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 = +∞,

\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 φi, i = 1, …, N, of corresponding eigenfunctions, i.e.,

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

Moreover, the functions φi 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

  • {{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}}

{{DEFAULTSORT:Hilbert-Schmidt theorem}}

Category:Operator theory

Category:Theorems in functional analysis