Fredholm Operators Research Articles

Decision-Making of Fredholm Operator on a New Variable Exponents Sequence Space of Supply Fuzzy Functions Defined by Leonardo Numbers

In this article, we will use a weighted regular matrix formed by Leonardo numbers and variable exponent sequence spaces to build a new stochastic space. We have proposed various geometric and topological structures for this new space, as well as the multiplication operator that operates on it.

Topological Anderson insulators by homogenization theory

. A central property of (Chern) topological insulators is the presence of robust asymmetric transport along interfaces separating two-dimensional insulating materials in different topological phases. A topological Anderson Insulator is an insulator whose topological phase is induced by spatial fluctuations. This article proposes a mathematical model of perturbed Dirac equations and shows that for sufficiently large and highly oscillatory perturbations, the system is in a different topological phase than the unperturbed model. In particular, a robust asymmetric transport indeed appears at an interface separating perturbed and unperturbed phases. The theoretical results are based on careful estimates of resolvent operators in the homogenization theory of Dirac equations and on the characterization of topological phases by the index of an appropriate Fredholm operator.

Ultrametric Fredholm operators and approximate pseudospectrum

PurposeThe paper deals with ultrametric bounded Fredholm operators and approximate pseudospectra of closed and densely defined (resp. bounded) linear operators on ultrametric Banach spaces.Design/methodology/approachThe author used the notions of ultrametric bounded Fredholm operators and approximate pseudospectra of operators.FindingsThe author established some results on ultrametric bounded Fredholm operators and approximate pseudospectra of closed and densely defined (resp. bounded) linear operators on ultrametric Banach spaces.Originality/valueThe results of the present manuscript are original.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Z2 classification of FTR symmetric differential operators and obstruction to Anderson localization

This paper concerns the Z2 classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.

On the class of almost order Fredholm operators

On the class of almost order Fredholm operators

Fredholm theory of the Toeplitz algebra on the space of all entire functions

AbstractWe prove that an aggregate Toeplitz operator on the Fréchet space of all entire functions is a Fredholm operator if and only if its symbol does not vanish. The result is motivated by and closely resembles the classical result of Gohberg and Douglas from the Hardy space theory of Toeplitz operators. There are however some subtle differences which we also discuss.

On a algebraic characterization of the generalized Fredholm operators

On a algebraic characterization of the generalized Fredholm operators

Equivalence after extension and Schur coupling for Fredholm operators on Banach spaces

Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC.Motivated by this result, we investigate when EAE and SC coincide for Fredholm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer k and every pair of Banach spaces (X,Y), either no pair of Fredholm operators of index k acting on X and Y, respectively, is SC, or every pair of this kind which is EAE is also SC. Consequently, whether EAE and SC coincide for Fredholm operators of index k depends only on the geometry of the underlying Banach spaces X and Y, not on the properties of the operators themselves.We quantify this finding by introducing two numerical indices which capture the coincidence of EAE and SC and provide a number of examples illustrating the possible values of these indices. Notably, this includes an example showing that the above-mentioned result of Ter Horst et al., which is based on a pair of essentially incomparable Banach spaces, does not extend to projectively incomparable Banach spaces.

Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas

The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for monatomic mixtures and polyatomic single species are more recently obtained. This work concerns the compactness of the operator for a multicomponent mixture of polyatomic species, where the polyatomicity is modeled by a discrete internal energy variable. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. The assumptions are essentially generalizations of the Grad's assumptions for monatomic single species. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model. Then it follows that the linearized collision operator is a Fredholm operator, and its domain is also obtained.

Asymptotically flat Fredholm bundles and assembly

Almost flat finitely generated projective Hilbert [Formula: see text]-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a [Formula: see text]-homology class [Formula: see text] twisted with an almost flat bundle in terms of the image of [Formula: see text] under Lafforgue’s assembly map and the almost representation associated with the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to generalize Dadarlat’s theorem to the case of an infinite-dimensional bundle equipped with a continuous family of Fredholm operators on the fibers. Along the way, we show that special cases of the Strong Novikov Conjecture can be proven if there exist sufficiently many almost flat bundles with Fredholm operator. To this end, we introduce the concept of an asymptotically flat Fredholm bundle and its associated asymptotic Fredholm representation, and prove an index theorem which relates the index of the asymptotic Fredholm bundle with the so-called asymptotic index of the associated asymptotic Fredholm representation.

