Fredholm Operators Research Articles

Notes on twisted equivariant K-theory for C*-algebras

In this paper, we study a generalization of twisted (groupoid) equivariant K-theory in the sense of Freed–Moore for [Formula: see text]-graded [Formula: see text]-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele’s K-theory for [Formula: see text]-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant K-group when the [Formula: see text]-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.

Functional Calculus and Joint Torsion of Pairs of Almost Commuting Operators

This paper investigates the transformation of determinants of pairs of Fredholm operators with trace class commutators. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we extend some results of Carey and Pincus on determinants of Toeplitz operators and Tate tame symbols. Additionally, we obtain variational formulas for joint torsion.

Fredholm solvability of time-periodic boundary value hyperbolic problems

We investigate a large class of linear boundary value problems for the general first-order one-dimensional hyperbolic systems in the strip [0,1]×R. We state rather broad natural conditions on the data under which the operators of the problems satisfy the Fredholm alternative in the spaces of continuous and time-periodic functions. A crucial ingredient of our analysis is a non-resonance condition, which is formulated in terms of the data responsible for the bijective part of the Fredholm operator. In the case of 2×2 systems with reflection boundary conditions, we provide a criterium for the non-resonant behavior of the system.

Canonical holomorphic sections of determinant line bundles

Abstract We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

Interpolation of Fredholm operators

We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters θ∈(0,1) and q∈[1,∞] under which an operator A is a Fredholm operator from the real interpolation space (X0,X1)θ,q to (Y0,Y1)θ,q for a given operator A:(X0,X1)→(Y0,Y1) between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corresponding indices generated by the K-functional of elements from the kernel of the operator A in the interpolation sum X0+X1. If in addition A is invertible operator on endpoint spaces, then these conditions are also necessary. We apply these results to present a solution of the variant of a long-standing problem of Lions–Magenes for the real interpolation method. We also discuss some applications to the spectral theory of operators as well as to perturbation of the Hardy operator by identity on weighted Lp-spaces.

On the completeness of root functions of a holomorphic family of non-coercive mixed problem

We consider a non-coercive mixed boundary value problem in a bounded domain D of complex space for a second order parameter-dependent elliptic differential operator with complex-valued essentially bounded measured coefficients and complex parameter . The differential operator is assumed to be of divergent form in D, the boundary operator is of Robin type. The boundary of D is assumed to be a Lipschitz surface. Under reasonable assumptions the pair (A, B) induces a family of non-coercive mixed problems and a holomorphic family of Fredholm operators in suitable Hilbert spaces , of Sobolev type (here are the Sobolev-Slobodetskii spaces over D). If there is a Lipschitz function close enough to the (possibly discontinuous) argument of the complex-valued multiplier of the parameter in then we prove that the operators are continuously invertible for all with sufficiently large modulus on each angle on the complex plane where the operator is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family to be (doubly) complete in the spaces , and the Lebesgue space .

Fredholm operators on the space of bounded sequences

Necessary and sufficient conditions are studied that a bounded operator Tx=(x1*x,x2*x,…) on the space ℓ∞, where xn*∈ℓ∞*, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1*,x2*,…} are investigated. Various estimates of the defect d(T) = codim R(T), where R(T) is the range of T, are given. The case of xn*=dnxtn*, where dn∈R and xtn*≥0 are extreme points of the unit ball Bℓ∞*, that is, tn∈βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R(T) are given and the value d(T) is calculated. For example, the condition {n : 0 < |dn| <Δ} = φ for some Δ is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R(T) (tn0 is isolated if there is a neighborhood u of tn0 satisfying tn∉u for all n ≠ n0). If {n : |dn| <Δ} = φ, then d(T) is equal to the defect Δ{tn} of {tn}. It is shown that if d(T) = ∞ and R(T) is closed, then there exists a sequence {An} of pairwise disjoint subsets of ▪ satisfying χAn∉R(T).

On the C *-Algebra Generated by the Bergman Operator, Carleman Second-Order Shift, and Piecewise Continuous Coefficients

We study the C * -algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space L 2 and extended by the Carleman rotation by an angle π. As a result, we obtain an efficient criterion for the operators from the indicated C * -algebra to be Fredholm operators.

Homotopy Equivalence of Hilbert $C$*-modules

&lt;p&gt;In this paper we introduce the concept of homotopy equivalence for Hilbert $C$*-modules and investigate some properties of this equivalence relation. We then&lt;br /&gt;present the homotopy equivalence in the context of Fredholm operators on Hilbert $C$*-modules and classify these operators in terms of their index.&lt;/p&gt;

A loop group extension of the odd Chern character

We show that the universal odd Chern form, defined on the stable unitary group U , extends to the loop group L U as an equivariantly closed differential form. This provides an odd analogue to the Bismut–Chern form that appears in supersymmetric field theories. We also describe the associated transgression form, the so-called Bismut–Chern–Simons form, and explicate some properties it inherits as a differential form on the space of maps of a cylinder into the stable unitary group. As one corollary, we show that in a precise sense the spectral flow of a loop of self adjoint Fredholm operators equals the lowest degree component of the Bismut–Chern–Simons form, and the latter, when restricted to cylinders which are tori, is an equivariantly closed extension of spectral flow. As another corollary, we construct the Chern character homomorphism from odd K -theory to the periodic cohomology of the free loop space, represented geometrically on the level of differential forms.

Multiplication operators on Cesàro function spaces

In this paper, we characterize the compact, invertible, Fredholm and closed range multiplication operators on Ces?ro function spaces.

