|
|
| Line 1: |
Line 1: |
| {{for|information about the operator ∘ of composition|function composition|composition of relations}}
| | The writer is called Wilber Pegues. To play lacross is one of the issues she enjoys most. Invoicing is my profession. Mississippi is where her house is but her spouse wants them to transfer.<br><br>Also visit my webpage :: accurate psychic predictions ([http://www.octionx.sinfauganda.co.ug/node/22469 just click www.octionx.sinfauganda.co.ug]) |
| 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]]
| |
The writer is called Wilber Pegues. To play lacross is one of the issues she enjoys most. Invoicing is my profession. Mississippi is where her house is but her spouse wants them to transfer.
Also visit my webpage :: accurate psychic predictions (just click www.octionx.sinfauganda.co.ug)