tailieunhanh - Báo cáo toán học: "Comparing functors classifying extensions of C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: So sánh functors phân loại phần mở rộng của C *- đại số. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 283 - 287 A VARIATION OF LOMONOSOV S THEOREM. II H. w. KJM R. L. MOORE c. M. PEARCY Let T be an infinite dimensional complex Banach space and let L T denote the algebra of all bounded linear operators on T. A closed subspace M of 3S is said to be invariant for an operator T in LtfK if Tx e M for all X 6 M M is hyperinvariant for T if M is invariant for every operator that commutes with T. The following fact has been proved in several places 1 3 5 Theorem a. Suppose that T and K are operators in L SP such that T is nonscalar and K is compact and nonzero. If there is a complex number Ả such that KT .TK then T has a nontrivial hyperinvariant subspace. In 5 we obtained a partial generalization of Theorem A Theorem B. Suppose that T and K are operators in L. T such that T is nonscalar and K is compact and nonzero. Let f be a function analytic on an open neighborhood 6U of ơ K the spectrum of K such that either KT Tf K or TK f K T. Then T has a nontrivial hyperinvariant subspace under any one of the following conditions i 1 ii 0 1 and K is quasinilpotent or iii 0 1 and K has trivial kernel. The proof of Theorem B given in 5 depended on Lomonosov s result 6 concerning transitive algebras and involved considerable manipulation of n-fold compositions of the function f. If in addition to the stated hypotheses one assumes that W c d then one can apply the notion of Schroeder functions which is of interest in stability theory to simplify the proof of Theorem B while at the same time allowing a slight relaxation of the other hypotheses. In fact in this case Theorem B does not extend Theorem A as much as it appears to if T and T have no eigenvalues if T satisfies the hypotheses of Theorem B and if flfli then unless 0 0 T also satisfies those of Theorem A with a different compact operator involved . The purpose of this note is to prove this rather surprising fact. 284 H. w. KIM R. L. MOORE and c. M. PEARCY We require some facts