Fredholm Conditions Research Articles

Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalize the Carey–Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Suppose α is an orientation preserving diffeomorphism (shift) of $${{\mathbb{R}}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. We establish sufficient conditions for the Fredholmness of the singular integral operator with shift $$(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ acting on $${L^p({\mathbb{R}}_+)}$$ with 1 < p < ∞, where P ± = (I ± S)/2, S is the Cauchy singular integral operator, and $${{{W_{\alpha}f=f\circ\alpha}}}$$ is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on $${{\mathbb{R}}_+}$$ and may admit discontinuities of slowly oscillating type at 0 and ∞.

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces \(L^p_N({\mathbb{R}}_{+},w)\) where 1 < p < ∞, w is a Muckenhoupt weight on \({\mathbb{R}}\) and \(N \in {\mathbb{N}}\). The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on \(L^{p}({\mathbb{R}},w)\) that are continuous on \({\mathbb{R}}\) and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SOp, w of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderon-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a.

B-Separable boundary value problems in Banach-valued function spaces

In this paper, the nonlocal boundary value problems for anisotropic partial differential-operator equations with dependent coefficients in Banach-valued Besov ( B) spaces are studied. The principal parts of the appropriate differential operators are non-self-adjoint. Several conditions for separability and Fredholmness are given. These results permit us to establish that the inverse of the corresponding differential operators belong to the Schatten q-class. The spectral properties of the appropriate differential operators are also investigated. In addition we study the maximal regularity of nonlocal initial boundary value problems for abstract parabolic equations, finite or infinite systems of parabolic equations and the separability of nonlocal boundary value problems for finite or infinite systems of quasi-elliptic equations in B spaces.

Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM

Let r = ( x 1 2 + x 2 2 ) 1 / 2 be the distance function to the origin O ∈ R 2 , and let us fix δ > 0 . We consider the “Schrödinger-type mixed boundary value problem” − Δ u + δ r − 2 u = f ∈ H m − 1 ( Ω ) on a bounded polygonal domain Ω ⊂ R 2 . The singularity in the potential δ r − 2 severely limits the regularity of the solution u . This affects the rate of convergence to u of the finite element approximations u S ∈ S obtained using a quasi-uniform sequence of meshes. We show that a suitable graded sequence of meshes recovers the quasi-optimal convergence rate ‖ u − u n ‖ H 1 ( Ω ) ≤ C dim ( S n ) − m / 2 ‖ f ‖ H m − 1 ( Ω ) , where S n are the FE spaces of continuous, piecewise polynomial functions of degree m ≥ 1 associated to our sequence of meshes and u n = u S n ∈ S n are the FE approximate solutions. This is in spite of the fact that u ⁄ ∈ H m + 1 ( Ω ) in general. One of the main results of our paper is to show that the singularities due to the potential and the singularities due to the singularities of the domain or to the change in boundary conditions can be treated in the same way. Our proof is based on regularity and well-posedness results in weighted Sobolev spaces, with the weight taking into account all singularities (including the ones coming from the potential). Our regularity results apply also to operators with weaker singularities, like the Schrödinger operator − Δ u + δ r − 1 , for which we also obtain Fredholm conditions and a formula for the index. Our a priori estimates also extend to piecewise smooth domains (i.e., curvilinear polygonal domains).

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.

Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions

We develop the Fredholm theory for Toeplitz operators, with symbols in the C*-algebra D = [SO, SAP]n, n generated by all slowly oscillating (SO) and semi-almost periodic (SAP) n × n matrix functions, on the Hardy spaces H n p (with 1 < p < ∞) over the upper half-plane. Using limit operator techniques, we get necessary Fredholm conditions for any operator in the Banach algebra alg(S, D) of singular integral operators with coefficients in D on the space [Lp (R)]n. Applying the Allan–Douglas local principle and the theory of Toeplitz operators with SAP matrix symbols, we also establish Fredholm criteria for Toeplitz operators with matrix symbols g ∈ D on the space H n p . An index formula for Fredholm Toeplitz operators with matrix symbols in D is obtained on the basis of an appropriate approximation of slowly oscillating components of the symbols. 2000 Mathematics Subject Classification 47B35 (primary), 47A53 (secondary).

Fredholmness of Singular Integral Operators with Piecewise Continuous Coefficients on Weighted Banach Function Spaces

We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $w$ is a Khvedelidze weight, and the variable exponent $p(t)$ satisfies the estimate $|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$.

On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let \(G \subset {\mathbb{C}}\) be a domain with smooth boundary and let α be a C2-diffeomorphism on \(\overline{G}\) satisfying the Carleman condition \(\alpha \circ \alpha = {\rm id}_{\overline{G}}\). We denote by \({\mathcal{R}}\) the C*-algebra generated by the Bergman projection of G, all multiplication operators aI \((a \in C(\overline{G}))\) and the operator \(W{\varphi} = \sqrt{|{\rm det} J_{\alpha}|}{\varphi} \circ \alpha\), where det Jα is the Jacobian of α. A symbol algebra of \({\mathcal{R}}\) is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator \((W{\varphi})(z) = \varphi(-\overline{z})\) is isomorphic and isometric to \({\mathbb{C^2}} \times M_2(mathbb{C})\)

On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [19] to the weighted caseX(Γ,w). These conditions are formulated in terms of indices α(Q t w) and β(Q t w) of a submultiplicative functionQ t w, which is associated with local properties of the space, of the curve, and of the weight at the pointt∈Γ. Using these results we obtain a lower estimate for the essential norm |S| of the Cauchy singular integral operatorS in reflexive weighted rearrangement-invariant spacesX(Γ,w) over arbitrary Carleson curves Γ: $$\left| S \right| \geqslant \cot \left( {\pi \lambda _\Gamma ,w/2} \right)$$ where\(\lambda _{\Gamma ,w} : = \begin{array}{*{20}c} {\inf } \\ {t \in \Gamma } \\ \end{array} \min \left\{ {\alpha \left( {Q_t w} \right),1 - \beta \left( {Q_t w} \right)} \right\}\). In some cases we give formulas for computation of α(Q t w) and β(Q t w).

