Adjoint of a Densely Defined Unbounded Operator is Unique

Question

Let $\mathcal H$ be a Hilbert space and $A: \mathcal D_A \to \mathcal H$ be an unbounded linear operator. Suppose also that $\mathcal D_A$ is dense in $\mathcal H$. We define the adjoint of $A$, $A^*$ by $A^*:\mathcal D_{A^*}\to \mathcal H$, where $$\mathcal D_{A^*}: = \{\psi \in \mathcal H| \exists \eta \in \mathcal H, \forall \alpha\in\mathcal D_A: \langle\eta, \alpha\rangle = \langle\psi, A\alpha\rangle\}, $$ and $A^*\psi = \eta$. I have seen a proof that this adjoint operator is well defined, but this proof made no reference at all to the fact that $A$ is densely defined. I have heard that this condition is necessary for $A^*$ to be unique, but I cannot find a proof of this. So my question is: Why must $A$ be densely defined in order for the adjoint to be unique/exist?

Answer 1

Suppose that $A^*$ and $B^*$ are both adjoints for the operator $A$. Then, $A^* \psi = \eta$ and $B^* \psi = \tilde \eta$. We then have, from the definition of the adjoint and sesquilinearity of the inner product, $$\langle \tilde \eta, \alpha\rangle = \langle \eta, \alpha\rangle\implies \langle\tilde \eta- \eta, \alpha\rangle = 0, \forall\alpha\in \mathcal D_A. $$ Now, a set $D$ is dense in the Hilbert space $\mathcal H$ if $\bar D = \mathcal H.$ There are several equivalent definitions of the topological closure, but the most useful one for our purposes is the following: $\bar D$ is the set consisting of $D$ and all of its limit points. If a point $x \in \mathcal H$ is a limit point for $D$, then there is a sequence $\{x_n\}_{n\in\mathbb N}$ in $D-\{x\}$ converging to $x$ (this is because every metric space, and thus every Hilbert space, is a Fréchet–Urysohn space). Now, since $\mathcal {\bar D}_A = \mathcal H,$ every point in $\mathcal H$ is the limit of some sequence in $\mathcal D_A$. Let $\{x_n\}_{n\in \mathbb N}$ be such a sequence converging to $\tilde \eta - \eta\in \mathcal H.$ By the above arguments, $$\langle \tilde \eta - \eta, x_n\rangle = 0. $$ It can be shown that the inner product is sequentially continuous, that is if $\lim_{n\to \infty} \phi_n = \phi$, then $\lim_{n\to \infty} \langle\psi, \phi_n\rangle = \langle \psi , \phi\rangle$ for all $\psi \in \mathcal H$. This implies that $$\langle \tilde \eta - \eta, \tilde \eta - \eta\rangle = 0 \implies \tilde \eta = \eta$$ by positive definiteness. Thus, the adjoint is unique.