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 = +∞,

$${\displaystyle \lim _{i\to +\infty }\lambda _{i}=0.}$$

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.,

$${\displaystyle A\varphi _{i}=\lambda _{i}\varphi _{i}{\mbox{ for }}i=1,\dots ,N.}$$

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

$${\displaystyle Au=\sum _{i=1}^{N}\lambda _{i}\langle \varphi _{i},u\rangle \varphi _{i}{\mbox{ for all }}u\in H.}$$

References

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

• Royden, Halsey; Fitzpatrick, Patrick (2017). Real Analysis (Fourth ed.). New York: MacMillan. ISBN 0134689496. (Section 16.6)