Most densely-defined unbounded linear operators on Hilbert spaces have a very large domain. In fact, for a lot of natural operators the intersection of their domains are still dense.
Let us consider some counterexamples:
Let $T : \mathbb Q^n \subset \mathbb R^n \rightarrow \mathbb R^m$ be a densely-defined unbounded operator.
Let $Q$ be orthogonal.
Then $T_Q : Q \mathbb Q^n \subset \mathbb R^n \rightarrow \mathbb R^m$, which maps $Qx$ to $Tx$ for $x \in \mathbb Q^n$, is a densely-defined unbounded operator, too.
Then $\mathbb Q^n \cap Q \mathbb Q^n = \{ 0 \}$ for most choices $Q$.
Let $\Omega \subseteq \mathbb R^n$ be a smoothly bounded domain.
Consider the differential operator $D : C_0^\infty(\Omega) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, which maps compactly-supported smooth functions over $\Omega$ to their total differential.
Pick an arbitrary weakly differentiable $\psi \in H^1(\Omega)$ that satisfies $\psi > 0$ essentially.
Then $\psi \cdot C_0^\infty(\Omega)$ is a linear subspace of weakly differentiable functions (partial differentiation), and the operator $D : \psi \cdot C_0^\infty(\Omega) \subset L^2(\Omega) \rightarrow L^2(\Omega)$ is densely defined.
However, for most $\psi$, we have $C_0^\infty(\Omega) \cap \psi \cdot C_0^\infty(\Omega) = \{ 0 \}$.
These examples are artificial. In both cases, the unbounded are not closed. Taking the closure provides a common intersection for the domains of the operators. In the first case, the closure is even bounded. Most Hilbert spaces of functions have a common subspace of their domains that is dense (e.g., compactly supported smooth functions).
I do not expect that "all closed densely-defined unbounded linear operators have a dense common domain". But is there a theory that provides an abstract point of view on situations where all relevant operators' domains have a large (i.e. dense) intersection?
