I just recently learned that it is good style to write constants like Euler's number $\mathrm e$ and also functions and operators in roman letters while reserving italic letters for variables. Example: $\mathrm f(x)=x^2$.
However, I wonder what applies to random variables. Technically, they are functions and thus should be written in roman letters. But we often treat and wright them down as variables, e.g. $\mathbb P(X=k)=\dots$
I can hardly imagine that the right way is to switch the notation, depending on how they are used. But what is the correct way then?
I'd call them functions than operators, since operator has the context of some vector space being involved, while a (real valued) random variable is a measurable mapping from the probability space $(\Omega, \mathcal{F}, P)$ to $(\mathbb{R}, \mathcal{B}_{\mathbb{R}})$ and you don't require any vector space structure on the probability space.
Most textbooks do the italicised X, as in $X$ as in Durrett's Probability: Theory and Examples 3e and in Billingsley's Probability and Measure, so that is what I would follow. I prefer just $P$ to $\mathbb{P}$ for probability measure as well (as in those books), just because its less annoying to write. However, so long as you're consistent within your writing (and preferably with other people in your area), its OK.