tailieunhanh - Báo cáo toán học: "Factoring trace-class operator-valued functions with applications to the class $A_{\aleph_0}$"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bao thanh toán theo dõi điều hành lớp học giá trị chức năng với các ứng dụng lớp A_ {\ aleph_0} $. | J. OPERATOR THEORY i4G9S5 35Ỉ-389 Copyright by INCREST. 1985 FACTORING TRACE-CLASS OPERATOR-VALUED FUNCTIONS WITH APPLICATIONS TO THE CLASS A o H. BERCOVICI c. FOIAS and c. PEARCY 1. INTRODUCTION Let AA be a separable infinite dimensional complex Hilbert space and let denote the algebra of all bounded linear operators on Ji. If T e AAAdA i let j r denote the dual aigebra generated by T . the smallest subalgebra of that contains T and Iff and is closed in the ultrawealc operator topology. Moreover let QT denote the quotient space where C JA denotes the trace-class ideal in under the trace norm and J- denotes the preannihilator of sdT in c j . One knows that sđ-Ị. is the dual space of Qr and that the duality is given by 1 A L tr XL A 6 L 6 where L denotes the image of L in QT. If .V and V are vectors in we write as usual .X X J for the rank-one operator in Cỵự A defined by x X j- u w v x u G JA. Then of course x X y e QT and it is easy to see that 2 T x y ẤX y A G sđT X 1 e JA. In a similar vein if T denotes the unit circle in c we denote by Lp Lp ĩ 1 p oo the usual Banach spaces of Lebesgue jP-integrable functions on T and by L L T the Banach space of essentially bounded measurable functions on T. One knows that L is the dual space of L1 under the pairing 3 g -- i di f e g 6 . 2 r .1 Õ 352 H. BERCOV1CI c. FOIAS c. PEARC Furthermore if for 1 í Í oc we denote by Hp P T the subspace of Lp consisting of those functions whose negative Fourier coefficients vanish then one knots - that is a weak -closed subspace of L and that the preannihilator of in Ll is the space Hg consisting of those functions g in H1 whose analytic extension g to D - . e c z 1 satisfies g 0 0. It follows easily that is the dual space of L1 lig under the pairing 2-J 4 g - - f ẽ gịé dt 2rt J 0 If T is an absolutely continuous contraction in . a contraction whose unitary part is either absent or absolutely continuous then the pairs of spaces Qt and z are related by the Foia functional .