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<p>Let <i>E</i> and <i>F</i> be Banach spaces. We establish conditions under which weak sequential completeness is equivalent to reflexivity in the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{I}(E;F)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">I</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of linear operators from <i>E</i> to <i>F</i> belonging to a given operator ideal <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal I\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">I</mi> </math></EquationSource> </InlineEquation>. Additionally, we investigate similar conditions for the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal{Q}(^nE;F)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Q</mi> <msup> <mo stretchy="false">(</mo> <mi>n</mi> </msup> <mi>E</mi> <mo>;</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of <i>n</i>-homogeneous polynomials from <i>E</i> to <i>F</i> belonging to a given polynomial ideal <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal Q\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Q</mi> </math></EquationSource> </InlineEquation>. Illustrative examples are provided.</p>

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Reflexivity and Weak Sequential Completeness in Operator Ideals and Polynomial Ideals

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月5日

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  • Sergio A. Pérez,
  • Michael A. Rincón-Villamizar

摘要

Let E and F be Banach spaces. We establish conditions under which weak sequential completeness is equivalent to reflexivity in the space \(\mathcal{I}(E;F)\) I ( E ; F ) of linear operators from E to F belonging to a given operator ideal \(\mathcal I\) I . Additionally, we investigate similar conditions for the space \(\mathcal{Q}(^nE;F)\) Q ( n E ; F ) of n-homogeneous polynomials from E to F belonging to a given polynomial ideal \(\mathcal Q\) Q . Illustrative examples are provided.