tailieunhanh - Báo cáo toán học: "Spectral multiplicity for direct integrals of normal operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đa dạng quang phổ cho tích phân trực tiếp của các nhà khai thác bình thường. | Copyright by INCREST 1980 J. OPERATOR THEORY 3 1980 213-235 SPECTRAL MULTIPLICITY FOR DIRECT INTEGRALS OF NORMAL OPERATORS EDWARD A. AZOFF and KEVIN F. CLANCEY INTRODUCTION The theory of spectral multiplicity solves the unitary equivalence problem for normal operators. Briefly recalling the theory one associates with each normal operator N a subset a N of c a measure V on c and a multiplicity function m defined on c and taking values in the extended natural numbers. The ordered triple consisting of ơ N the equivalence class v and the equivalence class m v then provide a complete set of unitary invariants for N. The statement that this set of invariants solves the unitary equivalence problem for normal operators requires these invariants to be computable . Suppose now that X 1 is a probability space and N is the operator of multiplication by a bounded Borel function p X - c. Then ơ N is the essential range of p and the scalar spectral measure V is ỊẰ 0 p-1 both of which can be regarded as computable. In their work 1 2 B. Abrahamse and T. Kriete consider the problem of computing the multiplicity function for V. The folk intuition is that W1 Ấ should be the cardinality of the preimage but Abrahamse and Kriete provide several illuminating examples to show that this intuition is not correct even when n is Lebesgue measure on 0 1 and p is smooth. They then introduce the notion of the essential preimage p X and show that the function 2- card P 1 Ấ is a multiplicity function for N. The definition of P XU depends on a limiting process so that the Abrahamse-Kriete multiplicity function though concrete is somewhat removed from the original function p. The function m card 9-1 - is in general too large to be a multiplicity function for N however it is always possible to find a Borel function p0 agreeing with p almost everywhere so that does provide a multiplicity function. This result which was independently derived by J. Howland 8 appears as Theorem below. This .