Abstract

Let \((\mathcal X,\mu)\) be a measure space. For any measurable set \(Y\subset\mathcal X\) let \(\mathbf 1_Y: \mathcal X\to{\mathbb R}\) be the indicator of \(Y\) and let \(\pi_Y^{}\) be the orthogonal projection \(L^2(\mathcal X)\ni f\mapsto{\pi_Y^{}}_{} f = \mathbf 1_Y f\). For any bounded operator \(W\) on \(L^2(\mathcal X,\mu)\) we define its \(\mu\)-norm \(\|W\|_\mu = \inf_\chi\sqrt{\sum\mu(Y_j)\|W\pi_Y^{}\|^2}\), where the infimum is taken over all measurable partitions \(\chi=\{Y_1,\dots,Y_J\}\) of \(\mathcal X\). We present some properties of the \(\mu\)-norm and some computations. Our main motivation is the problem of constructing a quantum entropy.

中文翻译：

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$$ \ mu $$-运营商范数

摘要

令\（（\ mathcal X，\ mu）\）为度量空间。对于任何可测量的集合\（Y \ subset \ mathcal X \），令\（\ mathbf 1_Y：\ mathcal X \ to {\ mathbb R} \）是\（Y \）的指标，并令\（\ pi_Y ^ { } \）为正交投影\（L ^ 2（\ mathcal X）\ ni f \ mapsto {\ pi_Y ^ {}} _ {} f = \ mathbf 1_Y f \）。对于\（L ^ 2（\ mathcal X，\ mu）\）上的任何有界算子\（W \），我们定义其\（\ mu \）- norm \（\ | W \ | __mu = \ inf_ \ chi \ sqrt {\ sum \ mu（Y_j）\ | W \ pi_Y ^ {} \ | ^ 2} \），在所有可测量分区\（\ chi = \ {Y_1，\ dots，Y_J \ } \）的\（\ mathcal X \）。我们介绍了\（\ mu \）- norm的一些属性和一些计算。我们的主要动机是构造量子熵的问题。