Let $D$ and $F$ be two unbounded on a Hilbert space $H$ whose dense domains of definition coincide. Moreover, assume that each is a symmetric operator. As is well-known, each operator is closable, i.e. extendable to a closed operator.
Will their extended domains coincide?
I don't see why. On $L^2(\mathbb{R})$ consider the operators $$ D = \frac{d^2}{dx^2}\quad \text{ and } \quad F = \frac{d^4}{dx^4} $$ each with the domain $C_c^\infty(\mathbb{R})$ (infinitely smooth functions with compact support). They are both symmetric. Both are closable, but the closure of $D$ is defined on the Sobolev space $H^2(\mathbb{R})$ while the closure of $F$ is defined on a smaller Sobolev space $H^4(\mathbb{R})$.