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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.10427 |
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| _version_ | 1866908318584274944 |
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| author | Stanković, Hranislav Kubrusly, Carlos |
| author_facet | Stanković, Hranislav Kubrusly, Carlos |
| contents | In this paper, we extend Ando's theorem on paranormal operators, which states that if $ T \in \mathfrak{B}(\mathcal{H}) $ is a paranormal operator and there exists $ n \in \mathbb{N} $ such that $ T^n $ is normal, then $ T $ is normal. We generalize this result to the broader classes of $ k $-paranormal operators and absolute-$ k $-paranormal operators. Furthermore, in the case of a separable Hilbert space $\mathcal{H}$, we show that if $ T \in \mathfrak{B}(\mathcal{H}) $ is a $ k $-quasi-paranormal operator for some $ k \in \mathbb{N} $, and there exists $ n \in \mathbb{N} $ such that $ T^n $ is normal, then $ T $ decomposes as $ T = T' \oplus T'' $, where $ T' $ is normal and $ T'' $ is nilpotent of nil-index at most $ \min\{n,k+1\} $, with either summand potentially absent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On roots of normal operators and extensions of Ando's Theorem Stanković, Hranislav Kubrusly, Carlos Functional Analysis 47B15, 47B20 In this paper, we extend Ando's theorem on paranormal operators, which states that if $ T \in \mathfrak{B}(\mathcal{H}) $ is a paranormal operator and there exists $ n \in \mathbb{N} $ such that $ T^n $ is normal, then $ T $ is normal. We generalize this result to the broader classes of $ k $-paranormal operators and absolute-$ k $-paranormal operators. Furthermore, in the case of a separable Hilbert space $\mathcal{H}$, we show that if $ T \in \mathfrak{B}(\mathcal{H}) $ is a $ k $-quasi-paranormal operator for some $ k \in \mathbb{N} $, and there exists $ n \in \mathbb{N} $ such that $ T^n $ is normal, then $ T $ decomposes as $ T = T' \oplus T'' $, where $ T' $ is normal and $ T'' $ is nilpotent of nil-index at most $ \min\{n,k+1\} $, with either summand potentially absent. |
| title | On roots of normal operators and extensions of Ando's Theorem |
| topic | Functional Analysis 47B15, 47B20 |
| url | https://arxiv.org/abs/2504.10427 |