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This article is a short survey of the notion of a collectively compact set of operators. A glimpse of basic results in the field with relevant references has been provided. As an application of this concept a theorem of Nayak, strengthening a result of Yamamoto, has been extended from matrices to compact operators. It is proved that if A is a compact operator on a complex separable Hilbert space then \(|A^n|^{\frac{1}{n}}\) converges in norm to some positive operator as n tends to infinity.

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A Survey of Collectively Compact Sets of Operators and Compact Operator Version of Nayak’s Theorem

  • 会议论文

  • 发表日期：

    2025年

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  • Neeru Bala,
  • B. V. Rajarama Bhat

摘要

This article is a short survey of the notion of a collectively compact set of operators. A glimpse of basic results in the field with relevant references has been provided. As an application of this concept a theorem of Nayak, strengthening a result of Yamamoto, has been extended from matrices to compact operators. It is proved that if A is a compact operator on a complex separable Hilbert space then \(|A^n|^{\frac{1}{n}}\) converges in norm to some positive operator as n tends to infinity.