数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (1): 173-188.doi: 10.1007/s10473-024-0109-z

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SOME PROPERTIES OF THE INTEGRATION OPERATORS ON THE SPACES ${F(p,q,s)}$*

Jiale Chen   

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, China
  • 收稿日期:2022-10-08 修回日期:2023-07-13 出版日期:2024-02-25 发布日期:2024-02-27
  • 作者简介:Jiale Chen, E-mail: jialechen@snnu.edu.cn
  • 基金资助:
    Fundamental Research Funds for the Central Universities (GK202207018) of China.

SOME PROPERTIES OF THE INTEGRATION OPERATORS ON THE SPACES ${F(p,q,s)}$*

Jiale Chen   

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, China
  • Received:2022-10-08 Revised:2023-07-13 Online:2024-02-25 Published:2024-02-27
  • About author:Jiale Chen, E-mail: jialechen@snnu.edu.cn
  • Supported by:
    Fundamental Research Funds for the Central Universities (GK202207018) of China.

摘要: We study the closed range property and the strict singularity of integration operators acting on the spaces $F(p,p\alpha-2,s)$. We completely characterize the closed range property of the Volterra companion operator $I_g$ on $F(p,p\alpha-2,s)$, which generalizes the existing results and answers a question raised in [A. Anderson, Integral Equations Operator Theory, 69 (2011), no. 1, 87-99]. For the Volterra operator $J_g$, we show that, for $0<\alpha\leq1$, $J_g$ never has a closed range on $F(p,p\alpha-2,s)$. We then prove that the notions of compactness, weak compactness and strict singularity coincide in the case of $J_g$ acting on $F(p,p-2,s)$.

关键词: integration operator, closed range property, strict singularity

Abstract: We study the closed range property and the strict singularity of integration operators acting on the spaces $F(p,p\alpha-2,s)$. We completely characterize the closed range property of the Volterra companion operator $I_g$ on $F(p,p\alpha-2,s)$, which generalizes the existing results and answers a question raised in [A. Anderson, Integral Equations Operator Theory, 69 (2011), no. 1, 87-99]. For the Volterra operator $J_g$, we show that, for $0<\alpha\leq1$, $J_g$ never has a closed range on $F(p,p\alpha-2,s)$. We then prove that the notions of compactness, weak compactness and strict singularity coincide in the case of $J_g$ acting on $F(p,p-2,s)$.

Key words: integration operator, closed range property, strict singularity

中图分类号: 

  • 47G10