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<p>In this note, we study a question introduced by Bourin [1] and extend the conclusion from [2] to the case of operators on noncommutative fully symmetric spaces. The conclusion is as follows. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0\leq x,y\in E(\cal{M})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="script">M</mi> </mrow> <mo class="MJX-tex-caligraphic" mathvariant="script" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, If <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t \in [0,{1 \over 4}] \cup [{3 \over 4},1]\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">]</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mrow> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </math></EquationSource> </InlineEquation>, then</p><p><Equation ID="Equa"> <EquationSource Format="TEX">\(\Vert {x^t}{y^{1 - t}} + {y^t}{x^{1 - t}} {\Vert_{E({\cal M})}} \le {2^{2t - {3 \over 2}}}\Vert x + y {\Vert_{E({\cal M})}}.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <mrow> <msup> <mi>x</mi> <mi>t</mi> </msup> </mrow> <mrow> <msup> <mi>y</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>y</mi> <mi>t</mi> </msup> </mrow> <mrow> <msup> <mi>x</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </msup> </mrow> <mrow> <msub> <mo>∥</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="script">M</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mo>≤</mo> <mrow> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>∥</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mrow> <msub> <mo>∥</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="script">M</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mo>.</mo> </math></EquationSource> </Equation></p>

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A norm inequality on noncommutative symmetric spaces related to a question of Bourin

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Jinchen Liu,
  • Kan He,
  • Xingpeng Zhao

摘要

In this note, we study a question introduced by Bourin [1] and extend the conclusion from [2] to the case of operators on noncommutative fully symmetric spaces. The conclusion is as follows. Let \(0\leq x,y\in E(\cal{M})\) 0 ≤ x , y ∈ E ( M ) , If \(t \in [0,{1 \over 4}] \cup [{3 \over 4},1]\) t ∈ [ 0 , 1 4 ] ∪ [ 3 4 , 1 ] , then

\(\Vert {x^t}{y^{1 - t}} + {y^t}{x^{1 - t}} {\Vert_{E({\cal M})}} \le {2^{2t - {3 \over 2}}}\Vert x + y {\Vert_{E({\cal M})}}.\) ∥ x t y 1 − t + y t x 1 − t ∥ E ( M ) ≤ 2 2 t − 3 2 ∥ x + y ∥ E ( M ) .