self-adjoint operator

A self-adjoint operator is the infinite-dimensional analogue of a symmetric (Hermitian) matrix — and it inherits all the good behavior: real eigenvalues, orthogonal eigenvectors, a clean diagonalization. These are the operators that model physical observables in quantum mechanics precisely because their spectra (the measurable values) are real.

On a Hilbert space H with inner product <.,.>, a bounded operator T is self-adjoint if it equals its own adjoint: <T x, y> = <x, T y> for all x, y in H, i.e. T = T*. Its spectrum is then a nonempty compact subset of the REAL line, and it has no residual spectrum. The numerical range <T x, x> is real for every x, which is the source of the real spectrum.

For unbounded operators one must be far more careful: 'symmetric' (meaning <T x, y> = <x, T y> on the domain) is strictly weaker than 'self-adjoint', because self-adjointness also demands that the domain of T equal the domain of its adjoint T*. A symmetric operator can fail to be self-adjoint, and the gap is governed by deficiency indices. The spectral theorem applies in full force only to genuinely self-adjoint operators, not merely symmetric ones.