We describe a procedure for extending the inner measure <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _{_{\mathcal {I}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>β</mi> <mmultiscripts> <mrow /> <mi mathvariant="script">I</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> associated to an operator ideal <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {I}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">I</mi> </math></EquationSource> </InlineEquation> to a measure <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _{_{\mathfrak {J}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>β</mi> <mmultiscripts> <mrow /> <mi mathvariant="fraktur">J</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> for bounded bilinear operators T. When <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {I}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">I</mi> </math></EquationSource> </InlineEquation> is injective and closed, we show that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _{_{\mathfrak {J}}}(T)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>β</mi> <mmultiscripts> <mrow /> <mi mathvariant="fraktur">J</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$T=RS$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mi>R</mi> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> for some bounded bilinear operator S and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$R\in \mathcal {I}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>∈</mo> <mi mathvariant="script">I</mi> </mrow> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {I}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">I</mi> </math></EquationSource> </InlineEquation> satisfies the <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma _r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>-condition, then we establish a convexity inequality for the measure <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_444_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _{_{\mathfrak {J}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>β</mi> <mmultiscripts> <mrow /> <mi mathvariant="fraktur">J</mi> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> of a bilinear operator interpolated by the real method.

Interpolation of the inner measure of bilinear operators by the real method

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2025年5月27日

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Fernando Cobos,
Luz M. Fernández-Cabrera,
Antón Martínez

摘要

We describe a procedure for extending the inner measure $\beta _{_{\mathcal {I}}}$ $β_{_{I}}$ associated to an operator ideal $\mathcal {I}$ $I$ to a measure $\beta _{_{\mathfrak {J}}}$ $β_{_{J}}$ for bounded bilinear operators T. When $\mathcal {I}$ $I$ is injective and closed, we show that $\beta _{_{\mathfrak {J}}}(T)=0$ $β_{_{J}} (T) = 0$ if and only if $T=RS$ $T = R S$ for some bounded bilinear operator S and $R\in \mathcal {I}$ $R \in I$ . If $\mathcal {I}$ $I$ satisfies the $\Sigma _r$ $Σ_{r}$ -condition, then we establish a convexity inequality for the measure $\beta _{_{\mathfrak {J}}}$ $β_{_{J}}$ of a bilinear operator interpolated by the real method.