Geodesic
$\mu $-Norm of an Operator
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 280-308.

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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 $-norm and some computations. Our main motivation is the problem of constructing a quantum entropy. 
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     author = {D. V. Treschev},
     title = {$\mu ${-Norm} of an {Operator}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {280--308},
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     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2020_310_a19/}
}
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D. V. Treschev. $\mu $-Norm of an Operator. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 280-308. http://geodesic.mathdoc.fr/item/TM_2020_310_a19/

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