International Journal of Pure and Applied Mathematics Volume 83 No. 1 2013, 7-11 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v83i1.2 P A ijpam.eu COMPOSITION OPERATORS ACTING ON HILBERT FUNCTION SPACES B. Yousefi 1 § , Gh.R. Moghimi 1,2 Department of Mathematics Payame Noor University P.O. Box 19395-3697, Tehran, IRAN Abstract: In this paper, we give some conditions for a weighted composition operator in the space of analytic functions on a region of the plane domain have an eigenvalue. AMS Subject Classification: 47B37, 47B33 Key Words: Riemann Mapping Theorem, Schwarz’s Lemma, weighted com- position operator 1. Introduction Let Ω be a domain in the complex plane, then the space H (Ω) of all complex- valued functions analytic on Ω can be made into a F-space by a complete metric for which a sequence {f n } in H (Ω) converges to f ∈ H (Ω) if and only if f n −→ f uniformly on every compact subsets of Ω. Each ϕ ∈ H (Ω) and analytic self-map ψ of Ω induces a linear weighted composition operator C ϕ,ψ : H (Ω) −→ H (Ω) by C ϕ,ψ (f )(z)= ϕ(z)f (ψ(z)) for every f ∈ H (Ω) and z ∈ Ω. Indeed, C ϕ,ψ = M ϕ C ψ where M ϕ denotes the operator of multiplication by ϕ and C ψ is a composition operator by means Received: December 12, 2011 c  2013 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author