A D-module is an algebraic structure with several differential operators acting on it.

Observables are represented by operators acting on a Hilbert space of such quantum states.

Working in standard spherical coordinates, we can define a particular operator acting on a function as:

's are assumed to be positive operators acting on appropriate state space and .

The key element of the operational calculus is to consider differentiation as an operator acting on functions.

In this theory, the kernel is understood to be a compact operator acting on a Banach space of functions.

Let be an operator acting on an -dimensional Hilbert space .

The spin raising and lowering operators acting on these eigenvectors give:

Quantization converts classical fields into operators acting on quantum states of the field theory.

Likewise, the operator acting on a state produces and .