数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (5): 2041-2050.doi: 10.1007/s10473-024-0524-1

• • 上一篇    下一篇

THE SCHUR TEST OF COMPACT OPERATORS*

Qijian Kang1,†, Maofa Wang2   

  1. 1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China;
    2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • 收稿日期:2022-06-20 修回日期:2024-06-13 出版日期:2024-10-25 发布日期:2024-10-22
  • 通讯作者: †Qijian Kang, E-mail: qjkang.math@whu.edu.cn
  • 作者简介:Maofa Wang, E-mail: mfwang.math@whu.edu.cn
  • 基金资助:
    NSFC (12171373).

THE SCHUR TEST OF COMPACT OPERATORS*

Qijian Kang1,†, Maofa Wang2   

  1. 1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China;
    2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2022-06-20 Revised:2024-06-13 Online:2024-10-25 Published:2024-10-22
  • Contact: †Qijian Kang, E-mail: qjkang.math@whu.edu.cn
  • About author:Maofa Wang, E-mail: mfwang.math@whu.edu.cn
  • Supported by:
    NSFC (12171373).

摘要: Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class $S_{2}$. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from $l_{p}$ into $l_{q}$.

关键词: Schur test, compact operator, infinite matrix

Abstract: Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class $S_{2}$. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from $l_{p}$ into $l_{q}$.

Key words: Schur test, compact operator, infinite matrix

中图分类号: 

  • 47B07