normal operator

A normal operator is one that 'gets along with' its own mirror image, the adjoint — the two commute. This single algebraic condition is exactly what is needed for the operator to be diagonalizable by an orthonormal basis (or, more generally, by a spectral measure). Self-adjoint operators, unitary operators, and positive operators are all special cases of normal ones.

On a Hilbert space, a bounded operator T is normal if it commutes with its adjoint: T T* = T* T. Equivalently, ||T x|| = ||T* x|| for all x. The class of normal operators is precisely the class for which the spectral theorem holds in its strongest form: every normal operator is unitarily equivalent to multiplication by a (complex-valued) function on some L^2 space.

Normality is what permits complex eigenvalues while still guaranteeing orthogonal eigenvectors. Self-adjoint means T = T* (real spectrum); unitary means T* = T^{-1} (spectrum on the unit circle); both automatically satisfy T T* = T* T. The honest caveat: a non-normal operator can have a perfectly fine spectrum yet fail utterly to be diagonalizable — its eigenvectors may not span and may be wildly non-orthogonal.