Fredholm Index Research Articles

Resolvent and logarithmic residues of a singular operator pencil in Hilbert spaces

The present paper considers the operator pencil A(λ)=A0+A1λ, where A0,A1≠0 are bounded linear mappings between complex Hilbert spaces and A0 is neither one-to-one nor onto. Assuming that 0 is an isolated singularity of A(λ) and that the image of A0 is closed, certain operators are defined recursively starting from A0 and A1 and they are shown to provide a characterization of the image and null space of the operators in the principal part of the resolvent and of the logarithmic residues of A(λ) at 0. The relations with the classical results based on ascent and descent in [10] are discussed. In the special case of A0 being Fredholm of index 0, the present results characterize the rank of the operators in the principal part of the resolvent, the dimension of the subspaces that define the ascent and descent, the partial multiplicities, and the algebraic multiplicity of A(λ) at 0.

Fredholm property of regular hypoelliptic operators on the scales of multianisotropic spaces

This paper studies the Fredholm properties for a class of regular hypoelliptic operators. We establish necessary and sufficient conditions for a priori estimates for differential operators acting in multianisotropic Sobolev spaces in Rn. Fredholm criteria and index invariance are obtained for a wide class of regular hypoelliptic operators on the special scales of multianisotropic weighted spaces.

Trivial endomorphisms of the Calkin algebra

We prove that it is consistent with ZFC that every unital endomorphism of the Calkin algebra \({\cal Q}(H)\) is unitarily equivalent to an endomorphism of \({\cal Q}(H)\) which is liftable to a unital endomorphism of \({\cal B}(H)\). We use this result to classify all unital endomorphisms of \({\cal Q}(H)\) up to unitary equivalence by the Fredholm index of the image of the unilateral shift. As a further application, we show that it is consistent with ZFC that the class of C*-algebras that embed into \({\cal Q}(H)\) is not closed under tensor product nor countable inductive limit.

Invariance of the Fredholm index and spectrum of non-smooth pseudodifferential operators

In this paper, we show the invariance of the Fredholm index of non-smooth pseudodifferential operators with coefficients in Hölder spaces. By means of this invariance, we improve previous spectral invariance results for non-smooth pseudodifferential operators P with coefficients in Hölder spaces. For this purpose, we approximate P with smooth pseudodifferential operators and use a spectral invariance result of smooth pseudodifferential operators. Then, we get the spectral invariance result in analogy to a proof of the spectral invariance result for non-smooth differential operators by Rabier.

A note on generalized invertibility and invariants of infinite matrices

The classes of band-dominated operators and the subclass of operators in the Wiener algebra {mathcal {W}} are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra {mathcal {W}} invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space l^p, 1le ple infty . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in {mathcal {W}} are invariant w.r.t. p.

Factorization of generalized holomorphic curve and homogeneity of operators

The present paper concerns the homogeneity and similarity of operators in Cowen-Douglas class $$B_n(\Omega )$$ . Let E be the Hermitian holomorphic vector bundle induced by $$T\in B_n(\mathbb {D})$$ , and $$E_{\alpha }$$ be the Hermitian holomorphic vector bundle induced by $$\phi _{\alpha }(T)$$ , where $$\phi _{\alpha }$$ is a M $$\ddot{o}$$ bius transformation of the unit disk $$\mathbb {D}$$ . Assume that the holomorphic Hermitian vector bundle $$E_{\alpha }$$ is congruent to $$E\otimes \mathcal {L}_{\alpha }$$ for some line bundle $$\mathcal {L}_{\alpha }$$ over $$\mathbb {D}$$ , for each $$\alpha \in \mathbb {D}$$ . Then it is shown that $$\mathcal {L}_{\alpha }$$ must be the trivial bundle and T is homogeneous. Furthermore, we investigate the similarity of operators with Fredholm index n associate with Hermitian holomorphic bundles. This characterization is given in terms of the factorization of generalized holomorphic curve induced by the corresponding holomorphic bundles.

On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid

We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, S, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside S), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In such a way, we can use the theory recently introduced by the author and give sufficient conditions for time-periodic bifurcation to take place.

P-Schatten commutators of projections

Let $${\mathcal {H}}={\mathcal {H}}_+\oplus {\mathcal {H}}_-$$ be a fixed orthogonal decomposition of the complex separable Hilbert space $${\mathcal {H}}$$ in two infinite-dimensional subspaces. We study the geometry of the set $${\mathcal {P}}^p$$ of selfadjoint projections in the Banach algebra $${\mathcal {A}}^p=\{A\in {\mathcal {B}}({\mathcal {H}}): [A,E_+]\in {\mathcal {B}}_p({\mathcal {H}})\},$$ where $$E_+$$ is the projection onto $${\mathcal {H}}_+$$ and $${\mathcal {B}}_p({\mathcal {H}})$$ is the Schatten ideal of p-summable operators ( $$1\le p <\infty$$ ). The norm in $${\mathcal {A}}^p$$ is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space $${\mathcal {P}}^p$$ is shown to be a differentiable $$C^\infty$$ submanifold of $${\mathcal {A}}^p$$ , and a homogeneous space of the group of unitary operators in $${\mathcal {A}}^p$$ . The connected components of $${\mathcal {P}}^p$$ are characterized, by means of a partition of $${\mathcal {P}}^p$$ in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.

