Operator domain ($\mathcal{D}(A)$)

The domain $\mathcal{D}(A)$ of an operator $A$ is the set of vectors on which $A$ is defined. For an unbounded differential operator, the domain includes regularity and boundary conditions and is part of the operator's definition.

A linear operator $T$ on a Hilbert space $H$ is densely defined when its operator domain is a dense subset of $H$. Density makes the adjoint operator unique wherever its defining inner-product identity has a solution.