Composition operator

For information about the operator ∘ of composition, see function composition and composition of relations.

In mathematics, the composition operator $C_\phi$ with symbol $\phi$ is a linear operator defined by the rule

C_\phi (f) = f \circ\phi

where $f \circ\phi$ denotes function composition.

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,^[1]^[2] 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 $C_\phi$ is compact or trace-class; answers typically depend on how the function φ behaves on the boundary of some domain.

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.

References

↑ B.O. Koopman, "Hamiltonian systems and transformations in Hilbert space", (1931) Proceedings of the National Academy of Sciences of the USA, 17, pp. 315–318.
↑ Pierre Gaspard, Chaos, scattering and statistical mechanics, (1998) Cambridge University Press

C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. xii+388 pp. ISBN 0-8493-8492-3.
J. H. Shapiro, Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. xvi+223 pp. ISBN 0-387-94067-7.

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