Hilbert-Schmidt theorem

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.

tatement 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 o + 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 angle varphi_{i} mbox{ for all } u in H.

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 356
id = ISBN 0-387-00444-0 (Theorem 8.94)


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