The S-functional calculus for the Clifford adjoint operator
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912541515448320 |
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| author | Colombo, F. Mantovani, F. Schlosser, P. |
| author_facet | Colombo, F. Mantovani, F. Schlosser, P. |
| contents | In this paper, we work within the framework of modules over the Clifford algebra $\mathbb{R}_d$. Our investigation focuses on the S-spectrum and the S-functional calculus in its various forms for the adjoint T* of a Clifford operator T. One of the key results we present is that the bisectoriality of T can be transferred to T*. This is grounded in the fact that, for Clifford operators, the S-spectrum of the adjoint operator T* is identical to that of T. Furthermore, we demonstrate that for the existing formulations of the S-functional calculus, including bounded, unbounded, $ω$, and $H^\infty$ versions, there is a clear connection between the left functional calculus of T and the right functional calculus of T*. This explicit link between the left functional calculus of T and the right functional calculus of T* and vice versa is obtained using the function $f^\#(s):=\overline{f(\overline{s})}$. Finally, we discuss the fact that the S-spectrum, related to the invertibility of the second-order operator T^2-2s_0T+|s|^2, can actually be characterized through the invertibility of the first-order $\mathbb{R}$-linear operator T-I^Rs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The S-functional calculus for the Clifford adjoint operator Colombo, F. Mantovani, F. Schlosser, P. Spectral Theory In this paper, we work within the framework of modules over the Clifford algebra $\mathbb{R}_d$. Our investigation focuses on the S-spectrum and the S-functional calculus in its various forms for the adjoint T* of a Clifford operator T. One of the key results we present is that the bisectoriality of T can be transferred to T*. This is grounded in the fact that, for Clifford operators, the S-spectrum of the adjoint operator T* is identical to that of T. Furthermore, we demonstrate that for the existing formulations of the S-functional calculus, including bounded, unbounded, $ω$, and $H^\infty$ versions, there is a clear connection between the left functional calculus of T and the right functional calculus of T*. This explicit link between the left functional calculus of T and the right functional calculus of T* and vice versa is obtained using the function $f^\#(s):=\overline{f(\overline{s})}$. Finally, we discuss the fact that the S-spectrum, related to the invertibility of the second-order operator T^2-2s_0T+|s|^2, can actually be characterized through the invertibility of the first-order $\mathbb{R}$-linear operator T-I^Rs. |
| title | The S-functional calculus for the Clifford adjoint operator |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2508.12958 |