To which classes of operators does the classical Spectral Theorem for real symmetric matrices generalize? State the key theorems.
Solution
The finite-dimensional spectral theorem extends in several standard directions.
- Over \(\mathbb C\), a matrix is unitarily diagonalizable iff it is normal. Hermitian matrices are the self-adjoint special case and have real eigenvalues.
- A compact self-adjoint operator on a Hilbert space has an orthonormal basis consisting of eigenvectors after adjoining a basis of its kernel; its nonzero eigenvalues are real, have finite multiplicity, and can accumulate only at \(0\). The analogous statement holds for compact normal operators with complex eigenvalues.
- For a bounded self-adjoint operator \(T\), there is a unique projection-valued spectral measure \(E\) on \(\sigma(T)\subset\mathbb R\) such that \[ T=\int_{\sigma(T)}\lambda\,dE(\lambda). \] Equivalently, one obtains the continuous functional calculus \(f\mapsto f(T)\); multiplication-operator models follow from the spectral theorem with the appropriate multiplicity data.
- Densely defined unbounded self-adjoint operators have the corresponding projection-valued-measure theorem, with the domain characterized by square-integrability of \(\lambda\) against the spectral measure.
Thus the eigenbasis theorem for symmetric matrices becomes, in infinite dimensions, a spectral decomposition; only compact normal operators retain a purely discrete eigenvector description in general.