Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916112080306176 |
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| author | Ghiloni, Riccardo Recupero, Vincenzo |
| author_facet | Ghiloni, Riccardo Recupero, Vincenzo |
| contents | In the present paper, we prove a resolvent equation for the $\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent operator in an open neighborhood of any given quaternion belonging to the $\mathcal{S}$-resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the $\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the $\mathcal{S}$-resolvent operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00511 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator Ghiloni, Riccardo Recupero, Vincenzo Spectral Theory Complex Variables Functional Analysis 30G35, 47D03, 47A60, 47A10 In the present paper, we prove a resolvent equation for the $\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent operator in an open neighborhood of any given quaternion belonging to the $\mathcal{S}$-resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the $\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the $\mathcal{S}$-resolvent operator. |
| title | Quaternionic resolvent equation and series expansion of the $\mathcal{S}$-resolvent operator |
| topic | Spectral Theory Complex Variables Functional Analysis 30G35, 47D03, 47A60, 47A10 |
| url | https://arxiv.org/abs/2402.00511 |