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 λi, i = 1, …, N, with N equal to the rank of A, such that |λi| is monotonically non-increasing and, if N = +∞,
Furthermore, if each eigenvalue of A is repeated in the sequence according to its multiplicity, then there exists an orthonormal set φi, i = 1, …, N, of corresponding eigenfunctions, i.e.,
Moreover, the functions φi form an orthonormal basis for the range of A and A can be written as
References
- {{cite book
|author1=Renardy, Michael |author2=Rogers, Robert C.
| title = An introduction to partial differential equations
|url=https://archive.org/details/introductiontopa00roge |url-access=limited | series = Texts in Applied Mathematics 13
| edition = Second
|publisher = Springer-Verlag
| location = New York
| year = 2004
| pages = [https://archive.org/details/introductiontopa00roge/page/n370 356]
| isbn = 0-387-00444-0
}} (Theorem 8.94)
- {{cite book
|author1=Royden, Halsey |author2=Fitzpatrick, Patrick
| title = Real Analysis
| edition= Fourth
|publisher = MacMillan
| location = New York
| year = 2017
| isbn = 978-0134689494
}} (Section 16.6)
{{Functional analysis}}
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