A constructive approach to topological invariants for one-dimensional strictly local operators

In this paper we shall focus on one-dimensional strictly local operators, the notion of which naturally arises in the context of discrete-time quantum walks on the one-dimensional integer lattice Z. In particular, we give an elementary constructive approach to the following two topological invariants associated with such operators: Fredholm index and essential spectrum. As a direct application, we shall explicitly compute and fully classify these topological invariants for a well-known quantum walk model.

Integrals Along Bimonoid Homomorphisms

We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is to characterize an existence of a specific integral, called a normalized generator integral, along a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism. We introduce a notion of a volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups. In applications, we show that there exists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. The volume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids, which gives an analogue of the Fredholm index theory. This paper gives a technical preliminary of our subsequent paper about a construction of TQFT’s.

Polyfold regularization of constrained moduli spaces

We introduce tame sc-Fredholm sections and slices of sc-Fredholm sections. A slice is a notion of subpolyfold that is compatible with the sc-Fredholm section and has finite locally constant codimension. We prove that the subspace of a tame polyfold that satisfies a transverse sc-smooth constraint in a finite dimensional smooth manifold is a slice of any tame sc-Fredholm section compatible with the constraint. Moreover, we prove that a sc-Fredholm section restricted to a slice is a tame sc-Fredholm section with a drop in Fredholm index given by the codimension of the slice. As a corollary, we obtain fiber products of tame sc-Fredholm sections. We describe applications to Gromov-Witten invariants, constructing the Piunikhin-Salamon-Schwarz maps for general closed symplectic manifolds, and avoiding sphere bubbles in moduli spaces of expected dimension $0$ and $1$.

The APS-index and the spectral flow

We study the Atiyah-Patodi-Singer (APS) index, and its equality to the spectral flow, in an abstract, functional analytic setting. More precisely, we consider a (suitably continuous or differentiable) family of self-adjoint Fredholm operators $A(t)$ on a Hilbert space, parametrised by $t$ in a finite interval. We then consider two different operators, namely $D := \frac{d}{dt}+A$ (the abstract analogue of a Riemannian Dirac operator) and $D := \frac{d}{dt}-iA$ (the abstract analogue of a Lorentzian Dirac operator). The latter case is inspired by a recent index theorem by Bar and Strohmaier (Amer. J. Math. 141 (2019), 1421--1455) for a Lorentzian Dirac operator equipped with APS boundary conditions. In both cases, we prove that Fredholm index of the operator $D$ equipped with APS boundary conditions is equal to the spectral flow of the family $A(t)$.

Many-body Fredholm index for ground-state spaces and Abelian anyons

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered $p$-dimensional ground state sector. The index is fractional with the denominator given by $p$. In particular, this yields a new short proof of the quantization of the Hall conductance and of Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the argument provides an explicit construction of Wilson loop operators exhibiting a non-trivial braiding and that can be used to create fractionally charged Abelian anyons.

On the Fredholm Property and Index of the Generalized Neumann Problem

For an elliptic equation of the $$2l$$th order with real coefficients, the coefficients in the principal part being constant, we consider the boundary value problem in which the normal derivatives of the $$(k_j-1)$$st order, $$j={1,\ldots ,l}$$, where $$1\le k_1<\ldots <k_l\le 2l$$, are prescribed. For $$k_j=j$$, this problem becomes the Dirichlet problem, and for $$k_j=j+1$$, it becomes the Neumann problem. A new easy-to-use form of the Fredholm property criterion is produced, and a formula for the index of this problem is provided. The equivalence of the newly found Fredholm property condition and the well-known complementarity condition is established. The cases of fourth- and sixth-order equations are considered in detail.

Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations

We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

Toeplitz operators on concave corners and topologically protected corner states

We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify that these topological invariants depend on the shape of the system. We discuss the Benalcazar--Bernevig--Hughes' 2-D Hamiltonian and see that there still exists topologically protected corner states even if we break some symmetries as long as the chiral symmetry is preserved.

Restricted diagonalization of finite spectrum normal operators and a theorem of Arveson

Kadison characterized the diagonals of projections and observed the presence of an integer. Arveson later recognized this integer as a Fredholm index obstruction applicable to any normal operator with finite spectrum coincident with its essential spectrum whose elements are the vertices of a convex polygon. Recently, in joint work with Kaftal, the author linked the Kadison integer to essential codimension of projections. This paper provides an analogous link between Arveson's obstruction and essential codimension as well as a new approach to Arveson's theorem which also allows for generalization to any finite spectrum normal operator. In fact, we prove that Arveson's theorem is a corollary of a trace invariance property of arbitrary normal operators. An essential ingredient is a formulation of Arveson's theorem in terms of diagonalization by a unitary which is a Hilbert--Schmidt perturbation of the identity.

An Invertibility Criterion in a C*-Algebra Acting on the Hardy Space with Applications to Composition Operators

In this paper, we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators, we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application, we prove that for “quasi-parabolic” composition operators the spectra and the essential spectra are equal.

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.

Fredholm theory of Toeplitz operators on standard weighted Fock spaces

We study the Fredholm properties of Toeplitz operators with bounded symbols of vanishing mean oscillation in the complex plane. In particular, we prove that the Toeplitz operator with such a symbol is Fredholm on a standard weighted Fock space if and only if the Berezin transform of the symbol is bounded away from zero outside a sufficiently large disk in the complex plane. We also show that the Fredholm index of the Toeplitz operator can be computed via the winding of the symbol along a sufficiently large circle. We finish by considering Toeplitz operators with matrix-valued symbols.