Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\Omega,\Sigma,\mu)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a finite measure space and X and Y be real Banach spaces. It is shown that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T \colon L^\infty(\mu,X)\rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo lspace="0pt">:</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-order continuousoperator and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X ^{} $</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>X</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> has the Radon-Nikodym property, Y is reflexive, then Tis a nuclear operator if and only if its conjugate operator<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T ^{} :Y ^{} \rightarrow L^1(\mu,X ^{} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>T</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>:</mo> <mmultiscripts> <mi>Y</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">→</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mmultiscripts> <mi>X</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is nuclear. In this case, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\|T\| _{\rm nuc}=\|T ^{*} \| _{\rm nuc}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">‖</mo> </mrow> <mi mathvariant="normal">nuc</mi> </msub> <mo>=</mo> <mmultiscripts> <mrow> <mo stretchy="false">‖</mo> <mmultiscripts> <mi>T</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">‖</mo> </mrow> <mi mathvariant="normal">nuc</mi> <mrow /> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>.

A note on the conjugates of nuclear operators

期刊论文
发表日期：

2026年2月23日

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M. Nowak

摘要

Let $(\Omega,\Sigma,\mu)$ $(Ω, Σ, μ)$ be a finite measure space and X and Y be real Banach spaces. It is shown that if $T \colon L^\infty(\mu,X)\rightarrow Y$ $T : L^{\infty} (μ, X) \to Y$ is a $\sigma $ $σ$ -order continuousoperator and $X ^{*} $ $X^{*}$ has the Radon-Nikodym property, Y is reflexive, then Tis a nuclear operator if and only if its conjugate operator $T ^{*} :Y ^{*} \rightarrow L^1(\mu,X ^{*} )$ $T^{*} : Y^{*} \to L^{1} (μ, X^{*})$ is nuclear. In this case, $\|T\| _{\rm nuc}=\|T ^{*} \| _{\rm nuc}$ ${‖ T ‖}_{nuc} = {‖ T^{*} ‖}_{nuc}$ .