AI 中文总结
研究矩阵布洛赫空间的Köthe对偶$\mathcal{B}(D,\ell_2)^K$，反驳其与特定空间一致的猜想，揭示对对角线元素位置敏感，通过托普利兹矩阵测试恢复迹范数变体，建立行向、列向及迹范数条件的双边估计。
AI 中文摘要
我们研究矩阵布洛赫空间的Köthe对偶$\mathcal{B}(D,\ell_2)^K$。2015年有一个猜想提出该空间与由对角线算子范数确定的二元混合范数空间一致。我们通过揭示一个结构障碍来反驳该猜想：$\mathcal{B}(D,\ell_2)^K$中的成员资格对对角线内元素的位置敏感，不能仅从对角线数据检测；特别是迹范数变体也不成立。然而，针对托普利兹矩阵进行测试能准确恢复迹范数变体，这是通过解析优控证明的矩阵形式的安德森 - 希尔兹定理。最后，我们建立了双边估计：行向和列向$\ell(2,1)$条件是充分的，而迹范数二元条件是必要的；后者包含关系是严格的。
英文摘要
We study the Köthe dual $\mathcal{B}(D,\ell_2)^K$ of the matricial Bloch space. A 2015 conjecture [Publ. Math. Debrecen \textbf{87} (2015), 351--370] proposed that this space coincides with the dyadic mixed-norm space determined by the operator norms of the diagonals. We disprove the conjecture by revealing a structural obstruction: membership in $\mathcal{B}(D,\ell_2)^K$ is sensitive to the placement of the entries within the diagonals and cannot be detected from diagonal data alone; in particular the trace-norm variant fails as well. However, testing against Toeplitz matrices exactly recovers the trace-norm variant, a matricial analogue of the Anderson--Shields theorem, proved via analytic majorants. Finally, we establish two-sided estimates: row-wise and column-wise $\ell(2,1)$ conditions are sufficient, while the trace-norm dyadic condition is necessary; the latter inclusion is strict.
