Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên các nhà khai thác S-phân huỷ. | Copyright by INCREST 1979 J. OPERATOR THEORY 2 1979 277-286 ON S-DECOMPOSABLE OPERATORS B. NAGY 1. INTRODUCTION Residually decomposable operators introduced by . Vasilescu 14 15 as well as bounded and unbounded -decomposable operators studied by 1. Bacalu 2 3 and the author 8 are linear operators that show a good spectral behaviour only outside a certain part of the spectrum. There are more related definitions of this good behaviour which are all connected with the concept of decomposability in the sense of c. Foias see . 7 . It was proved in 9 that for any closed operator there is a unique minimal closed subset of the spectrum called the strong spectral residuum outside which the operator has a spectral behaviour of this kind strong decomposability . The main result of this paper is that for any bounded operator there is a unique minimal closed subset of the spectrum called the spectral residuum outside which the operator shows a similar spectral behaviour decomposability . The spectral residuum is contained maybe properly in the strong spectral residuum and is in general different from the spectra residuum in Vasilescu s sense 15 p. 385 which was proved to exist only for a certain class of operators . As a preparation we extend a recent result by M. Radjabalipour 12 and prove the equivalence of bounded S 1 - and S-decomposable operators. In Section 2 we shall give the necessary definitions for closed operators but we shall restrict most of the discussion in Section 3 to the bounded case leaving open the question whether these results are valid if the operator is unbounded. 2. PRELIMINARIES Let X be a complex Banach space and let F X and ăỉ X denote the class of closed and bounded linear operators on X respectively c and c will stand for the complex plane and its compactification respectively. If Fc c then Fc denotes c F and F denotes the closure of F in c. For Te Fi JX S T is its domain and ơ T denotes its extended spectrum which is its spectrum s T if .
