Composition operator

{{Short description|Linear operator in mathematics}}

{{For|information about the operator ∘ of composition|function composition|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. It is also encountered in composition of permutations in permutations groups.

The study of composition operators is covered by [https://web.archive.org/web/20090418144227/https://www.ams.org/msc/47Bxx.html 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{{cite journal|doi=10.1073/pnas.17.5.315|title=Hamiltonian Systems and Transformation in Hilbert Space|journal=Proceedings of the National Academy of Sciences|volume=17|issue=5|pages=315–318|year=1931|last1=Koopman|first1=B. O.|author-link=Bernard Koopman|bibcode=1931PNAS...17..315K|pmc=1076052|pmid=16577368|doi-access=free}}{{Cite book|doi=10.1017/CBO9780511628856|title=Chaos, scattering and statistical mechanics|year=1998|last1=Gaspard|first1=Pierre|isbn=978-0-511-62885-6|publisher=Cambridge University Press|url=https://repositorio.unal.edu.co/handle/unal/80486 }} (and its wild surge in popularityBudišić, Marko, Ryan Mohr, and Igor Mezić. "Applied koopmanism." Chaos: An Interdisciplinary Journal of Nonlinear Science 22, no. 4 (2012): 047510. https://doi.org/10.1063/1.4772195 is sometimes jokingly called "Koopmania"Shervin Predrag Cvitanović, Roberto Artuso, Ronnie Mainieri, Gregor Tanner, Gábor Vattay, Niall Whelan and Andreas Wirzba, Chaos: Classical and Quantum Appendix H version 15.9, (2017), http://chaosbook.org/version15/chapters/appendMeasure.pdf), named after Bernard Koopman. It is the left-adjoint of the transfer operator of Frobenius–Perron.

In Borel functional calculus

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.

Since the domain considered here is that of Borel functions, the above describes the Koopman operator as it appears in Borel functional calculus.

In holomorphic functional calculus

The domain of a composition operator can be taken more narrowly, as some Banach space, often consisting of holomorphic functions: for example, some Hardy space or Bergman space. In this case, the composition operator lies in the realm of some functional calculus, such as the holomorphic functional calculus.

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 \varphi behaves on the boundary of some domain.

When the transfer operator is a left-shift operator, the Koopman operator, as its adjoint, can be taken to be the right-shift operator. 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.Gerald Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices" (2000) American Mathematical Society. https://www.mat.univie.ac.at/~gerald/ftp/book-jac/jacop.pdf {{isbn|978-0-8218-1940-1}} When the polynomials are orthogonal on some region of the complex plane (viz, in Bergman space), the Jacobi operator is replaced by a Hessenberg operator.{{Cite journal|doi=10.1016/j.laa.2011.04.027|title=Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials|journal=Linear Algebra and Its Applications|volume=435|issue=9|pages=2314–2320|year=2011|last1=Tomeo|first1=V.|last2=Torrano|first2=E.|doi-access=free}}

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.

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.

See also

References

{{reflist}}

  • C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, Florida, 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}}.

{{Functional analysis}}

{{Topological vector spaces}}

Category:Dynamical systems

Category:Linear operators

Category:Operator theory

Category:Topological vector spaces