I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some context first:
An old result of functional analysis tells us that a symmetric (in the sense that $(Ax,y)=(x,Ay)$, for all $x,y \in H$), unbounded operator $A$, acting on a Hilbert space $H$, cannot be defined on the whole space but only in a dense subspace of it. This is a direct consequence of the Hellinger-Toeplitz theorem. (see also: Riesz-Nagy, "Functional Analysis", 1955, p.296 and also Reed-Simon, "Methods of Modern Mathematical Physics", 1975, p.84). Since the operators of interest in physics are self-adjoint (and thus symmetric) they fall into this.
On the other hand, it is well known that any linear map from a subspace of a Banach space $X$ to another Banach space $Y$ can be extended to a linear map $X\to Y$ defined on the whole of $X$ using Zorn's Lemma (see for example: Unbounded linear operator defined on $l^2$).
Now the question is: Since the extension of a linear, unbounded operator to the whole of the space through the AC, will produce a -still- unbounded, linear operator, does the previous remark imply that the extension of linear, self-adjoint, unbounded operators on the whole of the space, produces non-self-adjoint operators? What would be a concrete relevant example?
Related question: Invertible unbounded linear maps defined on a Hilbert space