nLab adjoint operator

Definition

Let $A: H\to H$ be an unbounded operator on a Hilbert space $H$. An unbounded operator $A^*$ is its adjoint if

• $(A x|y) = (x|A^*y)$ for all $x\in dom(A)$ and $y\in dom(A^*)$; and
• every $B$ satisfying the above property for $A^*$ is a restriction of $A$.

An adjoint does not need to exist in general.

Related concepts

• self-adjoint operator