Malnormal矩阵

arXiv: Functional Analysis Pub Date : 2020-09-23 DOI:10.1090/proc/15821

Garrett Mulcahy, Thomas Sinclair

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引用次数: 0

摘要

我们展示了一个算子范数有界的$\{A_n\}$的$4n \ × 4n$复矩阵的无限序列$\{A_n\}$，其对易子映射$X\映射到XA_n - A_nX$是一致有界的，它是在具有Hilbert—Schmidt范数的迹零自伴随矩阵空间上的算子。这种结构是基于量子膨胀器家族的。我们给出了这些矩阵在量子膨胀器研究中的几种潜在应用。我们提出了几个与这种矩阵有关的自然猜想和问题，并提供了数值证据。

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Malnormal matrices

We exhibit an operator norm bounded, infinite sequence $\{A_n\}$ of $4n \times 4n$ complex matrices for which the commutator map $X\mapsto XA_n - A_nX$ is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert--Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and problems related to such matrices and provide numerical evidence.

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arXiv: Functional Analysis

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