Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một định lý nâng cho các mô hình điều hành của cấp bậc hữu hạn nhân-kết nối tên miền | Copyright by INCREST 1979 J. OPERATOR THEORY 1 1979 3 - 25 A LIFTING THEOREM FOR OPERATOR MODELS OF FINITE RANK ON MULTIPLY-CONNECTED DOMAINS JOSEPH A. BALL INTRODUCTION The purpose of this paper is to prove a version of the lifting theorem in the context of an operator model on a multiply connected domain R. When R is assumed to be the unit disk D various complications arising from the multiple connectedness of the underlying domain disappear there results a new duality proof of the general lifting theorem 15 which specializes to that of Sarason 14 for the scalar Coo case. We also obtain a version for multiply connected domains of the characterization of the compact operators in the commu-tant of a Coo contraction operator due to Muhly 12 . Here R is a bounded domain in the complex plane bounded by n 1 analytic nonintersecting Jordan curves and Rat jR is the uniform closure on R of the algebra of rational function with poles off of R. Let C dR be the c -algebra of functions continuous on dR. For JC a separable Hilbert space is the algebra of bounded linear operators on JC. Let p C dR -t be a -representation of C ỚR let Ji and JV subspaces of the Hilbert space X invariant under p for all f in Rat R such that andlet W Jt QJP. Dehneơ Rat R - jC by ơ P p J . Then Ơ defines a completely contractive unital . representation of Rat R and by a result of Arveson 6 any . representation arises in this way. The operator model discussed here first introduced by Abrahamse and Douglas 4 and studied further by Abrahamse 1 and the author 8 is a canonical model for such a . representation. A precise statement and sketch of the proof of this fact known to experts in the area but only hinted at in the literature see ref. 4 is given in Section 2 of this paper. When R is the unit disk D the model coincides with the canonical model of and Foias for a completely nonunitary contraction operator 16 . Let Ơ be a . .
