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
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
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