In mathematics, the composition operator
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where
The study of composition operators is covered by AMS category 47B33.
In physics
In physics, and especially the area of dynamical systems, the composition operator is usually referred to as the Koopman operator, named after Bernard Koopman. It is the left-adjoint of the transfer operator of Frobenius–Perron.
In category theory
In 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.
In functional analysis
The domain of a composition operator is usually taken to be some Banach space, often consisting of holomorphic functions: for example, some Hardy space or Bergman space. Interesting questions posed in the study of composition operators often relate to how the spectral properties of the operator depend on the function space. Other questions include whether
Applications
In mathematics, composition operators commonly occur in the study of shift operators, for example, in the Beurling–Lax theorem and the Wold decomposition. Shift operators can be studied as one-dimensional spin lattices. Composition operators appear in the theory of Aleksandrov–Clark measures.
The eigenvalue equation of the composition operator is Schröder's equation, and the principal eigenfunction f(x) is often called Schröder's function or Koenigs function.