composition operator
- For information about the operator "o" of composition, see function composition or composition of relations.
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 Lp space, 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.
The study of composition operators is covered by AMS category 47B33.
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.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
