Let $H_1$, $H_2$ be two possibly distinct real or complex Hilbert spaces, with linearity in the first coordinate of the inner product for concreteness. Let's think of passage to the adjoint as a map from the $A$ to $B$ where $A$ and $B$ are defined as follows:
$A:= \{S:D(S)\rightarrow H_2|S \text{ is a densely defined operator with domain in } H_1\}$
$B:= \{T:D(T)\rightarrow H_1|T \text{ is a closed operator with domain in } H_2\}$
I stress that I didn't say "densely defined" anywhere in the definition of $B$.
Is there an equivalence relation on $A$ roughly capturing the idea of being "densely equal" with the property that modding out by it would make passage to the adjoint injective? Is passage to the adjoint surjective?