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#### Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

##### External Resource

https://doi.org/10.7900/jot.2019feb26.2270

(Publisher version)

https://doi.org/10.48550/arXiv.1911.02070

(Preprint)

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##### Citation

Baldare, A., Côme, R., Lesch, M., & Nistor, V. (2021). Fredholm conditions for
invariant operators: finite abelian groups and boundary value problems.* Journal of Operator Theory,*
*85*(1), 229-256. doi:10.7900/jot.2019feb26.2270.

Cite as: https://hdl.handle.net/21.11116/0000-0008-0E67-6

##### Abstract

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 $E \to M$.

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 $E \to M$.