I have often seen, in the context of operator theory and operator algebras, the notation $\mathrm{Ad}(U)a=UaU^*$, where $U$ is a unitary operator on a Hilbert space $H$ and $a$ is a bounded linear operator on $H$. I have no idea what "Ad" stands for, where/how this notation came into common use, nor whether it fits into a more general context (e.g., for similarities or other automorphisms outside of the context of operator theory). Some Google searching revealed a use of "$\mathrm{Ad}$" in the theory of Lie groups that doesn't quite match with the above, but might have a common origin.
Where does "$\mathrm{Ad}$" come from, especially in the context of $\mathrm{Ad}(U)a=UaU^*$?