tailieunhanh - Báo cáo toán học: "Carlesons measures and operators on star-invariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Carleson biện pháp và các nhà khai thác trên sao bất biến subspaces. | Ji OPERATOR THEORY 15 1986 181-202 Copyright by INCREST 1986 CARLESON MEASURES AND OPERATORS ON STAR-INVARIANT SUBSPACES WILLIAM s. COHN INTRODUCTION Let D z 1 be the unit disk and let H1 denote the usual classes of functions analytic on D. A function s in Hx is called inner if -s ei0 1 . dớ . In this paper we study operators on the subspace K H2 0 pH2 where p belongs to a certain class of inner functions. Our main tools in this study are the results in 6 in which the Carleson measures for K were characterized under the hypothesis that p satisfy the connected level set condition that is there exist r0 0 rn 1 such that z r0 is connected. Recall that a measure on the closed disk D which assigns no mass to the singular support of p is called a Carleson measure for K if there is a constant c such that ịỉ Td 1 cil il for all f e K. See 6 p. 347. We will rely on Theorems and of 6 which characterize such measures. Although these theorems are difficult to state at this point we will record the following. Theorem 0. Let p satisfy the connected level set condition as above. Then the following conditions are equivalent i The measure n is a Carleson measure for K. ii There is a constant c such that I - z 2 II - w2 d 0 c 1 l p z for all z e D. 182 WILLIAM s. COHN Note that when p is supported on the unit circle T the integral in ii is the ordinary Poisson integral of p. As examples of inner functions satisfying the connected level set condition we give Pi z exp I V z 1 and 1 - KI 1 where 0 r Lit is true of course that most inner functions do not satisfy the connected level set condition. Thus it becomes an interesting perhaps difficult question as to whether our results extend to all inner functions s. We now give a somewhat simplified statement of our main results. In all of our theorems p denotes an inner function satisfying the connected level set condition. The main result of Section 1 is the following theorem. Theorem. Let f e K. Then there is a constant c such