{"title":"差距和相对尺寸","authors":"Chenfeng Liao, Chaofeng Zhu","doi":"arxiv-2408.13837","DOIUrl":null,"url":null,"abstract":"In this paper, the notion of Fredholm tetrad of closed linear subspaces in a\nBanach space is introduced. Then the stability of the Fredholm tetrad is\nproved. After that, the notion of semi-compact perturbation of a closed linear\nsubspace is introduced. Then for a of pair of closed linear subspace of a\nBanach space such that one is a semi-compact perturbation of the other, it is\nproved that the relative dimension between them is well-defined. If the\nperturbation is compact, the relative dimension is stable. Finally the\nperturbed argumented Morse index is studied.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"31 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Gaps and relative dimensions\",\"authors\":\"Chenfeng Liao, Chaofeng Zhu\",\"doi\":\"arxiv-2408.13837\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, the notion of Fredholm tetrad of closed linear subspaces in a\\nBanach space is introduced. Then the stability of the Fredholm tetrad is\\nproved. After that, the notion of semi-compact perturbation of a closed linear\\nsubspace is introduced. Then for a of pair of closed linear subspace of a\\nBanach space such that one is a semi-compact perturbation of the other, it is\\nproved that the relative dimension between them is well-defined. If the\\nperturbation is compact, the relative dimension is stable. Finally the\\nperturbed argumented Morse index is studied.\",\"PeriodicalId\":501036,\"journal\":{\"name\":\"arXiv - MATH - Functional Analysis\",\"volume\":\"31 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Functional Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.13837\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Functional Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.13837","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper, the notion of Fredholm tetrad of closed linear subspaces in a
Banach space is introduced. Then the stability of the Fredholm tetrad is
proved. After that, the notion of semi-compact perturbation of a closed linear
subspace is introduced. Then for a of pair of closed linear subspace of a
Banach space such that one is a semi-compact perturbation of the other, it is
proved that the relative dimension between them is well-defined. If the
perturbation is compact, the relative dimension is stable. Finally the
perturbed argumented Morse index is studied.