tailieunhanh - Báo cáo toán học: "AC functions on the circle and spectral families"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: AC chức năng trên vòng tròn và các gia đình phổ. | g Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 33-47 AC FUNCTIONS ON THE CIRCLE AND SPECTRAL FAMILIES EARL BERKSON and T. A. GILLESPIE 1. INTRODUCTION If J a i is a compact interval on the real line R we denote by AC J the Banach algebra of all complex-valued absolutely continuous functions on J with the norm II j given by ll llj ờ var J where var J is the total variation of f on J. In 10 11 12 Ringrose and Smart introduced the following notion of well-bounded operator. An operator T on a Banach space X is said to be well-bounded provided that for some compact interval J T has an AC J -functional calculus that is a bounded algebra homomorphism lỷ of AC J into ỔỈ X the algebra of bounded operators on X which sends the identity map to T and the constant 1 to the identity operator I . Ringrose showed that the well-bounded operators on X can be characterized by a certain representation reminiscent of the spectral theorem for self-adjoint operators but in terms of a not necessarily unique function on R whose values are projections acting in general in X the dual space of X 11 Theorem 2- i and Theorem 6- i . In order to ensure the existence of a spectral family of projections R Sd X such that T has a Stieltjes integral represen- tation T ị Ẳd Ẳ with uniquely determined the special class of well-bounded a operators of type B was introduced in 4 formal definitions will be given in 2 . The well-bounded operators of type B can be characterized 4 Theorem by additionally requiring that the AC J -functional calculus in the definition of well--boundedness be weakly compact that is map bounded subsets of AC J onto subsets of X whose closure in the weak operator topology is compact in that topology. In particular every well-bounded operator on a reflexive space is automatically of type B . A major limitation on direct applications of the theory of well-bounded operators is that the definition implicitly requires the spectrum of a well-bounded operator to be real in .