tailieunhanh - Báo cáo toán học: "A note on common invariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một lưu ý trên subspaces bất biến phổ biến. | J. OPERATOR THEORY 7 1982 . 335-339 Copyright by INCREST 1982 A NOTE ON COMMON INVARIANT SUBSPACES c. K. FONG It follows from the celebrated paper of V. Lomonosov 5 that two commuting operators on a Banach space have a common invariant subspace if one of them is compact and nonzero. In the present note we establish the following results which roughly say that two operators have a common invariant subspace if they are almost commuting and one of them is a nonzero compact operator. Theorem 1. Let Kbe a nonzero compact operator on a Banach space 3C and T be a bounded operator on 9 for which there exist a bounded open set D containing ơ T the spectrum of T and an analytic function p from D into D such that TK K f T . In case that ĨE is finite dimensional we assume further that 0 e o K . Then there is a nontrivial closed subspace of 3 which is invariant for both T and K. Theorem 2. If K and T are bounded operators on an infinite dimensional Banach space 3C such that K is compact and nonzero and if there exist a bounded open set D containing j K and an analytic function p from D into D such that KT T p K then there is a subspace of SL which is invariant for both T and K. Remark. The assumption that 0 e ơ K in case dimár oo in Theorem 1 is essential in view of the following example. Let Then KT TK but T and Á do not have a common invariant subspace except the trivial ones. It follows from 2 Corollary 1 that under the assumption of Theorem 1 T has a hyperinvariant subspace unless T is a scalar multiple of identity . However this does not ensure the existence of a subspace invariant for both T and K. Also from 3 it follows that under an assumption slightly different from that of Theorem 2 T has a hyperinvariant subspace. 336 c. K. FONG We remark that some related problems concerning existence of common invariant subspaces of two operators and their simultaneous triangulation have been studied by several authors see 1 4 . The proofs of Theorem 1 and Theorem 2 depend on the .