On the dynamics of a semigroup and its relation with the Riemann Hypothesis
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| Format: | Preprint |
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2025
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| _version_ | 1866912976911466496 |
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| author | Álvarez, Carlos F. Manzur, Juan |
| author_facet | Álvarez, Carlos F. Manzur, Juan |
| contents | The semigroup of weighted composition operators $(W_n)_{n\in \mathbb{N}}$, defined by $$W_nf(z)=(1+z+\cdots +z^n)f(z^n),$$ acts on the classical Hardy-Hilbert space $H^{2}(\mathbb{D})$, and exhibits intriguing connections with both the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP). In this paper, we prove that the adjoint operators $W^{\ast}_{n}$, for $n\geq 2$, are Devaney chaotic, frequently hypercyclic and mixing. In particular, these operators are hypercyclic and discuss connections with the RH and invariant subspaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_03625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the dynamics of a semigroup and its relation with the Riemann Hypothesis Álvarez, Carlos F. Manzur, Juan Functional Analysis 47A16, 47B33 (Primary) 46E20 (Secondary) The semigroup of weighted composition operators $(W_n)_{n\in \mathbb{N}}$, defined by $$W_nf(z)=(1+z+\cdots +z^n)f(z^n),$$ acts on the classical Hardy-Hilbert space $H^{2}(\mathbb{D})$, and exhibits intriguing connections with both the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP). In this paper, we prove that the adjoint operators $W^{\ast}_{n}$, for $n\geq 2$, are Devaney chaotic, frequently hypercyclic and mixing. In particular, these operators are hypercyclic and discuss connections with the RH and invariant subspaces. |
| title | On the dynamics of a semigroup and its relation with the Riemann Hypothesis |
| topic | Functional Analysis 47A16, 47B33 (Primary) 46E20 (Secondary) |
| url | https://arxiv.org/abs/2507.03625 |