tailieunhanh - Báo cáo toán học: "Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang and D. Sarason "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hai nhận xét Về lý của S. Axler, . A. Chang và D. Sarason. | Copyright by INCREST 1982 J. OPERATOR THEORY 7 1982 209 - 218 TWO REMARKS CONCERNING THE THEOREM OF s. AXLER . A. CHANG AND D. SARASON A. L. VOLBERG 1. INTRODUCTION This article is devoted to a proof of a conjecture of s. Axler . A. Chang and D. Sarason 2 and to an application of one of their results to bases consisting of rational fractions. Let D eC 1 be the unit disc and T be its boundary. For p e L T the Toeplitz operator Tt on the Hardy space H2 is defined by the equality Tvh p ph h G H2 where P is the orthogonal projection from L2 onto H2. The function p is called the symbol of this operator. The Hankel operator with the same symbol is defined by the formula Hfi P_ pA h e H2 where I P I being the identity operator. The following question arises rather naturally For what symbols f g is the product of two Toeplitz operators TfTg a compact perturbation of some Toeplitz operator Then it is well-known that this Toeplitz operator is Tfg 7 . So the question above can be reduced to the following one For what symbols f g is the operator Tfg TjTg H Hg compact Let us also recall that for f G L denotes the uniformly closed algebra generated by H and f. One of the most interesting algebras of this type is the algebra HK z . It is well-known that 7 z H c where c is the algebra of all continuous functions on the unit circle. Now we are in the position to state the following remarkable theorem proved in 2 Theorem a. If n H c then the operator H Hg is compact. The necessity of the condition was proved in 2 for a large class of functions f g but not in the general case. 210 A. L. VOLBERG 2. THE PROOF OF THE NECESSITY Suppose that the operator HfHg is compact. We have to prove . Without loss of generality we may suppose that Ịỉ ịI 1 1. Then it is well-known that there are unimodular functions uef Z V Eg H such that 1 dist zw Zf 1 dist zp Z x 1 see 8 14 . But dist u 2 1 dist t zz 1 and we conclude that the operators T Tv are invertible see 14 Ch. VIII . Now we .