By investigating the relationships between M-ideals of compact operators, the adjoint compact perturbation property, and the weak maximizing property, we generalize or strengthen some known facts related to these concepts. Notably, we show that if Y is an infinite dimensional Banach space such that the space of compact operators from <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1718_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> to Y is an M-ideal in the space of bounded linear operators, then Y has the local diameter 2 property. Additionally, for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1718_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, a pair <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1718_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$(\ell _p,Y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has the weak maximizing property whenever the space of compact operators from <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1718_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> to Y is an M-ideal in the space of bounded linear operators.

M-ideals of compact operators and norm attaining operators

期刊论文
发表日期：

2025年3月23日

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Manwook Han,
Sun Kwang Kim

摘要

By investigating the relationships between M-ideals of compact operators, the adjoint compact perturbation property, and the weak maximizing property, we generalize or strengthen some known facts related to these concepts. Notably, we show that if Y is an infinite dimensional Banach space such that the space of compact operators from $\ell _1$ $ℓ_{1}$ to Y is an M-ideal in the space of bounded linear operators, then Y has the local diameter 2 property. Additionally, for $1<p<\infty $ $1 < p < \infty$ , a pair $(\ell _p,Y)$ $(ℓ_{p}, Y)$ has the weak maximizing property whenever the space of compact operators from $\ell _p$ $ℓ_{p}$ to Y is an M-ideal in the space of bounded linear operators.