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<p>This paper studies the asymptotic behavior of the entropy numbers for the natural embeddings between finite-dimensional Schatten-Lorentz spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S_{p, q}^N(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>N</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our results extend the classical entropy estimates for the natural embeddings of Lorentz sequence spaces by Prochno, Sonnleitner, and Vybíral, and those for Schatten classes by Hinrichs, Prochno, and Vybíral. We also establish asymptotics for the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(N^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>N</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-th root of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{vol}(B_{S_{p, q}^N})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>vol</mtext> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <msubsup> <mi>S</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mi>N</mi> </msubsup> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, generalizing earlier work of Doležalová and Vybíral on Lorentz sequence spaces.</p>

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Entropy Numbers of Embeddings of Lorentz-Schatten Classes

  • 期刊论文

  • 发表日期：

    2026年3月7日

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  • Yazhou Han,
  • Ning Ma,
  • Xingpeng Zhao

摘要

This paper studies the asymptotic behavior of the entropy numbers for the natural embeddings between finite-dimensional Schatten-Lorentz spaces \(S_{p, q}^N(\mathbb {R})\) S p , q N ( R ) . Our results extend the classical entropy estimates for the natural embeddings of Lorentz sequence spaces by Prochno, Sonnleitner, and Vybíral, and those for Schatten classes by Hinrichs, Prochno, and Vybíral. We also establish asymptotics for the \(N^2\) N 2 -th root of \(\textrm{vol}(B_{S_{p, q}^N})\) vol ( B S p , q N ) , generalizing earlier work of Doležalová and Vybíral on Lorentz sequence spaces.