TAILIEUCHUNG - Báo cáo toán học: "A variation of Lomonosov's theorem "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một biến thể của định lý Lomonosov. | Copyright by INCREST 1979 J. OPERATOR THEORY 2 1979 131-140 A VARIATION OF LOMONOSOV S THEOREM H. w. KIM R. MOORE and c. M. PEARCY Let 3C be an infinite-dimensional complex Banach space and let denote the algebra of all bounded linear operators on 3C. One version of the pioneering theorem of Lomonosov 7 says that if T is a nonscalar operator in -SfX and T commutes with some nonzero compact operator then T has a nontrivial hyperinvariant subspace. Recall that a closed subspace of X is a nontrivial hyperinvariant subspace for an operator T in if 0 0 ểlẽ ĨE and T 911 c 11 for every operator T in JT ST that commutes with T. For expository accounts of ramifications of the Lomonosov technique the reader might consult 8 Chapter 7 or 9 . Additional results in this direction were obtained in 3 5 and 6 and in 1976 the first and third authors together with A. L. Shields proved the following theorem which was stated without proof as Theorem of 8 Theorem a. If T is a nonscalar operator in and there exists a nonzero compact operator K such that either a KT .TK for some scalar Ẳ b KT Tp K for some polynomial p satisfying 7 0 0 0 or c T is quasinilpotent and KT TnK for some positive integer n then T has a nontrivial hyperinvariant subspace. Somewhat later the authors improved upon Theorem A and also discovered that parts of Theorem A had been proved independently by others. In 1 Scott Brown showed among other things that a nonscalar operator T in has a nontrivial hyperinvariant subspace under hypothesis a and in 4 it was shown that such a T has a nontrivial hyperinvariant subspace under a weaker hypothesis than c . It is the main purpose of this note to prove that such an operator Thas a nontrivial hyperinvariant subspace under a hypothesis weaker than b thereby making available a complete proof of a stronger theorem than Theorem A. In what follows the spectrum of an operator T in will be denoted by ff T and the point spectrum of T . the set of eigenvalues of T by afT . .

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