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<p>In this paper, under a mild condition, we prove that the vanishing of Berezin type transform on the unit circle is equivalent to the compactness of a class of radial operators on the weighted Dirichlet spaces. Furthermore, we characterize the necessary and sufficient condition for Toeplitz operator induced by the bounded radial function to be compact. Finally, we study the relationship between radial operator and the essential commutant of the Toeplitz operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(T_z\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

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On the compactness of a class of radial operators on the weighted Dirichlet spaces

  • 期刊论文

  • 发表日期：

    2026年1月25日

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  • Bo Cui,
  • Yucheng Li,
  • Hexin Zhang

摘要

In this paper, under a mild condition, we prove that the vanishing of Berezin type transform on the unit circle is equivalent to the compactness of a class of radial operators on the weighted Dirichlet spaces. Furthermore, we characterize the necessary and sufficient condition for Toeplitz operator induced by the bounded radial function to be compact. Finally, we study the relationship between radial operator and the essential commutant of the Toeplitz operator \(T_z\) T z .