Composition operator
From Wikipedia, the free encyclopedia
In mathematics, the composition operator Cφ with symbol φ is defined by the rule
where 
denotes function composition. 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 Cφ is compact or trace-class; answers typically depend on how the function φ behaves on the boundary of some domain. Composition operators have found to be related to the theory of Aleksandrov-Clark measures.
The study of composition operators is covered by AMS category 47B33.
[edit] 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.
- 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.
