Fredholm Operators Research Articles

Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator

In this article, by popularization of the reproducing kernel Hilbert space method in the sense of the Atangana–Baleanu fractional operator; set of first-order integrodifferential equations are solved with respect to Fredholm operator and initial conditions of optimality. The solvability approach based on use of the generalized Mittag–Leffler function in order to avoid nonsingular and nonlocal kernel functions appears in the classical fractional operator's. The procedure of solution is studied and described in details under some hypotheses, which provides the theoretical structure behind the utilized numerical method. Indeed, error analysis and convergence of numerical solution for the identification of the method is introduced in Hilbert space. In this analysis, some computational results and graphical representations are presented to demonstrated the suitability and portability of the utilized new fractional operator. Finally, the gained results reach to that; the utilized method is simple, direct, and powerful tool in finding numerical solutions for considered fractional equations.

Existence of waves for a reaction–diffusion–dispersion system

Existence of travelling waves is studied for a bistable reaction–diffusion system of equations with linear integral terms (dispersion) and with some conditions on the nonlinearity. The proof is based on the Leray–Schauder method using the topological degree theory for Fredholm and proper operators with the zero index and a priori estimates of solutions in properly chosen weighted spaces.

Fredholm operators and essential S-spectrum in the quaternionic setting

For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate S-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact operator perturbations. Finally, in association with the Fredholm operators, we develop the theory of essential S-spectrum. We also characterize the S-spectrum in terms of the essential S-spectrum and Fredholm operators. In the sequel, we study left and right S-spectra as needed for the development of the theory presented in this note.

Rigorous Formulation of the Lasing Eigenvalue Problem as a Spectral Problem for a Fredholm Operator Function

We propose a new convenient for mathematical investigation formulation of the lasing eigenvalue problem as a spectral problem for an operator-valued function, which involves boundary integral operators. We prove that these integral operators are weakly singular and the operator of the problem is Fredholm with index zero.

Structure of the set of stationary solutions to the equations of motion of a class of generalized Newtonian fluids

We investigate the steady-state equations of motion of the generalized Newtonian fluid in a bounded domain Ω⊂RN, when N=2 or N=3. Applying the tools of nonlinear analysis (Smale’s theorem, theory of Fredholm operators, etc.), we show that if the dynamic stress tensor has the 2-structure then the solution set is finite and the solutions are C1-functions of the external volume force f for generic f. We also derive a series of properties of related operators in the case of a more general p-structure, show that the solution set is compact if p>3N∕(N+2) and explain why the same approach as in the case p=2 cannot be applied if p≠2.

A Trace Formula for the Index of B-Fredholm Operators

AbstractIn this paper we define B-Fredholm elements in a Banach algebraAmodulo an idealJofA. When a trace function is given on the idealJ, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operatorTacting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], whereT0is a Drazin inverse ofTmodulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

3D Current Algebra and Twisted K Theory

Equivariant twisted K theory classes on compact Lie groups [Formula: see text] can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra [Formula: see text] using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators.

Equivalence after extension and Schur coupling coincide for inessential operators

In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE ⇒ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators.

On a Banach Algebra Generated by the Bergman Operator, Constant Coefficients, and Finitely Generated Groups of Shifts

We study a Banach algebra generated by the Bergman operator, constant coefficients, and shifts formed by finitely generated commutative groups of hyperbolic transformations of a unit disk acting in the Lebesgue space L p , p > 1, and obtain an efficient criterion for the operators from the considered Banach algebra to be Fredholm operators.

Global Continuation of Homoclinic Solutions

When extending bifurcation theory of dynamical systems to nonautonomous problems, it is a central observation that hyperbolic equilibria persist as bounded entire solutions under small temporally varying perturbations. In this paper, we abandon the smallness assumption and aim to investigate the global structure of the entity of all such bounded entire solutions in the situation of nonautonomous difference equations. Our tools are global implicit function theorems based on an ambient degree theory for Fredholm operators due to Fitzpatrick, Pejsachowicz and Rabier. For this we yet have to restrict to so-called homoclinic solutions, whose limit is 0 in both time directions.

Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case

The paper is devoted to the study of the solution of the Cauchy problem for a first-order differential equation in a Banach space with a small parameter on the right-hand side perturbing the equation. The coefficient of the derivative of the unknown function is a Fredholm operator with index zero and one-dimensional kernel. The case of a regular pair of operator coefficients is considered. An asymptotic expansion of the solution of the problem is constructed by using a method due to Vasil’eva, Vishik, and Lyusternik. In calculating the components of the regular and boundary-layer parts of the expansion, the cascade decomposition of the equations is used. It is proved that this expansion is asymptotic. Conditions for regular degeneration are found. The behavior of the solution as the parameter tends to zero is studied.

