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August, 2021 Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions
Jiale Chen, Maofa Wang
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Taiwanese J. Math. 25(4): 757-774 (August, 2021). DOI: 10.11650/tjm/201208

Abstract

For a fixed nonnegative integer $m$, an analytic map $\varphi$ and an analytic function $\psi$, the generalized integration operator $I^{(m)}_{\varphi,\psi}$ is defined by \[ I^{(m)}_{\varphi,\psi} f(z) = \int_0^z f^{(m)}(\varphi(\zeta)) \psi(\zeta) \, d\zeta \] for $X$-valued analytic function $f$, where $X$ is a Banach space. Some estimates for the norm of the operator $I^{(m)}_{\varphi,\psi} \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ are obtained. In particular, it is shown that the Volterra operator $J_b \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ is bounded if and only if $J_b \colon A^2_{\alpha} \to A^2_{\alpha}$ is in the Schatten class $S_p(A^2_{\alpha})$ for $2 \leq p \lt \infty$ and $\alpha \gt -1$. Some corresponding results are established for $X$-valued Hardy spaces and $X$-valued Fock spaces.

Funding Statement

This work was partially supported by NSFC (No. 11771340) of China.

Acknowledgments

The authors thank the referees who provided numerous valuable comments that improved the overall presentation of the paper and informed us the relevant reference [1].

Citation

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Jiale Chen. Maofa Wang. "Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions." Taiwanese J. Math. 25 (4) 757 - 774, August, 2021. https://doi.org/10.11650/tjm/201208

Information

Received: 21 October 2020; Revised: 8 December 2020; Accepted: 24 December 2020; Published: August, 2021
First available in Project Euclid: 30 December 2020

Digital Object Identifier: 10.11650/tjm/201208

Subjects:
Primary: 47B38
Secondary: 46E40

Keywords: Generalized integration operator , vector-valued Bergman space , vector-valued Fock spaces , vector-valued Hardy space

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 4 • August, 2021
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