[[Functional analysis MOC]]
# Adjoint operator

Let $T : X \to X$ be an [[unbounded operator]] on a [[Hilbert space]].
An unbounded operator $A^{\dagger}$ is its **adjoint** iff #m/def/anal/fun 
$$
\begin{align*}
\braket{ x | Ay } = \braket{ A^{\dagger}x | y } 
\end{align*}
$$
for all $x \in \opn{dom}A^{\dagger}$ and $y \in \opn{dom}A$,
and any $(B^{\dagger})^{\dagger}$ is a restriction of $A$.

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