TAILIEUCHUNG - Báo cáo toán học: "Nuclear operators in nest algebra "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hạt nhân các nhà khai thác trong đại số tổ. | J. OPERATOR THEORY 10 1983 337-352 Copyright by INCREST 1983 NUCLEAR OPERATORS IN NEST ALGEBRAS s. c. POWER 1. INTRODUCTION The main result shows that each nuclear operator T in a nest algebra Alg f admits a representation T TjEdr s where T is a finite positive Borel measure on the nest and T - TE is a nuclear operator valued function on s such that TE ETe I E-. almost everywhere. This representation leads to conditions under which T can be decomposed into an exact sum of rank one operators in Alg s in the following sense T ỵJRi II Till s 11 Ik i l i l with 7 7 2 . . rank one operators in Alg . We call this property exact decomposability and it is shown in particular that T is exactly decomposable if s is countable or if T is dissipative. A basic result required in the analysis is a construction of Lance Lemmas of 11 which splits an upper triangular 2x2 operator matrix into a sum of two operators of the form loo 0 0 power of this decomposition is given in the fact that it leads naturally to a useful result of Parrott 14 In 11 it is used to derive Arveson s distance formula 1 to which Parrott s result is closely related 15 . In Section 3 we make inductive use of the lemma and an inherent left and . An indication of some of the continuity in order to associate with each positive operator c and nest s a positive operator valued Borel measure C J on S- If this construction is applied to the positive part c of an operator T uc in Alg f then the operator measure UC A on ế provides the appropriate generalisation of Lance s construction and in case 338 s. c. POWER has three elements coincides with this construction. In Section 4 we give a Radon-Ni-kodym theorem for nuclear operator valued measures. For a nuclear operator T this allows us to form the derivative TE of the measure i C d with respect to the scalar measure T J trace C d and thereby obtain the main result. The relationship between c and C J seems to be worthy of further analysis. In Section 5 we complete

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