Hilbert–Schmidt theorem

In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.

Statement of the theorem

Let (H, ⟨ , ⟩) be a real or complex Hilbert space and let A : H → H be a bounded, compact, self-adjoint operator. Then there is a sequence of non-zero real eigenvalues λ_i, i = 1, ..., N, with N equal to the rank of A, such that |λ_i| is monotonically non-increasing and, if N = +∞,

$\lim_{i \to + \infty} \lambda_{i} = 0.$

Furthermore, if each eigenvalue of A is repeated in the sequence according to its multiplicity, then there exists an orthonormal set φ_i, i = 1, ..., N, of corresponding eigenfunctions, i.e.

$A \varphi_{i} = \lambda_{i} \varphi_{i} \mbox{ for } i = 1, \dots, N.$

Moreover, the functions φ_i form an orthonormal basis for the range of A and A can be written as

$A u = \sum_{i = 1}^{N} \lambda_{i} \langle \varphi_{i}, u \rangle \varphi_{i} \mbox{ for all } u \in H.$

References

• Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Theorem 8.94)