tailieunhanh - Báo cáo toán học: "On Toeplitz operators with loops. II."

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nó Toeplitz nhà khai thác với các vòng. | J. OPERATOR THEORY 7 1982 109-123 Copyright by INCREST. 1982 ON TOEPLITZ OPERATORS WITH LOOPS. II DOUGLAS N. CLARK INTRODUCTION Let TF denote the Toeplitz operator associated with a rational function F e t of e with the poles of F z lying off T z G C z 1 . Suppose that the bounded components of C F T are denoted ỉ if the index of TF ẢI for 2 G JSfj is negative and 2 if that index for Ả EẾị is positive. Label the index of Tp kl as Ni for Ằ e i and v for 2 e A-. The and tị are called the loops of F. In 3 the following similarity theorem for Tp is proved. Theorem 1. Suppose F has the further properties I The intersection of the closures of any two loops consists of a finite number of points called the multiple points of F . II The boundary of each loop is an analytic curve except at the multiple points where it is piecewise smooth with inner angle 0 0 7T 2n. No distinct arcs of dF T meet at angle 0 - 0. III No multiple point of F is the image F z0 of a point z0 G T where F lz 0. IV F never backs up. That is if Tj and Tj are the Riemann mapping functions from z 1 to ị and ỈTJ respectively a bar over a set denotes conjugate then the arguments arg If1 F e and arg Tfỵ F e are monotone decreasing. Then Tp is similar to 1 ỵ Tt. ỵ r . on . . where H is the vector H2 space based on a Hilbert space of dimension V. The purpose of this paper is to improve upon Theorem 1 in such a way as to show that similarity properties of Tp depend upon more than the geometry of F T the index of Tp and the backing up of F. 110 DOUGLAS N. CLARK The improved version of Theorem 1 involves first the removal of the condition 0 n 2iT and the last sentence of II . Specifically 11 will be replaced by IT The boundary of each loop is an analytic curve except at the multiple points where it is piecewise smooth with inner angle 0 0. The generalization is accomplished by overcoming the need for nontangential approach of the i f e as proved for the case of Theorem 1 in 3 Lemma . This same tangential .