Multiplicative operators on analytic function spaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908695082827776 |
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| author | Behera, Kanha Liu, Junming Muthukumar, P. |
| author_facet | Behera, Kanha Liu, Junming Muthukumar, P. |
| contents | H. J. Schwartz proved in his thesis (1969) that a nonzero bounded operator on Hardy spaces $(H^p, 1\leq p\leq\infty)$ is almost multiplicative if and only if it is a composition operator. But, his proof has a gap. In this article, we show that his result is not correct for $H^\infty$ and we fill the gap for $H^p, 1\leq p<\infty.$ Further, we prove that on several classical spaces such as the Bloch space, the little Bloch space, Besov spaces $B_p$ for $p>1$, and weighted Bergman spaces an operator is almost multiplicative if and only if it is a composition operator. Finally, we give a complete characterization of those composition operators that are multiplicative with respect to the Duhamel product of analytic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05798 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative operators on analytic function spaces Behera, Kanha Liu, Junming Muthukumar, P. Functional Analysis Complex Variables Primary: 30H50, 47B33, Secondary: 30H10, 30H20, 30H25, 30H30 H. J. Schwartz proved in his thesis (1969) that a nonzero bounded operator on Hardy spaces $(H^p, 1\leq p\leq\infty)$ is almost multiplicative if and only if it is a composition operator. But, his proof has a gap. In this article, we show that his result is not correct for $H^\infty$ and we fill the gap for $H^p, 1\leq p<\infty.$ Further, we prove that on several classical spaces such as the Bloch space, the little Bloch space, Besov spaces $B_p$ for $p>1$, and weighted Bergman spaces an operator is almost multiplicative if and only if it is a composition operator. Finally, we give a complete characterization of those composition operators that are multiplicative with respect to the Duhamel product of analytic functions. |
| title | Multiplicative operators on analytic function spaces |
| topic | Functional Analysis Complex Variables Primary: 30H50, 47B33, Secondary: 30H10, 30H20, 30H25, 30H30 |
| url | https://arxiv.org/abs/2512.05798 |