Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Quasitriangular phần mở rộng của C *- đại số và vấn đề là các nhà khai thác chung của quasitriangularity. | J. OPERATOR THEORY 10 1983 167-205 Copyright by INCREST 1983 QUASITRIANGULAR EXTENSIONS OF C -ALGEBRAS AND PROBLEMS ON JOINT QUASITRIANGULARITY OF OPERATORS NORBERTO SALINAS 1. INTRODUCTION In this paper we shall be concerned with the question of when an n-tuple of operators on Hilbert space is jointly quasitriangular. We recall that an 71-tuple Tỵ . T of operators on a Hilbert space 34 is said to be jointly quasitriangular cf. 5 2 19 if there exists an increasing sequence p of finite rank projections on .TC that tends strongly to the identity operator Ijp and such that lim l PmỴTkPm 0 for 1 k n. Since the separability m- oo of 34 is implicit in this definition and if 24 is finite dimensional all n-tuples of operators on 24 are jointly quasitriangular we assume throughout the paper that 24 is an infinite dimensional separable Hilbert space. The set of all jointly quasitriangular n-tuples of operators on 34 will be denoted by QT . One of our main objectives in the present paper is to generalize to the case on n-tuples of operators various results obtained in 2 3 and 4 for the case of single operators. This program was started in 29 where attention was focused in proving invariant subspace theorems for commuting n-tuples of operators. The n-tu-ples that we consider there are not necessarily commuting and since our aim in this paper is different we obtain other generalizations that were not given in 29 . Given an operator T on 34 let CO_ T be the set of those complex numbers Ấ such that T Ấ is essentially left invertible and of negative Fredholm index. Alternatively Aea _ T if and only if T Ấ is essentially left invertible but T Ấ K is right invertible for no compact operator K. It was shown by Douglas and Pearcy 16 Theorem that OJ_ T 0 whenever Te QTr Apostol Foias and Voicu-lescu in 2 Theorem completed the spectral characterization of quasitriangularity by proving that the converse of this result is valid. They also succeeded in computing the distance from a
