An introduction to the fine structures on the $S$-spectrum
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910643611762688 |
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| author | Colombo, Fabrizio De Martino, Antonino Pinton, Stefano Sabadini, Irene Schlosser, Peter |
| author_facet | Colombo, Fabrizio De Martino, Antonino Pinton, Stefano Sabadini, Irene Schlosser, Peter |
| contents | Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_07251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An introduction to the fine structures on the $S$-spectrum Colombo, Fabrizio De Martino, Antonino Pinton, Stefano Sabadini, Irene Schlosser, Peter Spectral Theory Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum. |
| title | An introduction to the fine structures on the $S$-spectrum |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2410.07251 |