On the index of a non-Fredholm model operator

Let $\{A(t)\}_{t \in \mathbb{R}}$ be a path of self-adjoint Fredholm operators in a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$ as $t \to \pm \infty$. Computing the index of the operator $D_A= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A = \int_{\mathbb{R}}^{\oplus} dt \, A(t)$, and its relation to spectral flow along this path, has a long history. While most of the latter focuses on the case where $A(t)$ all have purely discrete spectrum, we now particularly study situations permitting essential spectra. Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we consider spectral shift functions $\xi(\, \cdot \,; A_+, A_-)$ and $\xi(\, \cdot \, ; H_2, H_1)$ associated with the pairs $(A_+, A_-)$ and $(H_2,H_1)$. Assuming $A_+$ to be a relatively trace class perturbation of $A_-$ and $A_{\pm}$ to be Fredholm, the value $\xi(0; A_-, A_+)$ was shown in [14] to represent the spectral flow along the path $\{A(t)\}_{t\in \mathbb{R}}$ while that of $\xi(0_+; H_1,H_2)$ yields the Fredholm index of $D_A$. The fact, proved in [14], that these values of the two spectral functions are equal, resolves the index = spectral flow question in this case. When the path $\{A(t)\}_{t \in \mathbb{R}}$ consists of differential operators, the relatively trace class perturbation assumption is violated. The simplest assumption that applies (to differential operators in (1+1)-dimensions) is a relatively Hilbert-Schmidt perturbation. This is not just an incremental improvement. In fact, the approximation method we employ here to make this extension is of interest in any dimension. Moreover we consider $A_\pm$ which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions for the pairs $(A_+, A_-)$ and $(H_2,H_1)$ found in all of the previous papers [9], [14], and [22] can be proved in the non-Fredholm case.

Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices

We study the probability density function (PDF) of the smallest eigenvalue of Laguerre–Wishart matrices [Formula: see text] where [Formula: see text] is a random [Formula: see text] ([Formula: see text]) matrix, with complex Gaussian independent entries. We compute this PDF in terms of semi-classical orthogonal polynomials, which are deformations of Laguerre polynomials. By analyzing these polynomials, and their associated recurrence relations, in the limit of large [Formula: see text], large [Formula: see text] with [Formula: see text] — i.e. for quasi-square large matrices [Formula: see text] — we show that this PDF, in the hard edge limit, can be expressed in terms of the solution of a Painlevé III equation, as found by Tracy and Widom, using Fredholm operator techniques. Furthermore, our method allows us to compute explicitly the first [Formula: see text] corrections to this limiting distribution at the hard edge. Our computations confirm a recent conjecture by Edelman, Guionnet and Péché. We also study the soft edge limit, when [Formula: see text], for which we conjecture the form of the first correction to the limiting distribution of the smallest eigenvalue.

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rn for a second order elliptic differential operator A(x, ∂). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ∂) is of Robin type on ∂D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ⊂ ∂D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.

On a spectral flow formula for the homological index

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)|t∈R} that converges in norm to asymptotes A± at ±∞. Then under certain conditions [22] that include the assumption that the operators {D(t)=D+A(t),t∈R} all have discrete spectrum the spectral flow along the path {D(t)} can be shown to be equal to the index of ∂t+D(t) when the latter is an unbounded Fredholm operator on L2(R,H). In [16] an investigation of the index = spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In [11] the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index.In this paper we take the investigation begun in [11] much further. We show how to generalize a formula known from the setting of the L2 index theorem to the non-Fredholm setting. Restricting back to the case of selfadjoint Fredholm operators our formula extends the result of [22] in the sense of relaxing the discrete spectrum condition. It also generalizes some other Fredholm operator results of [21,16,11] that permit essential spectrum for the operators in the path. Our result may also apply however when the operators {D(t)} have essential spectrum equal to the whole real line.Our main theorem gives a trace formula relating the homological index of [7] to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalization of spectral flow to the non-Fredholm setting.

On a characterization of the S-essential spectra of the sum and the product of two operators and application to a transport operator

In this paper, we develop some operational calculus inspired from the Fredholm operator theory to investigate the S-essential spectra of the sum and the product of two operators acting on a Banach space. Furthermore, we apply the obtained results to determine the S-essential spectra of an integro-differential operator with abstract boundary conditions in L1 ([–a, a] × [–1, 1]) (a > 0).

On the transmission problems for the Oseen and Brinkman systems on Lipschitz domains in compact Riemannian manifolds

The purpose of this work is to show the well‐posedness in L2‐Sobolev spaces of the Poisson‐transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi‐Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the Levi‐Civita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds.

Existence of Pulses for Local and Nonlocal Reaction-Diffusion Equations

Reaction-diffusion equations with a space dependent nonlinearity are considered on the whole axis. Existence of pulses, stationary solutions which vanish at infinity, is studied by the Leray–Schauder method. It is based on the topological degree for Fredholm and proper operators with the zero index in some special weighted spaces and on a priori estimates of solutions in these spaces. Existence of solutions is related to the speed of travelling wave solutions for the corresponding autonomous equations with the limiting nonlinearity.

Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble**Dedicated to Professor R J Baxter on the occasion of his birthday.

It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled limit of the annihilation process . Furthermore, deleting each particle at random in the rescaled limit of the coalescence process , a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to , where s denotes the gap size and denotes the Riemann zeta function. It is shown how this can be rigorously established using an asymptotic formula for matrix Fredholm operators. A determinant formula is derived for the gap probability in the finite N case, and this is used to illustrate the asymptotic formulas against numerical computations.