Normal Operator - Unbounded Normal Operators

Unbounded Normal Operators

The definition of normal operators naturally generalizes to some class of unbounded operators. Explicitly, a closed operator N is said to be normal if

Here, the existence of the adjoint implies that the domain of is dense, and the equality implies that the domain of equals that of, which is not necessarily the case in general.

The spectral theorem still holds for unbounded normal operators, but usually requires a different proof.