The self-adjoint operator $A^*A$ (for a generic linear operator $A$) is used in functional analysis, linear algebra, statistics, physics, and probably many other fields. I am curious if there is a standard way to refer to this operator? There's the Gramian of course, but that seems specifically to refer to a collection of finite-dimensional vectors. "Gram operator" seems natural to me, but Googling suggests that it's not widely used.
In the context of Quantum Mechanics the operaror a*a with the same characreristis of your given operator is called the Number operator (in the framework of creation and anihilation operators). The number in this case accounts for the number of particles in a given quantum state.