For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$.
Is there a notation for unbounded operators (incorparating partially defined ones)?
Do you use some especially for your own notes?
There is not a notation that anyone would recognize without you explaining it. In order to acquire established notation, an object needs to have some properties making it worth talking about. The set of densely defined operators is just that, a set. No vector space structure, no semigroup structure, no norm, no topology. A collection of disparate objects that has no interesting properties as a whole.
It is standard to write $D(T)$ for the domain of operator $T$. That's about it.