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The essential norm of the integral type operators

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Abstract

On the basis of the characterizations of the boundedness and compactness of the Volterra type operator \(I_{g, \varphi }\) from mixed-norm spaces \(H(p,\, q,\, \phi )\) to Zygmund spaces \( \mathcal {Z}\), the authors provide a function-theoretic estimate for the essential norm of Volterra type operator \(I_{g, \varphi }\) by means of the definition of the essential norm of an operator and the properties of the analytic function. An estimate for the essential norm of the generalized integration operator

$$\begin{aligned} I^{(n)}_{g, \varphi }f(z)=\int ^{z}_{0}f^{(n)}(\varphi (\xi ))g(\xi )d\xi , \ z\in \mathbb {D}, \end{aligned}$$

from Bloch-type spaces to F(p, q, s) spaces is also obtained.

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Acknowledgements

The authors are very grateful to anonymous referees and editors for their valuable and detailed suggestions and insightful comments to improve the original manuscript. The project is supported by the Natural Science Foundation of China (Grant nos. 11771184, 11771188) and the Foundation Research Project of Jiangsu Province of China (no. BK20161158).

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Authors and Affiliations

  1. College of Sciences, Nanjing Agricultural University, Nanjing, 210095, China

    Xiaoman Liu

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Yongmin Liu & Lina Xia

  3. School of Mathematics and Physics Science, Xuzhou Institute of Technology, Xuzhou, 221018, China

    Yanyan Yu

Authors
  1. Xiaoman Liu
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  2. Yongmin Liu
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  3. Lina Xia
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  4. Yanyan Yu
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All authors contributed equally to the writing of this paper. They also read and approved the final manuscript.

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Correspondence to Yongmin Liu.

Additional information

Communicated by Orr Moshe Shalit.

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Liu, X., Liu, Y., Xia, L. et al. The essential norm of the integral type operators. Banach J. Math. Anal. 14, 181–202 (2020). https://doi.org/10.1007/s43037-019-00028-y

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