The study of composition operators is covered by AMS category 47B33.
In physics, and especially the area of dynamical systems, the composition operator is usually referred to as the Koopman operator[1][2] (and its wild surge in popularity[3] is sometimes jokingly called "Koopmania"[4]), named after Bernard Koopman.
Using the language of category theory, the composition operator is a pull-back on the space of measurable functions; it is adjoint to the transfer operator in the same way that the pull-back is adjoint to the push-forward; the composition operator is the inverse image functor.
is compact or trace-class; answers typically depend on how the function
An appropriate basis, explicitly manifesting the shift, can often be found in the orthogonal polynomials.
When these are orthogonal on the real number line, the shift is given by the Jacobi operator.
Shift operators can be studied as one-dimensional spin lattices.
The composition operator has been used in data-driven techniques for dynamical systems in the context of dynamic mode decomposition algorithms, which approximate the modes and eigenvalues of the composition operator.