Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1\leqslant p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, E, W be Banach spaces, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> a non-empty set, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">$F\subseteqq l_{\infty }( \Omega ,W) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>⫅</mo> <msub> <mi>l</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> a linear subspace, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$S:E\rightarrow F$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation> a positive homogeneous operator and Z a Banach space. In this paper we introduce the concept of (p, S) -summing operator from E into Z and on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$E\hspace{1.111pt}{\otimes }\hspace{1.111pt}Z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mspace width="1.111pt" /> <mo>⊗</mo> <mspace width="1.111pt" /> <mi>Z</mi> </mrow> </math></EquationSource> </InlineEquation> we define <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$d_{p^{}}^{S}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>d</mi> <mrow> <mmultiscripts> <mi>p</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </mrow> <mi>S</mi> </mmultiscripts> </math></EquationSource> </InlineEquation>, the general droite Saphar seminorm associated to the operator S. We prove that the spaces <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$\Pi _{p}^{S}( E,Z^{}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Π</mi> <mrow> <mi>p</mi> </mrow> <mi>S</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mmultiscripts> <mi>Z</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40879_2025_811_Article_IEq8.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$( E\hspace{1.111pt}{\otimes }\hspace{1.111pt}Z,d_{p^{}}^{S}) ^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mspace width="1.111pt" /> <mo>⊗</mo> <mspace width="1.111pt" /> <mi>Z</mi> <mo>,</mo> <mmultiscripts> <mi>d</mi> <mrow> <mmultiscripts> <mi>p</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </mrow> <mi>S</mi> </mmultiscripts> <mo stretchy="false">)</mo> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> are isometrically isomorphic. Various applications and illustrative examples are given.

A general version of the Saphar theorem for linear summing operators

期刊论文
出版模式：

开放获取
发表日期：

2025年2月26日

查看全文

Dumitru Popa

摘要

Let $1\leqslant p<\infty $ $1 ⩽ p < \infty$ , E, W be Banach spaces, $\Omega $ $Ω$ a non-empty set, $F\subseteqq l_{\infty }( \Omega ,W) $ $F ⫅ l_{\infty} (Ω, W)$ a linear subspace, $S:E\rightarrow F$ $S : E \to F$ a positive homogeneous operator and Z a Banach space. In this paper we introduce the concept of (p, S) -summing operator from E into Z and on $E\hspace{1.111pt}{\otimes }\hspace{1.111pt}Z$ $E \otimes Z$ we define $d_{p^{*}}^{S}$ $d_{p^{*}}^{S}$ , the general droite Saphar seminorm associated to the operator S. We prove that the spaces $\Pi _{p}^{S}( E,Z^{*}) $ $Π_{p}^{S} (E, Z^{*})$ and $( E\hspace{1.111pt}{\otimes }\hspace{1.111pt}Z,d_{p^{*}}^{S}) ^{*}$ ${(E \otimes Z, d_{p^{*}}^{S})}^{*}$ are isometrically isomorphic. Various applications and illustrative examples are given.