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arXiv · 2405.13662

Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights

Ruishen Qian·Fanglei Wu·Hasi Wulan
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Abstract

Suppose $(C_t)_{t\geq0}$ is the composition semigroup induced by a one-parameter semigroup $(\varphi_t)_{t\geq0}$ of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of $(C_t)_{t\geq0}$ acting on the weighted Bergman space induced by doubling weights, provided $(\varphi_t)_{t\geq0}$ is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of $(C_t)_{t\geq0}$ also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of $(C_t)_{t\geq0}$.

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Ruishen Qian, Fanglei Wu, Hasi Wulan. 2024-05-22. Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights. https://arxiv.org/abs/2405.13662

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