A characterization of differential operators in the ring of complex polynomials
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929683593953280 |
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| author | Fechner, Włodzimierz Gselmann, Eszter |
| author_facet | Fechner, Włodzimierz Gselmann, Eszter |
| contents | The paper aims to provide a full characterization of all operators $T\colon \mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot g+f\cdot T(g) \] for all $f, g\in \mathscr{P}(\mathbb{C})$. We do not assume the linearity of $T$. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_04671 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A characterization of differential operators in the ring of complex polynomials Fechner, Włodzimierz Gselmann, Eszter Classical Analysis and ODEs Primary: 47B92, Secondary: 13B25, 39B22, 47J05, 47A62, 47B38 The paper aims to provide a full characterization of all operators $T\colon \mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot g+f\cdot T(g) \] for all $f, g\in \mathscr{P}(\mathbb{C})$. We do not assume the linearity of $T$. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form. |
| title | A characterization of differential operators in the ring of complex polynomials |
| topic | Classical Analysis and ODEs Primary: 47B92, Secondary: 13B25, 39B22, 47J05, 47A62, 47B38 |
| url | https://arxiv.org/abs/2311.04671 |