Pseudodifferential operators on manifolds with fibred boundaries

Let X be a compact manifold with boundary. Suppose that the boundary is fibred, : @X! Y, and let x2C 1 (X) be a boundary defining function. This data fixes the space of 'fibred cusp' vector fields, consisting of those vector fields V on X satisfying V x = O(x 2 ) and which are tangent to the fibres of ; it is a Lie algebra and C 1 (X) module. This Lie algebra is quantized to the 'small calculus' of pseudodierential operators (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an exact fibred cusp' metric is analyzed as is the wavefront set associated to the calculus.

Necessary Conditions for Fredholmness of Partial Differential Operators of Irregular Singular Type

The convergence of formal power series solutions of ordinary differential equations are extensively studied by many authors in connection with irregularities, (cf. [6] and [10].) In case of partial differential equations of regular singular type, several sufficient conditions are known, (cf. [4] and [1].) In the preceeding paper [9], we gave sufficient conditions for the Fredholmness of partial differential operators of irregular singular type of two independent variables in analytic and Gevrey spaces. Then we deduced the convergence of formal power series solutions from the Fredholmness of the operators. These conditions are expressed in terms of Toeplitz symbols, and they are equivalent to a Riemann-Hilbert factorization condition, (cf. [3] and (2.5), (2.6) , (2.7) which follow.) In this paper, we shall show the necessity of these sufficient conditions. More precisely, we will prove that the Riemann-Hilbert factorization conditions (2.6) and (2.7) are necessary and sufficient for the partial differential operators of irregular singular type to be Fredholm operators on certain analytic and Gevrey spaces. The proof of our theorem is based on the analysis of the main (principal) part of irregular singular type operators via Toeplitz operators on the torus T • = R/27rZ. In fact, we will show that the essential parts of these operators in studying Fredholm properties are precisely Toeplitz operators. This

Singular integral equations in douglis algebra

In this paper, it is considered for some twodimensional singular integral equations of the hypercomplex functions in he Douglis scnse. In some special cases. the Fredholm conditions and index formula of such equations are obtained.

SUPER KRICHEVER FUNCTOR

A supersymmetric generalization of the Krichever map is proposed. This map assigns injectively a point of an infinite dimensional super Grassmannian to a set of geometric data consisting of an arbitrary algebraic super manifold of dimension 1|1 defined over a field of any characteristic and a line bundle on it. The naturality of this map comes from the fact that it is obtained as the restriction of a contravariant functor to certain special objects. This functor gives an antiequivalence between the category of infinite dimensional super vector spaces satisfying the Fredholm condition together with their stabilizers, and the category of algebraic super curves and certain sheaves on them including all even vector bundles of arbitrary rank.

The Transformation of Vector-Functions, Scaling and Bifurcation

Various known methods for studying the bifurcation of zeros of a Banach-space-valued mapping are unified under a single idea, akin to using a coordinate transformation to obtain a simple form of the function under consideration. The general nature of the hypotheses permits the dropping of the pervasive "Fredholm condition" of bifurcation theory.

Fredholmness and finite section method for Toeplitz operators in $l^p (Z_+ \times Z_+)$ with piecewise continuous symbols II

In this paper we prove sufficient conditions for Fredholmness of discrete Toeplitz operators with piecewise continuous symbols on the space l^p over the quarter-plane and for the applicability of the finite section method to such operators. The methods used here are based on a bilocalization technique and the local principle of Douglas and Krupnik. Part I of this work contained the proofs of the necessity of the corresponding conditions, the necessary definitions, and the formulation of the main results.

Fredholmness and finite section method for Toeplitz operators in $l^p (Z_+ \times Z_+)$ with piecewise continuous symbols I

We consider discrete Toeplitz operators on the space l^p over the quarter-plane for a class of piecewise continuous symbols. This class of symbols is usually denoted by PC_p(\mathbb T^2) and it contains, in particular, all finite sums of the form \Sigma a_i (\xi) b_i (\eta), (\xi, \eta) \in \mathbb T^2 , where a_i and b_i are of bounded variation. Necessary and sufficient conditions for Fredholmness of such operators and for the applicability of the finite section method to them are obtained. The present part I contains the necessary definitions, the formulation of the main results, and the proofs of the necessity of the given conditions. Their sufficiency will be proved in part II of this work.

The transformation of vector-functions, scaling and bifurcation

Various known methods for studying the bifurcation of zeros of a Banach-space-valued mapping are unified under a single idea, akin to using a coordinate transformation to obtain a simple form of the function under consideration. The general nature of the hypotheses permits the dropping of the pervasive "Fredholm condition" of bifurcation theory.

The role of Fredholm conditions in signorin's perturbation method

The role of Fredholm conditions in signorin's perturbation method