Generalized essential spectra involving the class of g-g-Riesz operators

Abstract In this paper, we explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X. The results are formulated in terms of some topological conditions made on X or on its dual X * {X^{*}} . In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided.

Solvability in the sense of sequences for some linear and nonlinear Fredholm operators with the logarithmic Laplacian

We study the solvability of certain linear and nonlinear nonhomogeneous equations in one dimension involving the logarithmic Laplacian and the transport term. In the linear case we show that the convergence in L 2 ( R ) of their right sides yields the existence and the convergence in L 2 ( R ) of the solutions. We generalize the results obtained in the earlier article of Efendiev and Vougalter [Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian. Monatsh Math. 2023] in the non-Fredholm case without the drift. In the nonlinear part of the work we demonstrate that, under the reasonable technical assumptions, the convergence in L 1 ( R ) of the integral kernels implies the existence and the convergence in L 2 ( R ) of the solutions.

On ellipticity of operators with shear mappings

The nonlocal boundary value problems are considered, in which the main operator and the operators in the boundary conditions include the differential operators and twisting operators. The de nition of the trajectory symbols for this class of problems is given. We show that the elliptic problems de ne the Fredholm operators in the corresponding Sobolev spaces. The ellipticity condition of such nonlocal boundary value problem is given.

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) H of entire functions associated with operator valued kernel functions. de Branges operators E=(E−,E+) analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on X, where X is a complex separable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS B(E) based on the de Branges operator E=(E−,E+) has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces B(E) has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces B(E) has been given.

Operators that are Weakly Coercive and a Compact Perturbation

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

Fredholm and Schatten Class Operators on Bergman Spaces with Exponential Weights

In this paper, we give a characterization of Fredholmness of the Toeplitz operators on the Bergman spaces A φ p with exponential weights in D when 0 < p < ∞ . Also, we obtain the sufficient and necessary conditions which the Toeplitz and Hankel operators on A φ 2 belong to h -Schatten class, where h is a continuous increasing convex function.

Orientation flow for skew-adjoint Fredholm operators with odd-dimensional kernel

The orientation flow of paths of real skew-adjoint Fredholm operators with invertible endpoints was studied by Carey, Phillips and Schulz-Baldes. For paths of real skew-adjoint Fredholm operators with odd-dimensional kernel the orientation flow is defined with respect to a real one-dimensional reference projection. It is homotopy invariant and fulfills a concatenation property. When applied to closed paths it is independent of the reference projection and provides an isomorphism of the fundamental group of the space of real skew-adjoint Fredholm operators with odd-dimensional kernel to Z2. As an example the orientation flow of the magnetic flux inserted in a half-sided Kitaev chain is studied.

A comparative study for the spectral properties of Toeplitz and Hankel operators

In this introductory review, we study Hankel and Toeplitz operators considering them as acting on certain spaces of analytic functions, namely Hardy spaces and compare their spectral properties such as their compactness criteria. In contrast to Toeplitz operators, the symbol of a Hankel operator is not uniquely determined by the operator. We also connect Toeplitz operators with Fredholm operators and give some of the most beautiful properties of Toeplitz operators such as the essential spectrum of Toeplitz operator with continuous symbol and the index of Toeplitz operator introducing Fredholm operators firstly.

LOCAL MODEL OF THE THERMOELASTIC STATE OF A POROUS MATERIAL

In the paper, a local stationary model of the thermoelastic state of a porous material is constructed. The model is based on uncoupled boundary value problems for stationary equations of heat conduction and thermoelasticity for a space with two spherical cavities. The temperature field is determined by the constant temperature on the surfaces of the cavities, which are considered free of forces. The problem was solved by the generalized Fourier method (GFM), for which the paper presents its further development for a certain class of thermoelasticity problems. For this purpose, systems of equally directed spherical coordinates were introduced, the origins of which are connected with the centers of the cavities. In the work, a new set of axisymmetric basic solutions of the Lamé equation for a sphere is constructed and addition theorems are proved for it and for solutions of the vector biharmonic equation in the given coordinate systems. The GFM formalism made it possible to reduce the boundary value problems to algebraic solvable systems with Fredholm operators in the space l2 under the condition that the spherical surfaces do not intersect. The reduction method was used for the numerical solution of the systems. Graphs of stresses on the surface of one of the cavities and stresses on the axis of symmetry of the problem between the cavities at different relative sizes of the cavities and at different temperatures of their heating were obtained. The obtained results agree with those known for one cavity. The convergence of the reduction method is verified numerically.