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Báo cáo toán học: "Unitary equivalence of restricted shifts "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tương đương đơn nhất của sự thay đổi hạn chế. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 17-28 UNITARY EQUIVALENCE OF RESTRICTED SHIFTS WILLIAM s. COHN INTRODUCTION Suppose D is a finitely connected domain in the plane bounded by n 1 analytic Jordan curves. Letting dD denote the boundary of D we write dD 71 u J . u y 1. Let H H D be the Banach space of holomorphic functions bounded on D and suppose 7 2 H D is the Hilbert space of holomorphic functions f defined on D such that f has a harmonic majorant. If Ớ e e . eifl is a point on the n-torus then by Ho we mean the Hardy class of multiple valued functions modulus automorphic of index Ỡ. A subspace M of H2 is called invariant if fg e M for all f e M and g 6 H . Any such subspace has the form M cpHị where p is a modulus automorphic inner function and 0 is the appropriate index. See 10 . In the language of the Abrahamse-Douglas model theory a multiplicity one bundle shift on D is the multiplication operator sfl on Ho given by s6 f z-f. A scalar Co model over D is an operator sị on the subspace Ho pHy given by s Pzf compression of s8 . Here p denotes orthogonal projection onto HỒ pH2 cp is a modulus automorphic inner function and y is an index chosen so pHy c Ho. See 2 and 3 . The main problem in constructing a successful model theory is nonuniqueness. It can occur that s and are unitarily equivalent but Ỡ In this paper we prove the following theorem. 2 - 2692 18 WILLIAM s. COHN Theorem. Suppose p is singular inner and nonatomic. Then s is unitarily equivalent to S ị if and only if 0 y. Thus we describe a situation where the c0 model is unique. We greatly prefer to work with single valued functions. This is accomplished by replacing the Ho spaces by H2 spaces with weighted inner products. We give the details below. In terms of single valued functions an invariant subspace M H2 has the form M sH2 where the single valued function J is a rigid function or simply an inner function . That is s e Hm s 1 a.e. ds on y 1 and there are constants Cj i 1 . n such .