tailieunhanh - Báo cáo toán học: "A trace formula for Wiener-Hopf operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một công thức theo dõi cho các nhà khai thác Wiener-Hopf. | J. OPERATOR THEORY 8 1982 279-298 Copyright by INCREST 1982 A TRACE FORMULA FOR WIENER-HOPF OPERATORS HAROLD WIDOM INTRODUCTION For a function Ơ e oo R the Wiener-Hopf operator W ơ on L2 R is defined by W ff p - P ơíỊ y where CO co Ặ ễ -- e-iiJt p x dx 2 x e ỈY X c dẹ J 2 t J CO co and p is the projection from L2 R to 2 R . We consider here the following questions For which functions f and Ơ is the operator 1 W r -W . 7 of trace class If it is trace class what is its trace First it is not hard to show that if Ơ is real-valued then a necessary condition that W ff 2 - W 72 be trace class is that the distributional inverse Fourier transform a be equal on R 0 to a function satisfying CO 2 ị xi ơự 2dx oo co this integral and analogous ones are understood to be taken over R 0 and that if 7 J and ơ2 real-valued or not satisfy 2 then W Ơ1 W ƠS - W ơlff2 280 HAROLD WIDOM is trace ciass. The Toepl itz analogue of this is very well known and quite trivial. It is less straightforward here since cr is a distribution rather than a function. Proofs will be given in the next section. It follows that if 2 holds then W ơ K - W ơ is trace class for all n and so 1 is trace class for all polynomials f. We shall assume about Ơ only that 2 holds and of course that Ơ e Loo To state the main result of the paper we introduce two pieces of notation. First dm denotes the measure on R2 given by dm 2 o - - - I-M dộ . 8tt2 The usefulness of this measure lies in the identity 3 ịị bi Ci - - ffaOdmfe É2 - ỷ ị IUiCỳlèự -x dx satisfied by all ffj a- for which 2 holds which will also be derived in the next section. Second for complex numbers a and j and a function f belonging to C1 on the line segment joining a and p we write U a p f -. 1 0 a w - 1 - 0 ef p dỡ J 0 1-0 0 Theorem 1. Assume 2 holds and that either of the following conditions is satisfied a f is analytic on a neighborhood of the closed convex hull Ỉ of the essential range of a b Ơ is real valued f e L R and t f f e L R . Then the operator