关于其伴随算子或二次伴随算子达到范数的算子
On operators whose adjoints or second adjoints attain their norms
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中文总结 AI 辅助
针对无限维巴拿赫空间上有界线性算子的范数达到问题，证明了$c_0$和$\boldsymbol{\textit{ℓ}}_1$上所有算子的二次伴随都达到范数，还得到$c$相关空间的对应刻画及Holub--Mujica型定理等结果。
中文摘要 AI 辅助
一个长期存在的开放问题是：是否存在一个无限维巴拿赫空间，使得其上的每个有界线性算子都能达到其范数。我们用$\text{NA}_1(X,Y)$表示其伴随算子达到范数的算子类，$\text{NA}_2(X,Y)$表示其二次伴随算子达到范数的算子类，证明了$\text{NA}_2(c_0,c_0)=\text{mathcal{L}}(c_0,c_0)$和$\text{NA}_2(\boldsymbol{\textit{ℓ}}_1,\boldsymbol{\textit{ℓ}}_1)=\text{mathcal{L}}(\boldsymbol{\textit{ℓ}}_1,\boldsymbol{\textit{ℓ}}_1)$，据我们所知，这提供了首批已知的无限维空间，其上的每个算子都具有达到范数的二次伴随算子。在$c_0$相关结果的基础上，我们对由$c$的超平面给出的$\boldsymbol{\textit{ℓ}}_1$预对偶的自然族内的这类等式开展系统研究，在此背景下获得了完整刻画。特别地，我们证明$\text{NA}_2(c,c)\neq\text{mathcal{L}}(c,c)$，而$\text{NA}_3(c,c)=\text{mathcal{L}}(c,c)$。我们还为$\text{NA}_1$和$\text{NA}_2$类建立了Holub--Mujica型定理，更准确地说，在适当的可分性和逼近性质假设下，等式$\text{mathcal{L}}(X,Y)=\text{NA}_1(X,Y)$迫使从$X$到$Y$的每个算子都是紧算子，而$\text{mathcal{L}}(X,Y)=\text{NA}_2(X,Y)$迫使从$X$到$Y$的每个弱紧算子都是紧算子。最后，我们强化了Ostrovskii的构造，证明每个无限维巴拿赫空间都存在等价范数和一个投影，其二次伴随算子不达到其范数。
英文摘要
A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With $\mathrm{NA}_1(X,Y)$ and $\mathrm{NA}_2(X,Y)$ denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that \[ \mathrm{NA}_2(c_0,c_0)=\mathcal{L}(c_0,c_0) \qquad\text{and}\qquad \mathrm{NA}_2(\ell_1,\ell_1)=\mathcal{L}(\ell_1,\ell_1), \] providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for $c_0$, we undertake a systematic study of this equality within a natural family of $\ell_1$-preduals given by hyperplanes of $c$, obtaining a complete characterization in this setting. In particular, we prove that \[ \mathrm{NA}_2(c,c)\neq \mathcal{L}(c,c), \qquad\text{whereas}\qquad \mathrm{NA}_3(c,c)=\mathcal{L}(c,c). \] We also establish Holub--Mujica-type theorems for the classes $\mathrm{NA}_1$ and $\mathrm{NA}_2$. More precisely, under suitable separability and approximation property assumptions, the identity $\mathcal L(X,Y)=\mathrm{NA}_1(X,Y)$ forces every operator from $X$ into $Y$ to be compact, whereas $\mathcal L(X,Y)=\mathrm{NA}_2(X,Y)$ forces every weakly compact operator from $X$ into $Y$ to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.
发表机构
- Czech Technical University in Prague（布拉格捷克理工大学）
- Hanyang University（汉阳大学）
- University of Granada（格拉纳达大学）
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