TAILIEUCHUNG - Báo cáo toán học: "Are commuting systems of decomposable operators decomposable? "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hệ thống đi lại của các nhà khai thác phân huỷ trong tự phân huỷ?. | OPERATOR THEORY 12 1984 213 - 219 Copyright by INCREST 1984 ARE COMMUTING SYSTEMS OF DECOMPOSABLE OPERATORS DECOMPOSABLE JỒRG ESCHMEIER The notion of decomposability has been introduced for a single continuous linear operator on a complex Banach space by c. Foia in 9 . Later it was generalized to the case of several commuting operators by s. Frunza in 11 . In the sense of s. Frunza a commuting system a v of continuous linear operators on a complex Banach space X is called decomposable if there is a spectral capacity E for a . a map E F CN - Inv a from the system of all closed subsets of cw into the set of all closed subspaces of X invariant under Oj .aN which satisfies i 0 0 E CN X ii n Fi n E Fi for each family F i6r in F CN ie iei iii X EịUy . E ũn for each open cover CN Ờ1 u . u u iv a a E F F for each F eF Cn . Here ff E F denotes the Taylor spectrum see 17 of a restricted to E F . If a a1 .aN is decomposable its spectral capacity is given by 10 11 E F Xa F xeX aa x F . Here att x denotes the local spectrum of a relative to XG X . the smallest closed subset of CN such that on the complement X is locally of the form X zL ứ1 l Ẳ . ữJv jvơ- with analytic A -valued functions A . . fN see 1 11 6 . If a . aN is decomposable with spectral capacity E then due to the projection property of the Taylor spectrum each ữị 1 c i N is decomposable with spectral capacity . F E tiĩ F F F c c where Til . CN - c denotes the projection of CN onto its z -th component. One of the most natural questions concerns the converse of this statement. Is it true that each 2 2294 214 - JỒRG ESCHMEIER commuting system a ứ . af with decomposable components fl 1 i N is decomposable This question was formulated by E. Albrecht s. Frunza and F. -H. Vasilescu in 3 . We are going to prove that there is a Banach space X and a non-decompo-sable system a ữỵ fl2 consisting of two commuting decomposable operators fli aa on A. The construction is based on a result concerning the solvability of .

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