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Register a new userFredholm conditions for operators invariant with respect to compact Lie group actions
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Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $G$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components. We prove that the map $pi_alpha(P)$ is Fredholm if, and only if, $P$ is {em transversally $alpha$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.
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Let $Gamma$ be a compact group acting on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. When $Gamma$ is finite, we explicitly characterize the operators $P$ for which the map $pi_alpha(P)$ is Fredholm in terms of the principal symbol of $P$ and the action of $Gamma$ on the vector bundles $E_i$. When $Gamma = {1}$, that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol $C^*$-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. Whenever our results also hold for non-discrete groups, we prove them in this greater generality. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.
We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces by restriction a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $pi_alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $Gamma$ is finite abelian. We prove then that the restriction $pi_alpha(P)$ is Fredholm if, and only if, $P$ is $alpha$-elliptic, a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $alpha$-elliptic, but the converse is not true in general. However, if $Gamma$ acts freely on a dense open subset of $M$, then $P$ is $alpha$-elliptic for the given fixed $alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $psi^{m}(M; E)^Gamma$ of classical, $Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E to M$ and of the structure of its restrictions to the isotypical components of $Gamma$. These structures are described in terms of the isotropy groups of the action of the group $Gamma$ on $E to M$.
Let $mathfrak{n}$ be a nonempty, proper, convex subset of $mathbb{C}$. The $mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the official equivariant K-homology groups) and show that these are isomorphism.
In cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon cite{RS95}. In cite{GLMST}, a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper cite{Pu08}. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and {its perturbation by a relatively trace-class operator}. In this paper we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher $p^{th}$ Schatten class condition for $0leq p<infty$, thus allowing differential operators on manifolds of any dimension $d<p+1$. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by cite{BCPRSW, CGK16}. This leads us to introduce a notion of `generalised spectral flow for such paths and to investigate its properties.
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