Usually, unbounded operators on a Hilbert space $\mathcal{H}$ are defined as linear operators from a subset of $\mathcal{H}$ to $\mathcal{H}$. My question is why is it necessary to restrict the domain.
Suppose I have a (densely defined, if you want) linear operator $T: D(T) \subseteq \mathcal{H} \to \mathcal{H}$ such that
$$ \| T \| := \sup \big\{ \| Tx\| : \, \| x \| \leq 1 , \, x \in D(T) \big\} = \infty \, . $$
Can you show that $\exists x\in \mathcal{H}$ such that $Tx \notin \mathcal{H}$, justifying the need to restrict the domain?