TAILIEUCHUNG - Báo cáo toán học: "A new proof of the Szegö limit theorem and new results for Toeplitz operators with discontinuous symbol "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một bằng chứng mới của định lý giới hạn Szegö và các kết quả mới cho các nhà khai thác Toeplitz với biểu tượng không liên tục. | J. OPERATOR THEORY 3 1980 23-39 Copyright by INCREST 1980 A NEW PROOF OF THE SZEGO LIMIT THEOREM AND NEW RESULTS FOR TOEPLITZ OPERATORS WITH DISCONTINUOUS SYMBOL E. BASOR and J. WILLIAM HELTON 1. INTRODUCTION The Strong Szego Limit Theorem computes the limit as N - oo of det PffTyPff after appropriate normalization. Here Tv is the Toeplitz operator with nice generating function p and PN is the projection onto the first N 1 coordinates. To an operator theorist the result seems strange since Toeplitz operators are not generally determinant class operators or even close to them. Thus even existence of the limit seems bizarre. In this paper we show that PNTVPN has a direct connection with a determinant class operator and in fact splits into two canonical and extremely stable parts. From this we can swiftly obtain the Strong Szego Limit Theorem. The basic argument which illustrates the above claim is as follows. Suppose p P P- where p p_ 5 6 H03. Then Tv TV_TV . Now it is very P p- simple to verify that PnT PnT Pn and TtpPtf PNT p PN. Thus if the order of Ti _7 in O v pJ were reversed we would have det p Tj _Pn det PNTtỊỈ PN det PflTp Pff . Let fk denote the Lth Fourier coefficient of the function f. Then the above term is easily computed to be p 0 p_0 Ar 1. The order however can be reversed if we are willing to compensate with a multiplicative commutator. To wit TnTp-Pa PNTv PtlT Tv_Tv T PiiT9_Pli. 24 E. BASOR and J. w. HELTON Note that the middle term is a multiplicative commutator namely it has the form A B A I A SỊA B 1. Furthermore multiplicative commutators of Toeplitz operators with smooth generating function are determinant class since Ty TJ is trace class. Therefore lira IW p o P-o v 1 N- OO lim det PN T l Tv Pn N- co det T T _ . This argument gives US existence of the limit provided a nice outer factorization p p q - exists. Furthermore a general theory of traces of additive commutators and determinants of multiplicative commutators was developed in H-H .

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