{"title":"The Lomonosov type theorems and the invariant subspace problem for non-archimedean Banach spaces","authors":"A. El Asri , A. Kubzdela , M. Babahmed","doi":"10.1016/j.jmaa.2024.129043","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we study the existence of invariant (and even hyperinvariant) subspaces of bounded operators on a non-archimedean Banach space <span><math><mi>E</mi><mo>=</mo><mo>(</mo><mi>E</mi><mo>,</mo><mo>‖</mo><mo>.</mo><mo>‖</mo><mo>)</mo></math></span> over a valued field <span><math><mi>K</mi></math></span> equipped with a non-trivial non-archimedean valuation <span><math><mo>|</mo><mo>.</mo><mo>|</mo></math></span>. Specifically, we consider compact operators and operators that commute with a compact operator. First we show that if <strong>E</strong> has a base, then any compact operator <strong>T</strong> such that <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi></mrow></msub><mo></mo><msup><mrow><mo>‖</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>‖</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup><mo>></mo><mn>0</mn></math></span> has a finite-dimensional hyperinvariant subspace. Next we show that if <span><math><mi>K</mi></math></span> is locally compact, then every compact operator <strong>T</strong> on <strong>E</strong> has a hyperinvariant subspace. Afterward, assuming that <span><math><mi>K</mi></math></span> is spherically complete or <strong>E</strong> is of countable type, we provide a necessary condition for a bounded operator on <strong>E</strong> to have a hyperinvariant subspace. We demonstrate that the classical Lomonosov Invariant Subspace theorem does not hold in the case where <span><math><mi>K</mi></math></span> is non-spherically complete. Finally, we prove Lomonosov type theorem for spectral quasinilpotent operators, when <span><math><mi>K</mi></math></span> is locally compact.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129043"},"PeriodicalIF":1.2000,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X2400965X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the existence of invariant (and even hyperinvariant) subspaces of bounded operators on a non-archimedean Banach space over a valued field equipped with a non-trivial non-archimedean valuation . Specifically, we consider compact operators and operators that commute with a compact operator. First we show that if E has a base, then any compact operator T such that has a finite-dimensional hyperinvariant subspace. Next we show that if is locally compact, then every compact operator T on E has a hyperinvariant subspace. Afterward, assuming that is spherically complete or E is of countable type, we provide a necessary condition for a bounded operator on E to have a hyperinvariant subspace. We demonstrate that the classical Lomonosov Invariant Subspace theorem does not hold in the case where is non-spherically complete. Finally, we prove Lomonosov type theorem for spectral quasinilpotent operators, when is locally compact.
在本文中,我们研究了有界算子在一个有价域 K 上的非archimedean Banach 空间 E=(E,‖.‖)上的不变(甚至超不变)子空间的存在性,该有价域 K 配备了一个非三元非archimedean 估值|.|。具体来说,我们考虑紧凑算子和与紧凑算子相乘的算子。首先,我们证明,如果 E 有一个基,那么任何使 limn‖Tn‖1n>0 的紧凑算子 T 都有一个有限维的超变子空间。接下来我们证明,如果 K 是局部紧凑的,那么 E 上的每个紧凑算子 T 都有一个超不变子空间。之后,假设 K 是球面完备的或 E 是可数类型的,我们提供了 E 上有界算子具有超不变子空间的必要条件。我们证明了经典的罗蒙诺索夫不变子空间定理在 K 非球面完备的情况下不成立。最后,当 K 局部紧凑时,我们证明了谱准无偶算子的罗蒙诺索夫类型定理。
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