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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài:Đi lại thay đổi trọng và lý thuyết chức năng phân tích nhiều biến. | J. OPERATOR THEORY 1 1979 207-223 Copyright by INCREST 1979 COMMUTING WEIGHTED SHIFTS AND ANALYTIC FUNCTION THEORY IN SEVERAL VARIABLES NICHOLAS p. JEWELL and A. R. LUBIN 1. INTRODUCTION Given a separable complex Hilbert space H with orthonormal basis e and a bounded sequence of complex numbers w a weighted shift operator T is a bounded linear operator which satisfies Te w e 1 for all n. T is called unilateral or bilateral according as the index n ranges over the non-negative integers or over all the integers. An excellent introduction to the theory of such operators and an extensive bibliography can be found in the recent comprehensive survey article by A. L. Shields 12 It is shown there that each weighted shift is unitarily equivalent to multiplication by the function z on a weighted H2 or L2 space. This identification has been the cornerstone of an extensive interplay between operator theory and analytic function theory and weighted shift operators have been a rich source of examples and counter-examples in both areas. In this paper we begin to extend the theory of single i.e. one-variable weighted shifts to systems of A-variable weighted shifts which we define below and we show an analogous identification between such systems and multiplications on certain H2 or L2 spaces in several variables. We concentrate our attention on the unilateral case where we will develop the basic analytic function theory. We will follow the outline of 12 as a model for our theory and we omit the details of proofs which are obvious extensions of the single operator case. We assume that the reader is familiar with the basic theory of operators in Hilbert space. We present several applications of our theory to the theory of general commuting contractions commuting subnormal operators Toeplitz operators in several complex variables and several variable analytic function theory. It is these applications which to a great extent motivate this presentation of the general theory. We also .
