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.
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Notes
In fact, a seminorm.
This subadditivity is important for the existence of the limit (1.8).
The quantity on the right-hand side of (3.2) equals the trace of \(W^*W\) divided by \(J\). In particular, if \(W\) is unitary then \(\|W\|_\mu=1\). Below such and analogous quantities will be called the average trace of \(W^*W\).
The number \(K\in{\mathbb N}\) and the points \(y_k\) depend on \(j\), but for brevity we do not indicate this in the notation.
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Funding
The work was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00887.
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Treschev, D.V. \(\mu\)-Norm of an Operator. Proc. Steklov Inst. Math. 310, 262–290 (2020). https://doi.org/10.1134/S008154382005020X
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DOI: https://doi.org/10.1134/S008154382005020X