Weighted inversion of vector valued Dirichlet series
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912595992117248 |
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| author | Dabhi, Prakash A. Solanki, Karishman B. |
| author_facet | Dabhi, Prakash A. Solanki, Karishman B. |
| contents | Let $Λ\subset[0,\infty)$ be an additive semigroup with $0\inΛ$, $ω$ be an admissible weight on $Λ$, $\mathcal A$ be a unital Banach algebra, and let $f(s)=\sum_{λ\inΛ} f_λe^{-λs}$ for $s\in\mathcal{H}=\{j+it\in\mathbb{C}:j\geq0\}$ be a generalized Dirichlet series satisfying $\|f\|_ω=\sum_{λ\inΛ}\|f_λ\|ω(λ)<\infty,$ where $f_λ\in\mathcal{A}$ for all $λ\inΛ$. We take $\mathcal{A}$ to be a commutative complex Banach algebra (with $Λ=\log\mathbb{N}$) and $M_d(\mathcal{X})$ - the Banach algebra of $d \times d$ matrices having entries from $\mathcal{X}$, where $\mathcal{X}$ is either the complex plane or the real algebra of bicomplex numbers or quaternions, and show that $f$ is invertible if and only if the closure of the image of $f$ is contained in the set of all invertible elements of $\mathcal{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted inversion of vector valued Dirichlet series Dabhi, Prakash A. Solanki, Karishman B. Functional Analysis Spectral Theory Primary 11M41, Secondary 46H99, 15B33 Let $Λ\subset[0,\infty)$ be an additive semigroup with $0\inΛ$, $ω$ be an admissible weight on $Λ$, $\mathcal A$ be a unital Banach algebra, and let $f(s)=\sum_{λ\inΛ} f_λe^{-λs}$ for $s\in\mathcal{H}=\{j+it\in\mathbb{C}:j\geq0\}$ be a generalized Dirichlet series satisfying $\|f\|_ω=\sum_{λ\inΛ}\|f_λ\|ω(λ)<\infty,$ where $f_λ\in\mathcal{A}$ for all $λ\inΛ$. We take $\mathcal{A}$ to be a commutative complex Banach algebra (with $Λ=\log\mathbb{N}$) and $M_d(\mathcal{X})$ - the Banach algebra of $d \times d$ matrices having entries from $\mathcal{X}$, where $\mathcal{X}$ is either the complex plane or the real algebra of bicomplex numbers or quaternions, and show that $f$ is invertible if and only if the closure of the image of $f$ is contained in the set of all invertible elements of $\mathcal{A}$. |
| title | Weighted inversion of vector valued Dirichlet series |
| topic | Functional Analysis Spectral Theory Primary 11M41, Secondary 46H99, 15B33 |
| url | https://arxiv.org/abs/2509.16658 |