Liviu I. Nicolaescu
We describe two topologies on the space of unbounded Fredholm operators and we explain their K-theoretic relevance. In the process we also prove a very general result concerning the continuity of families of first order, elliptic boundary value problems.
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This note relates the notion of a discrete-spectrum family of Fredholm operators, recently introduced by N. Ivanov in arXiv:2111.15081, with the classical continuity properties in the space of operators.
Family index for Fredholm extensions of semi-Fredholm operators
本文研究希尔伯特空间中半Fredholm算子的Fredholm实现（抽象边值问题），通过将算子转化为图并与水平子空间构成Fredholm对，变形水平子空间而非算子，证明了抽象边值问题族的K指标公式。
Fredholm Criteria for $G$-pseudodifferential Operators
本文基于Simonenko原理，研究了$G$-伪微分算子在Sobolev空间中的Fredholm性质，针对有限群情况进一步探讨了其符号的可逆性与Fredholm性质的关系。
The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.
We prove some Fredholm conditions for many algebras of differential operators on particular classes of open manifolds, which include asymptotically Euclidean or asymptotically hyperbolic manifolds. Our typical result is that an operator $P$ is Fredholm if, and only if, it is elliptic and some limit operators $(P_α)_{α\in A}$ are invertible. The operators $P_α$ are right-invariant operators on amenable Lie groups $G_α$, and are of the same type of $P$. To obtain this result, we consider algebras of differential operators that are generated by groupoids. We study a general gluing procedure for g…
The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an elementary derivation of sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds.
Spaces of unbounded Fredholm operators: I. Homotopy equivalences
本文研究图拓扑下无界Fredholm算子空间及其子空间，证明它们与经典有界算子空间的自然映射是同伦等价，揭示无界本质正Fredholm算子子空间表示奇K理论，且各空间的可逆算子子空间可缩。
We consider the mapping properties of generalized Laplace-type operators ${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such operators are Fredholm when between certain weighted Sobolev or weighted Hölder spaces. These are generalizations of well-known theorems in the asymptotically conical (or asymptotically Euclidean) setting, and also sharpen and extend corresponding theorems by Joyce. The methods here a…
We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possible systems of compatible determinant line bundles and compare the conventions and approaches used elsewhere in the literature.
Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P \in ψ^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i \to M$, $i = 0,1$, and let $α$ be an irreducible representation of the group $G$. Then $P$ induces a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components. We prove that the map $π_α(P)$ is Fredholm if, and only if, $P$ is {\em transversally $α$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the…
Let $Γ$ be a compact group acting on a smooth, compact manifold $M$, let $P \in ψ^m(M; E_0, E_1)$ be a $Γ$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i \to M$, $i = 0,1$, and let $α$ be an irreducible representation of the group $Γ$. Then $P$ induces a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components of the corresponding Sobolev spaces of sections. When $Γ$ is finite, we explicitly characterize the operators $P$ for which the map $π_α(P)$ is Fredholm in terms of the principal symbol of $P$…
This is an expository paper which gives a proof of the Atiyah-Singer index theorem for elliptic operators. Specifcally, we compute the geometric K-cycle that corresponds to the analytic K-cycle determined by the operator. This paper and its companion ("K-homology and index theory II: Dirac Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. …
This is an expository paper which gives a proof of the Atiyah-Singer index theorem for Dirac operators, presenting the theorem as a computation of the K-homology of a point. This paper and its follow up ("K-homology and index theory II: Elliptic Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. Math. 2014.
Fredholm properties of the jacobi Operator of minimal conical hypersurfaces
本文研究了极小锥面的Jacobi算子的Fredholm性质，证明了在一定条件下该算子的可解性，并讨论了应用实例。
Let $\Y$ be a smooth connected manifold, $Σ\subset\C$ an open set and $(σ,y)\to\scrP_y(σ)$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,σ)\in \Y\times Σ$ and holomorphically on $σ$. We show how to associate to $\scrP$, under mild hypotheses, a smooth vector bundle $\kerb\to\Y$ whose fiber over a given $y\in \Y$ consists of classes, modulo holomorphic elements, of meromorphic elements $ϕ$ with $\scrP_yϕ$ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.
Naively, the analytic index of a family of self-adjoint Fredholm operators ought to be (an equivalence class of) the family of the kernels of these operators. The present paper is devoted to a rigorous version of this idea based on ideas of Segal as developed by the author in arXiv:2111.14313 [math.KT]. The resulting new definition of the analytic index makes sense under much weaker continuity assumptions than the Atiyah-Singer one and can be easily adjusted to families of operators in fibers of a Hilbert bundle. We prove the correctness of the new definition and show that it agrees with the A…
We consider two calculi of pseudodifferential operators on manifolds with fibered boundary: Mazzeo's edge calculus, which has as local model the operators associated to products of closed manifolds with asymptotically hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which is similarly modeled on products of closed manifolds with asymptotically Euclidean spaces. We construct an adiabatic calculus of operators interpolating between them, and use this to compute the `smooth' K-theory groups of the edge calculus, determine the existence of Fredholm quantizations of elliptic…
In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H^1(L), the space of harmonic 1-forms on L. We proved this first by showing that the linearized operator for the deformation map is surjective and then applying the Banach space implicit function theorem. In this paper, we obtain the same surjectivity result by using a different method, the Fredholm Alternative, which is a powerful tool for compact operat…
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