AI 中文总结
该论文研究投影范数可达性，证明核范数可达算子在积分算子空间的$w^*$-稠密性，推导II-多面体空间上核算子的范数可达性结论，并应用于解决积分投影范数可达张量的开放问题。
AI 中文摘要
我们证明，核范数可达算子（对应多项式）在积分算子（对应多项式）空间中总是$w^*$-稠密的。此外，若预对偶空间不含$\boldsymbol{\textit{ℓ}_1}$的任何同构副本，则该稠密性在范数意义下成立。我们还证明，存在自反空间，其投影张量积中的投影范数可达元素集合与整个投影张量积（此处也为自反空间）不重合。接下来，我们证明若$Y$是II-多面体空间，则从任意空间$X$到$Y^*$的每个核算子都可达其核范数。作为推论，若$X^*$或$Y^*$具有逼近性质，则从$X^*$到$Y^{**}$的范数可达算子集合是稠密的。最后，我们研究投影张量积的一个自然子空间的近性结果，并将其应用于积分投影范数可达张量，解决了一个提出的开放问题。
英文摘要
We show that nuclear norm-attaining operators (resp.\ polynomials) are always $w^*$-dense in the space of integral operators (resp.\ polynomials). Besides, the denseness is in norm if the predual space does not contain any isomorphic copy of $\ell_1$. We also show that there are reflexive spaces for which the set of projective norm-attaining elements does not coincide with the whole projective tensor product (which is indeed also reflexive here). Next, we show that if $Y$ is a II-polyhedral space, then every nuclear operator from an arbitrary space $X$ to $Y^*$ attains its nuclear norm. As a consequence, if $X^*$ or $Y^*$ has the approximation property, then the set of norm-attaining operators from $X^*$ to $Y^{**}$ is dense. Finally, we study proximinality results of a natural subspace of the projective tensor product and obtain an application to integral projective norm-attaining tensors which solves a proposed open question.
Comments20 pages
