semi-Fredholm Operators Research Articles

Periodic solutions for semilinear partial differential equation with lack of compactness

Through this work, we examine the periodicity of solutions for the following semilinear retarded equation defined as ddtw(t)=B˜w(t)+D˜(wt)+G˜(t,wt). Summarizing the theory of perturbation of semi-Fredholm operators and the contraction mapping theorem, we propose some adequate conditions on the bounded linear operator D˜ and the function G to establish the periodicity of solutions on the interval [0,+∞) without taking into consideration the compactness condition on the semigroup generated by the part of B˜. Furthermore, we present an application that includes numerical simulations to illustrate the applicability of the theoretical findings.

On the criteria of a measure of non-strict cosingularity in the description of spectral properties of operator matrix

Abstract In this paper, we are interested to formulate new assumptions on the entries of an unbounded 3 × 3 3\times 3 block operator matrix defined with a maximal domain on the product of Banach spaces guaranteeing its corresponding Frobenius–Schur formula. Our approach allows us to derive some original stability results intervening in the theory of perturbed lower semi-Fredholm operators involving the concept of a measure of non-strict cosingularity perturbation. A new technique is presented to investigate the Weidmann and defect essential spectra of the closure of such model of operator matrix via new criterion of perturbation.

Some results on upper semi-Fredholm operators on Banach lattices

Some results on upper semi-Fredholm operators on Banach lattices

Two classes of operators related to the perturbation classes problem

Let {{mathcal {S}}}{{mathcal {S}}} and {{mathcal {S}}}{{mathcal {C}}} be the strictly singular and the strictly cosingular operators acting between Banach spaces, and let PPhi _+ and PPhi _+ be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators Phi {mathcal {S}} and Phi {mathcal {C}} that satisfy {{mathcal {S}}}{{mathcal {S}}}subset Phi {mathcal {S}}subset PPhi _+ and {{mathcal {S}}}{{mathcal {C}}}subset Phi {mathcal {C}}subset PPhi _-. We give some conditions under which these inclusions become equalities, from which we derive some positive solutions to the perturbation classes problem for semi-Fredholm operators.

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

New results on perturbations of p-adic linear operators

Abstract In this research paper, we develop some aspects of the theory of Fredholm linear operators acting in p-adic (or non-Archimedean) Banach spaces. In this regard, we establish sufficient conditions for the p-adic Fredholmness of the algebraic sum of unbounded linear operators. Next, we study the perturbation of p-adic upper semi-Fredholm operators under strictly singular operators. Moreover, we give some results concerning quasi-compact and p-adic lower semi-Fredholm operators.

Essential spectra in non-Archimedean fields

In the paper we extend some aspects of the essential spectra theory of linear operators acting in non-Archimedean (or p-adic) Banach spaces. In particular, we establish sufficient conditions for the relations between the essential spectra of the sum of two bounded linear operators and the union of their essential spectra. Moreover, we give essential prerequisites by studying the duality between p-adic upper and p-adic lower semi-Fredholm operators. We close this paper by giving some properties of the essential spectra.

Nonlinear maps preserving semi-Fredholm operators with bounded nullity

Let be an infinite dimensional complex Banach space, and let be the algebra of all bounded linear operators on . In this paper, given any positive integer n, we provide a characterization of all bijective maps on preserving the difference of semi-Fredholm operators with nullity less than n, and describe completely the structure of these maps.

Pseudo-generalized inverse I

The purpose of this paper is to introduce and study a new class of operators that we call pseudogeneralized invertible operators. This class includes both the class of generalized invertible operators and the class of Drazin invertible operators. Some results connected with ascent and essential ascent are also obtained for this new class. The relationship between pseudo-generalized invertible operators, semi-Fredholm operators and generalized invertible operators is explored.

Periodic Solution for some Class of Linear Partial Differential Equation with infinite Delay using Semi-Fredholm perturbations

Abstract In this work, we study the existence of periodic solutions for a class of linear partial functional differential equations with infinite delay. Inspiring by an existing study, by applying the perturbation theory of semi-Fredholm operators, we introduce a suitable a priori estimate on the norm of the operator L to establish the periodicity of solutions in the case where the linear part is nondensely defined and satisfies the Hille-Yosida condition and without considering the exponential stability condition on the semigroup generated by the part of this operator on the closure of it’s domain. Moreover, in the special case where the linear part generates a strongly continuous semigroup and perturbed by a compact linear operator, we give some sufficient conditions to derive periodic solution from bounded ones. Finally, our theoretical results are illustrated by applications in both densely and nondensely cases.

