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<p>We study <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ell _\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>-valued and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\ell _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-valued noncommutative symmetric spaces. We define related Hardy spaces of noncommutative martingales, show duality and interpolation properties of these spaces. Using these results, we give a direct proof of algebraic atomic decompositions and Davis-type decompositions within the context of Hardy spaces affiliated with separable noncommutative symmetric spaces, which constitute an interpolation of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((L_p, L_q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>L</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> pair for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(1&lt; p \le q &lt; 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> <mo>&lt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We also derive a version of Doob maximal inequality associated with symmetric spaces of measurable operators.</p>

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On \(\ell _\infty \)-valued symmetric spaces of noncommutative martingales

  • 期刊论文

  • 发表日期：

    2026年7月22日

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  • Turdebek N. Bekjan

摘要

We study \(\ell _\infty \) ℓ ∞ -valued and \(\ell _1\) ℓ 1 -valued noncommutative symmetric spaces. We define related Hardy spaces of noncommutative martingales, show duality and interpolation properties of these spaces. Using these results, we give a direct proof of algebraic atomic decompositions and Davis-type decompositions within the context of Hardy spaces affiliated with separable noncommutative symmetric spaces, which constitute an interpolation of the \((L_p, L_q)\) ( L p , L q ) pair for \(1< p \le q < 2\) 1 < p ≤ q < 2 . We also derive a version of Doob maximal inequality associated with symmetric spaces of measurable operators.