tailieunhanh - Báo cáo toán học: "An extension of Scott Brown's invariant subspace theorem: K-spectral sets "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một phần mở rộng của định lý không gian con bất biến của Scott Brown: bộ K-quang phổ. | J. OPERATOR THEORY 3 1980 3-21 Copyright by INCREST 1980 AN EXTENSION OF SCOTT BROWN S INVARIANT SUBSPACE THEOREM -SPECTRAL SETS JOSEPH G. STAMPFLI Recently Scott Brown showed that every subnormal operator has an invariant subspace. Similar techniques can be used to show that every operator in J J the algebra of bounded linear operators on a separable Hilbert space for which ƠÍT is a A -spectral set also has an invariant subspace. This result has been proved by J. Agler 30 when a T is a spectral set for T. There are more differences between spectral sets and X-spectral sets than might be apparent at first glance. First dilation theory is available in the former case but not in the latter. Second orthogonality disappears in several places as one moves from spectral to Tf-spectral sets. Third there are several interesting special cases such as polynomially bounded operators and unitary p-dilations which are not covered by spectral sets. Definition. The compact set M a T is a K-spectral set for T e if 11 7 II jq for all f e R M where ll llẩ sup z zeM . R M denotes the uniform closure of the rational functions with poles off M. To begin with we need an extension of the orthogonal direct sum decomposition for operators proved independently by Mlak 21 and Lautzenheiser 17 in the spectral set case. It should be mentioned that although ÁT-spectral sets are never mentioned in 17 and only very briefly in 21 still many of the techniques carry over from their work. See also 23 . Theorem 1. Let T e Assume M is a K-spectral set for T. Let Gỵ G2 . . be the nontrivial Gleason parts of R M . Then there exists an invertible operator Q such that QTQ-1 s Si Í 0 4 J. G. STAMPFLI where so is normal and a Si Gtfor i 1 2 . . Thus T -j- Tị direct sum . Note Some of the terms may be absent. Because of its length we have relegated the proof of Theorem 1 to the Appendix. However we will invoke the notation and techniques in Proposition 5. From now on we will be using Af-spectral sets M where