On the periodic solutions for some retarded partial differential equations by the use of semi-Fredholm operators

The main goal of this work is to examine the periodic dynamic behavior of some retarded periodic partial differential equations (PDE). Taking into consideration that the linear part realizes the Hille-Yosida condition, we discuss the Massera’s problem to this class of equations. Especially, we use the perturbation theory of semi-Fredholm operators and the Chow and Hale’s fixed point theorem to study the relation between the boundedness and the periodicity of solutions for some inhomogeneous linear retarded PDE. An example is also given at the end of this work to show the applicability of our theoretical results.

Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology

In this article we investigate disjointly non-singular (DNS) operators. Following [9] we say that an operator T from a Banach lattice F into a Banach space E is DNS, if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [9] by showing that this class of operators forms an open subset of L(F,E) as soon as F is order continuous. Moreover, we show that in this case T is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology and the topology generated by T as a map (we say that T “complements” the unbounded norm topology in F). Since the class of DNS operators plays a similar role in the category of Banach lattices as the upper semi-Fredholm operators play in the category of Banach spaces, we investigate and indeed uncover a similar characterization of the latter class of operators, but this time they have to complement the weak topology.

Different completions of $A + CX$

In this paper we use Takahasi’s idea of using properties of spectral measures in addressing the question of the existence of an operator X such that A+CX is of appropriate types. In particular, we consider the class of right semi-Fredholm operators. Also, in the case of right invertibility we will show how the results of this type can be used to address the appropriate completion problem of the operator matrix M_X= \begin{bmatrix} A & C \\ X & B \end{bmatrix} .

Singular Sylvester equation in Banach spaces and its applications: Fredholm theory approach

We prove that, by assuming the existence of at least one left upper semi-Fredholm operator, then under some natural conditions, the singular operator equation AX−XB=C is solvable if the appropriate matrix equation is solvable. This characterization is convenient because the matrix version of the problem has been closed in [14] and [17]. In addition, we obtain sufficient conditions for A, B and X such that the generalized derivation AX−XB is a compact operator. A connection is established with Fréchet derivatives and commutators of idempotents. Applications to Schur coupling and linear time-invariant systems are mentioned.

The perturbation classes problem for subprojective and superprojective Banach spaces

We show that the perturbation class for the upper semi-Fredholm operators between two Banach spaces X and Y coincides with the strictly singular operators when X is subprojective and that the perturbation class for the lower semi-Fredholm operators coincides with the strictly cosingular operators when Y is superprojective. Similar results were previously obtained under stronger conditions for X and Y.

Additive maps preserving semi-Fredholm operators with bounded ascent on $\mathcal B(\mathcal X)$

Additive maps preserving semi-Fredholm operators with bounded ascent on $\mathcal B(\mathcal X)$

On Operators with Closed Range and Semi-Fredholm Operators Over W*-Algebras

In this paper, we consider $${\mathcal A}$$-Fredholm and semi-$${\mathcal A}$$-Fredholm operators on Hilbert C*-modules over a W*-algebra $${\mathcal A}$$ defined in [3] and [9]. Using the assumption that $${\mathcal A}$$ is a W*-algebra (rather than an arbitrary C*-algebra), we obtain a generalization of Schechter—Lebow characterization of semi-Fredholm operators and a generalization of the “punctured neighborhood” theorem, as well as some other results generalizing their classical counterparts. We consider both adjointable and nonadjointable semi-Fredholm operators over W*-algebras. Moreover, we also work with general bounded adjointable operators with closed ranges over C*-algebras and prove a generalization of a Bouldin result for Hilbert spaces to Hilbert C*-modules.

Semi-Fredholm theory on Hilbert $C^{*}$ -modules

We establish the semi-Fredholm theory on Hilbert C∗-modules as a continuation of Fredholm theory on Hilbert C∗-modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert C∗-module, and we prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open, and many other results that generalize their classical counterparts.

Spectral Properties Involving Generalized Weakly Demicompact Operators

In this paper, we investigate the concept of generalized weakly demicompact operators with respect to weakly closed densely defined linear operators. We give their relationship with Fredholm and upper semi-Fredholm operators. In particular a characterization by means of upper semi-Browder spectrum is given. Moreover, we provide some sufficient conditions on the inputs of a closable block operator matrix to ensure the generalized weak demicompactness of its closure. Our results generalize many known ones in the literature.

Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation results

In this paper, we study the essential and the structured essential pseudospectra of closed densely defined linear operators acting on a Banach space X. We start by giving a refinement and investigating the stability of these essential pseudospectra by means of the class of demicompact linear operators. Moreover, we introduce the notion of pseudo demicompactness and we study its relationship with pseudo upper semi-Fredholm operators. Some stability results for the Gustafson essential pseudospectrum involving pseudo demicompact operators is given.