International Journal of Pure and Applied Mathematics ————————————————————————– Volume 7 No. 1 2003, 1-5 A NEW METHOD FOR OBTAINING THE SPECTRUM OF CERTAIN COMPACT WEIGHTED COMPOSITION OPERATOR Ali Zohri §1 , A.I. Shahbazov 2 1,2 Payame Noor Univresity IMM Academy of Sciences of Ardebil Center Baku, AZERBAYJAN 1 e-mail: zohri2000@yahoo. com Abstract: In this work we consider the weighted composition operators in Banach modules on the multidimensional vector-valued analogous to the disc algebra. When this operator induced by compressly mapping, we show that it is compact and describe its spectrum. AMS Subject Classification: 47B38 Key Words: compact operator, weighted composition operator, compact open topology spectrum, Banach modules, Banach algebra, resonancing eigen- value, nilpotent, quasinilpotent 1. Introduction Let D is bounded domain in C n , and ¯ D is its closure and ϕ : ¯ D → D is continuous mapping that is holomorphic into D. It is well-known that (e.g. see [1]) the mapping ϕ has unique attractive point z ◦ such that z ◦ ∈ D and iteration ϕ k of mapping ϕ (ϕ ◦ = id ¯ D ,ϕ k+1 = ϕ k ◦ ϕ,k ≥ 0) converges uniformly Received: February 15, 2003 c  2003, Academic Publications Ltd. § Correspondence author