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arXiv · 1903.10261

Strict singularity of Volterra type operators on Hardy spaces

Qingze Lin·Junming Liu·Yutian Wu
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Abstract

In this paper, we first characterize the boundedness and compactness of Volterra type operator $S_gf(z) = \int_0^z f'(\zeta)g(\zeta)d\zeta, \ z \in \mathbb{D},$ defined on Hardy spaces $H^p, \, 0< p <\infty$. The spectrum of $S_g$ is also obtained. Then we prove that $S_g$ fixes an isomorphic copy of $\ell^p$ and an isomorphic copy of $\ell^2$ if the operator $S_g$ is not compact on $H^p (1\leq p<\infty)$. In particular, this implies that the strict singularity of the operator $S_g$ coincides with the compactness of the operator $S_g$ on $H^p$. At last, we post an open question for further study.

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Qingze Lin, Junming Liu, Yutian Wu. 2019-03-25. Strict singularity of Volterra type operators on Hardy spaces. https://arxiv.org/abs/1903.10261

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