A note on the $A$-numerical range of semi-Hilbertian operators
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914671090466816 |
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| author | Sen, Anirban Birbonshi, Riddhick Paul, Kallol |
| author_facet | Sen, Anirban Birbonshi, Riddhick Paul, Kallol |
| contents | In this paper we explore the relation between the $A$-numerical range and the $A$-spectrum of $A$-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of $A$-normal operator and prove that closure of the $A$-numerical range of an $A$-normal operator is the convex hull of the $A$-spectrum. We further prove Anderson's theorem for the sum of $A$-normal and $A$-compact operators which improves and generalizes the existing result on Anderson's theorem for $A$-compact operators. Finally we introduce strongly $A$-numerically closed class of operators and along with other results prove that the class of $A$-normal operators is strongly $A$-numerically closed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05360 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the $A$-numerical range of semi-Hilbertian operators Sen, Anirban Birbonshi, Riddhick Paul, Kallol Functional Analysis Primary: 46C05, Secondary: 47A05 In this paper we explore the relation between the $A$-numerical range and the $A$-spectrum of $A$-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of $A$-normal operator and prove that closure of the $A$-numerical range of an $A$-normal operator is the convex hull of the $A$-spectrum. We further prove Anderson's theorem for the sum of $A$-normal and $A$-compact operators which improves and generalizes the existing result on Anderson's theorem for $A$-compact operators. Finally we introduce strongly $A$-numerically closed class of operators and along with other results prove that the class of $A$-normal operators is strongly $A$-numerically closed. |
| title | A note on the $A$-numerical range of semi-Hilbertian operators |
| topic | Functional Analysis Primary: 46C05, Secondary: 47A05 |
| url | https://arxiv.org/abs/2402.05360 |