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<p>In this paper, we establish some results on convergence rates for weighted sums of pairwise independent measurable operators. More precisely, we prove the convergence rate given in (<InternalRef RefID="Equ1">1.1</InternalRef>) under certain conditions for a sequence <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_636_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{ X_n, n\geqslant 1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of pairwise independent measurable operators and positive functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_636_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(l,\phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo>,</mo> <mi>ϕ</mi> </mrow> </math></EquationSource> </InlineEquation>. As applications, we investigate the convergence rates for weighted sums in noncommutative Lorentz and Marcinkiewicz spaces, and weighted additive convolution sums in free probability.</p>

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Convergence rates for weighted sums of pairwise independent operators with applications

  • 期刊论文

  • 发表日期：

    2025年2月17日

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  • Un Cig Ji,
  • Nguyen Van Quang

摘要

In this paper, we establish some results on convergence rates for weighted sums of pairwise independent measurable operators. More precisely, we prove the convergence rate given in (1.1) under certain conditions for a sequence \(\{ X_n, n\geqslant 1\}\) { X n , n ⩾ 1 } of pairwise independent measurable operators and positive functions \(l,\phi \) l , ϕ . As applications, we investigate the convergence rates for weighted sums in noncommutative Lorentz and Marcinkiewicz spaces, and weighted additive convolution sums in free probability.