<p>In this paper we continue the investigation of classes of vector-valued sequences that are represented by Banach operator ideals. By a procedure we mean a correspondence <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_421_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(X \mapsto X^{\textrm{new}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>↦</mo> <msup> <mi>X</mi> <mtext>new</mtext> </msup> </mrow> </math></EquationSource> </InlineEquation> that assigns a sequence class <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_421_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(X^{\textrm{new}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mtext>new</mtext> </msup> </math></EquationSource> </InlineEquation> built upon a given sequence class <i>X</i>. The general question is whether or not <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_421_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(X^{\textrm{new}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mtext>new</mtext> </msup> </math></EquationSource> </InlineEquation> is ideal-representable whenever <i>X</i> is. We address this question for three already studied procedures, namely, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_421_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(X \mapsto X^{\textrm{u}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>↦</mo> <msup> <mi>X</mi> <mtext>u</mtext> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_421_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">\(X \mapsto X^{\textrm{dual}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>↦</mo> <msup> <mi>X</mi> <mtext>dual</mtext> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_421_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(X \mapsto X^{\textrm{fd}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>↦</mo> <msup> <mi>X</mi> <mtext>fd</mtext> </msup> </mrow> </math></EquationSource> </InlineEquation>. Applications of the solutions of these problem will provide new concrete examples of ideal-representable sequence classes.</p>

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Representation of sequence classes by operator ideals: Part II

  • Geraldo Botelho,
  • Ariel S. Santiago

摘要

In this paper we continue the investigation of classes of vector-valued sequences that are represented by Banach operator ideals. By a procedure we mean a correspondence \(X \mapsto X^{\textrm{new}}\) X X new that assigns a sequence class \(X^{\textrm{new}}\) X new built upon a given sequence class X. The general question is whether or not \(X^{\textrm{new}}\) X new is ideal-representable whenever X is. We address this question for three already studied procedures, namely, \(X \mapsto X^{\textrm{u}}\) X X u , \(X \mapsto X^{\textrm{dual}}\) X X dual and \(X \mapsto X^{\textrm{fd}}\) X X fd . Applications of the solutions of these problem will provide new concrete examples of ideal-representable sequence classes.