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We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.

If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for $H^k(M)$? References/more detail would be appreciated. Thanks.

(Crossposted from http://math.stackexchange.com/questions/324777/do-eigenfunctions-of-elliptic-operator-form-basis-of-hkmhttps://math.stackexchange.com/questions/324777/do-eigenfunctions-of-elliptic-operator-form-basis-of-hkm due to lack of replies)

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.

If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for $H^k(M)$? References/more detail would be appreciated. Thanks.

(Crossposted from http://math.stackexchange.com/questions/324777/do-eigenfunctions-of-elliptic-operator-form-basis-of-hkm due to lack of replies)

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.

If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for $H^k(M)$? References/more detail would be appreciated. Thanks.

(Crossposted from https://math.stackexchange.com/questions/324777/do-eigenfunctions-of-elliptic-operator-form-basis-of-hkm due to lack of replies)

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michael faber
michael faber
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Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.

If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for $H^k(M)$? References/more detail would be appreciated. Thanks.

(Crossposted from http://math.stackexchange.com/questions/324777/do-eigenfunctions-of-elliptic-operator-form-basis-of-hkm due to lack of replies)