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{{for|information about the operator ∘ of composition|function composition|composition of relations}} | |||
In [[mathematics]], the '''composition operator''' <math>C_\phi</math> with symbol <math>\phi</math> is a [[linear operator]] defined by the rule | |||
:<math>C_\phi (f) = f \circ\phi</math> | |||
where <math>f \circ\phi</math> denotes [[function composition]]. In [[physics]], and especially the area of [[dynamical systems]], the composition operator is usually referred to as the '''Koopman operator''',<ref>[[Bernard Koopman|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.</ref><ref>Pierre Gaspard, ''Chaos, scattering and statistical mechanics'', (1998) Cambridge University Press</ref> named after [[Bernard Koopman]]. It is the [[left-adjoint]] of the Frobenius-Perron or [[transfer operator]]. In the language of [[category theory]], the composition operator is a [[pull-back]] on the space of [[measurable function]]s; 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]]. | |||
The [[domain (mathematics)|domain]] of a composition operator is usually taken to be some [[Banach space]], often consisting of [[holomorphic function]]s: for example, some [[Hardy space]] or [[Bergman space]]. Interesting questions posed in the study of composition operators often relate to how the [[Spectrum (functional analysis)|spectral properties]] of the operator depend on the [[function space]]. Other questions include whether <math>C_\phi</math> is [[compact operator|compact]] or [[trace-class]]; answers typically depend on how the function ''φ'' behaves on the boundary of some domain. | |||
In mathematics, composition operators commonly occur in the study of [[shift operator]]s, for example, in the [[Beurling-Lax theorem]] and the [[Wold decomposition]]. Shift operators can be studied as one-dimensional [[spin lattice]]s. Composition operators appear in the theory of [[Aleksandrov-Clark measure]]s. | |||
The [[eigenvalue]] equation of the composition operator is [[Schröder's equation]], and the principal [[eigenfunction]] ''f(x)'' is often called [[Schröder's equation|Schröder's function]] or [[Koenigs function]]. | |||
The study of composition operators is covered by [http://www.ams.org/msc/47Bxx.html AMS category 47B33]. | |||
==See also== | |||
* [[Multiplication operator]] | |||
* [[Composition ring]] | |||
* [[Carleman matrix]] | |||
==References== | |||
<references/> | |||
* 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. | |||
* [[Joel Shapiro (mathematician)|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. | |||
[[Category:Operator theory]] | |||
[[Category:Functional analysis]] | |||
[[Category:Dynamical systems]] | |||
Revision as of 11:55, 8 November 2013
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, the composition operator with symbol is a linear operator defined by the rule
where denotes function composition. 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 Frobenius-Perron or transfer operator. 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.
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 is compact or trace-class; answers typically depend on how the function φ behaves on the boundary of some domain.
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 study of composition operators is covered by AMS category 47B33.
See also
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.