The Friedrichs extension for elliptic wedge operators of second order

Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.

The Friedrichs extension for elliptic wedge operators of second order

Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.

Fredholmness of multiplication of a weighted composition operator with its adjoint on H^{2} and A_{alpha}^{2}

In this paper, we obtain that C_{psi,varphi}^{ast} is bounded below on H^{2} or A_{alpha}^{2} if and only if C_{psi,varphi } is invertible. Moreover, we investigate the Fredholm operators C_{psi_{1},varphi_{1}}C_{psi_{2},varphi_{2}}^{ast} and C_{psi_{1},varphi_{1}}^{ast}C_{psi_{2},varphi_{2}} on H^{2} and A_{alpha}^{2}.

Toeplitz operators on the space of real analytic functions: The Fredholm property

We completely characterize those continuous operators on the space of real analytic functions on the real line for which the associated matrix is Toeplitz (that is, we describe Toeplitz operators on this space). We also prove a necessary and sufficient condition for such operators to be Fredholm operators. While the space of real analytic functions is neither Banach space nor has a basis which makes available methods completely different from classical cases of Hardy spaces or Bergman spaces, nevertheless the results themselves show surprisingly strong similarity to the classical Hardy-space theory.

АСИМПТОТИЧЕСКОЕ РЕШЕНИЕ УРАВНЕНИЯ ПЕРВОГО ПОРЯДКА С МАЛЫМ ПАРАМЕТРОМ ПРИ ПРОИЗВОДНОЙ С ВОЗМУЩЕННЫМ ОПЕРАТОРОМ

The paper is devoted to the Cauchy problem for a differential equation with a small parameter when using a Fredholm operator in a Banach space with a certain method. The investigated effect of this parameter. The solution is in the form of an asymptotic expansion. When solving the problems of using the cascade decomposition method for equations, which allows us to split the equation into equations in subspaces.

Compactness of multiplication, composition and weighted composition operators between some classical sequence spaces: A new approach

In a previous paper the author studied the compactness of multiplication, composition and weighted composition operators among some sequence spaces. We were motivated by these results and present two different approaches for obtaining some of the results in a previous paper. The first approach is to apply the theory of matrix transformations and the Hausdorff measure of noncompactness, and the second one is to use known results on multiplier spaces and the Hausdorff measure of noncompactness. We also use our techniques and methods from our proofs of the existing results to establish some new results related to the class of Fredholm operators and some classes of operators considered here and in the previous paper.

Fredholm operators on C*-algebras

The aim of this note is to generalize the notion of Fredholm operator to an arbitrary $C^*$-algebra. Namely, we define "finite type" elements in an axiomatic way, and also we define Fredholm type element $a$ as such element of a given $C^*$-algebra for which there are finite type elements $p$ and $q$ such that $(1-q)a(1-p)$ is "invertible". We derive index theorem for such operators. In applications we show that classical Fredholm operators on a Hilbert space, Fredholm operators in the sense of Breuer, Atiyah and Singer on a properly infinite von Neumann algebra, and Fredholm operators on Hilbert $C^*$-modules over an unital $C^*$-algebra in the sense of Mishchenko and Fomenko are special cases of our theory.

The index map in algebraic K-theory

For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group theory has features of $K$-theory (e.g. determinant bundles, tame symbol cocycle for Kac-Moody extension), the $K$-theory torsor relates higher loop groups with higher $K$-theory. We study the classifying "index" map of this torsor in detail. We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the $K$-theory torsor to previously studied dimension and determinant torsors.

Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies

Elastic bodies with cuspidal singularities at the surface are known to support wave processes in a finite volume which lead to absorption of elastic and acoustic oscillations (this effect is recognized in the engineering literature as Vibration Black Holes). With a simple argument we will give the complete description of the phenomenon of wave propagation towards the tip of a three-dimensional cusp and provide a formulation of the Mandelstam (energy) radiation conditions based on the calculation of the Umov-Poynting vector of energy transfer. Outside thresholds, in particular, above the lower bound of the continuous spectrum, these conditions coincide with ones due to the traditional Sommerfeld principle but also work at the threshold frequencies where the latter principle cannot indicate the direction of wave propagation. The energy radiation conditions supply the problem with a Fredholm operator of index zero so that a solution is determined up to a trapped mode with a finite elastic energy and exists provided the external loading is orthogonal to these modes. An example of a symmetric cuspidal body is presented which supports an infinite sequence of eigenfrequencies embedded into the continuous spectrum and the corresponding trapped modes with the exponential decay at the cusp tip. We also determine (unitary and symmetric) scattering matrices of two types and derive a criterion for the existence of a trapped mode with the power-low behavior near the tip.