adjoint

Let $\mathscr{H}$ be a Hilbert space and let $A:𝒟(A)\subset \mathscr{H}\to \mathscr{H}$ be a densely defined linear operator. Suppose that for some $y\in \mathscr{H}$, there exists$z\in \mathscr{H}$ such that $(Ax,y)=(x,z)$ for all $x\in 𝒟(A)$. Then such $z$ is unique, for if $z^{\prime }$ is another element of $\mathscr{H}$ satisfying that condition, we have $(x,z-z^{\prime })=0$ for all $x\in 𝒟(A)$, which implies $z-z^{\prime }=0$ since $𝒟(A)$ is dense (http://planetmath.org/Dense). Hence we may define a new operator $A^{*}:𝒟(A^{*})\subset \mathscr{H}\to \mathscr{H}$ by

$𝒟(A^{*})=$$\mathrm{\{}y\in \mathscr{H}:\text{there is}z\in \mathscr{H}\text{such that}(Ax,y)=(x,z)\},$

$A^{*}(y)=$$z.$

It is easy to see that $A^{*}$ is linear, and it is called the adjoint of $A$.

Remark. The requirement for $A$ to be densely defined is essential, for otherwise we cannot guarantee $A^{*}$ to be well defined.