Normal operator

A bounded operator $T$ on a complex Hilbert space is normal if it commutes with its adjoint:

Self-adjoint, unitary, and orthogonal projection operators are normal. In finite dimension, normality is equivalent to unitary diagonalizability.

The spectral theorem extends this structure to infinite-dimensional Hilbert spaces through a projection-valued spectral measure. This extra structure implies that a non-scalar normal operator on a Hilbert space has nontrivial closed invariant subspaces.