Spectral Theorem
Theorem For every compact self-adjoint operator T on a real or complex Hilbert space H, there exists an orthonormal basis of H consisting of eigenvectors of T. More specifically, the orthogonal complement of the kernel of T admits, either a finite orthonormal basis of eigenvectors of T, or a countably infinite orthonormal basis {en} of eigenvectors of T, with corresponding eigenvalues {λn} ⊂ R, such that λn → 0.
In other words, a compact self-adjoint operator can be unitarily diagonalized. This is the spectral theorem.
When H is separable, one can mix the basis {en} with a countable orthonormal basis for the kernel of T, and obtain an orthonormal basis {fn} for H, consisting of eigenvectors of T with real eigenvalues {μn} such that μn → 0.
Corollary For every compact self-adjoint operator T on a real or complex separable infinite dimensional Hilbert space H, there exists a countably infinite orthonormal basis {fn} of H consisting of eigenvectors of T, with corresponding eigenvalues {μn} ⊂ R, such that μn → 0.
Read more about this topic: Compact Operator On Hilbert Space, Compact Self Adjoint Operator
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