terms of a limiting process. The examples presented are for illustrative purposes, some of which are provided for testing the considered iterative processes

INTEGRO-DIFFERENTIAL EQUATIONS EMBODYING POWERS OF A DIFFERENTIAL OPERATOR

We establish solvability and correctness criteria for two Fredholm type linear integro-differential operators B2, B4 encompassing up to second and fourth powers, respectively, of a differential operator A�with aknown inverse I = A�1. We also derive explicit solution formulae to corresponding initial and boundary value problems by using the inverse of the differential operator. The approach is based on the theory of the extensions of linear operators in Banach spaces. Three example problems for ordinary and partial integro-differential operators are solved.

On uniform exponential trisplitting for cocycles of linear operators in Banach spaces

Abstract The aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.

On Stability of Perturbed Semigroups in Partially Ordered Banach Spaces

We prove necessary and sufficient stability conditions for perturbed semigroups of linear operators in Banach spaces with cones and consider examples using these conditions. In particular, we consider an example where the boundary-value problem is perturbed by a linear operator with a delayed independent variable and establish stability conditions for such a perturbed semigroup.

Some remarks on spectra of nuclear operators

Abstract We give criteria for the spectra of some nuclear operators in subspaces of quotients of Lp -spaces to be central-symmetric, as well as for the spectra of linear operators in Banach spaces to be Zd -symmetric in the sense of B. Mityagin. Also, we present a short proof of a corresponding Mityagin’s theorem.

Almost periodic solutions at infinity of differential equations and inclusions

We study the notion of a function almost periodic at infinity and present some spectral conditions of the almost periodicity at infinity for bounded solutions of a differential inclusions governed by closed multivalued linear operators in Banach spaces. Applications to degenerate differential equations and semilinear differential inclusions are given. As example, we consider the problem of stabilization of solutions to a heat equation.

On the perturbation of outer inverses of linear operators in Banach spaces

The main concern of this article is the perturbation problem for outer inverses of linear bounded operators in Banach spaces. We consider the following perturbed problem. Let T∈B(X,Y) with an outer inverse T{2}∈B(Y,X) and δT∈B(X,Y) with ‖δTT{2}‖<1. What condition on the small perturbation δT can guarantee that the simplest possible expression B=T{2}(I+δTT{2})−1 is a generalized inverse, Moore–Penrose inverse, group inverse, or Drazin inverse of T+δT? In this article, we give a complete solution to this problem. Since the generalized inverse, Moore–Penrose inverse, group inverse, and Drazin inverse are outer inverses, our results extend and improve many previous results in this area.

A new perturbation theorem for Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is “the generalized Neumann lemma” which is quite different from the method in [12] where “the generalized Banach lemma” was used. By the method of the perturbation analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

Generalized Kato linear relations

For a Banach space the notions of generalized Kato linear relation and the corresponding spectrum are introduced and studied. We show that the symmetric difference between the generalized Kato spectrum and the Goldberg spectrum of multivalued linear operators in Banach spaces is at most countable. The obtained results are used to describe the generalized Kato spectrum of the inverse of the left shift operator regarded as a linear relation.

On the law of large numbers for compositions of independent random semigroups

We study random linear operators in Banach spaces and random one-parameter semigroups of such operators. For compositions of independent random semigroups of linear operators in the Hilbert space we obtain sufficient conditions for fulfilment of the law of large numbers and give examples of its violation.

Series representation of compact linear operators in Banach spaces

Let \(p\in(1,\infty)\) and \(I=(0,1)\); suppose that \(T\colon L_{p}(I)\rightarrow L_{p}(I)\) is a~compact linear map with trivial kernel and range dense in \(L_{p}(I)\). It is shown that if the Gelfand numbers of \(T\) decay sufficiently quickly, then the action of \(T\) is given by a series with calculable coefficients. The special properties of \(L_{p}(I)\) enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.

On the essential minimum modulus of linear operators in Banach spaces

On the essential minimum modulus of linear operators in Banach spaces