Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on lp-Spaces |
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Authors: | Mingxue Liu Peide Liu |
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Affiliation: | (1) Departement of Mathematics, Guangdong Polytechnic Normal University, Guangzhou, 510665, People’s Republic of China;(2) College of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China |
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Abstract: | In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an lp-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on lp that is quasinilpotent at a non-zero positive vector x0 has a non-trivial invariant closed subspace. In this paper, it is proved that if is a collection of continuous operators with moduli on lp that is finitely modulus-quasinilpotent at a non-zero positive vector x 0 then and its right modulus sub-commutant have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on l p whose moduli commute with a non-zero positive operator I on l p that is quasinilpotent at a non-zero positive vector x 0 have a common non-trivial invariant closed subspace, so that all positive operators on l p which commute with a non-zero positive operator S on l p that is quasinilpotent at a non-zero positive vector x 0 have a common non-trivial invariant closed subspace. This research was supported by the Natural Science Foundation of Hunan Province of P. R. China (04JJ6004), the Foundation of Education Department of Hunan Province of P. R. China (04C002) and the Natural Science Foundation of P. R. China (10671147). Received: 4 December 2005 Revised: 19 June 2006 |
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Keywords: | 47A15 47B65 46B40 |
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