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<p>A classical result of Calkin [Ann. of Math. (2) 42 (1941), pp.&#xa0;839–873] says that an inner derivation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1723_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="169" /> </InlineMediaObject> <EquationSource Format="TEX">\(S\mapsto [T,S] = TS-ST\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>↦</mo> <mo stretchy="false">[</mo> <mi>T</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>T</mi> <mi>S</mi> <mo>-</mo> <mi>S</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> maps the algebra of bounded operators on a Hilbert space into the ideal of compact operators if and only if <i>T</i> is a compact perturbation of the multiplication by a scalar. In general, an analogous statement fails for operators on Banach spaces. To complement Calkin’s result, we characterize Volterra-type inner derivations on Hardy spaces using generalized area operators and compact intertwining relations for Volterra and composition operators. Further, we characterize the compact intertwining relations for multiplication and composition operators between Hardy and Bergman spaces.</p>

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Volterra-type inner derivations on Hardy spaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月4日

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  • H. Arroussi,
  • C. Tong,
  • J. A. Virtanen,
  • Z. Yuan

摘要

A classical result of Calkin [Ann. of Math. (2) 42 (1941), pp. 839–873] says that an inner derivation \(S\mapsto [T,S] = TS-ST\) S ↦ [ T , S ] = T S - S T maps the algebra of bounded operators on a Hilbert space into the ideal of compact operators if and only if T is a compact perturbation of the multiplication by a scalar. In general, an analogous statement fails for operators on Banach spaces. To complement Calkin’s result, we characterize Volterra-type inner derivations on Hardy spaces using generalized area operators and compact intertwining relations for Volterra and composition operators. Further, we characterize the compact intertwining relations for multiplication and composition operators between Hardy and Bergman spaces.