A several variables Kowalski-S\lodkowski theorem for topological spaces
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| Format: | Preprint |
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2024
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| author | Jaikishan Lata, Sneh Singh, Dinesh |
| author_facet | Jaikishan Lata, Sneh Singh, Dinesh |
| contents | In this paper, we provide a version of the classical result of Kowalski and Słodkowski that generalizes the famous Gleason-Kahane-$\dot{\rm Z}$elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on $\mathcal{A}$-valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and Słodkowski, as a composition of a multiplicative linear functional on $\mathcal{A}$ and a point evaluation on the polynomials, where $\mathcal{A}$ is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and Słodkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_04131 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A several variables Kowalski-S\lodkowski theorem for topological spaces Jaikishan Lata, Sneh Singh, Dinesh Functional Analysis Primary 32A65, Secondary 32A35, 46E40, 47B38 In this paper, we provide a version of the classical result of Kowalski and Słodkowski that generalizes the famous Gleason-Kahane-$\dot{\rm Z}$elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on $\mathcal{A}$-valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and Słodkowski, as a composition of a multiplicative linear functional on $\mathcal{A}$ and a point evaluation on the polynomials, where $\mathcal{A}$ is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and Słodkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces. |
| title | A several variables Kowalski-S\lodkowski theorem for topological spaces |
| topic | Functional Analysis Primary 32A65, Secondary 32A35, 46E40, 47B38 |
| url | https://arxiv.org/abs/2410.04131 |