article · Proceedings of the Steklov Institute of Mathematics · 2020
$$\mu$$-Norm of an Operator
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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.
Cite this paper
Treschev, D. V. (2020). $$\mu$$-Norm of an Operator. Proceedings of the Steklov Institute of Mathematics, 310(1), 262–290. https://doi.org/10.1134/s008154382005020x
Treschev, Dmitrii Valer'evich. “$$\mu$$-Norm of an Operator.” Proceedings of the Steklov Institute of Mathematics, vol. 310, no. 1, 2020, pp. 262–90, https://doi.org/10.1134/s008154382005020x.
Treschev, Dmitrii Valer'evich. 2020. “$$\mu$$-Norm of an Operator.” Proceedings of the Steklov Institute of Mathematics 310 (1): 262–90. https://doi.org/10.1134/s008154382005020x.
Treschev, D.V. (2020) ‘$$\mu$$-Norm of an Operator’, Proceedings of the Steklov Institute of Mathematics, 310(1), pp. 262–290. Available at: https://doi.org/10.1134/s008154382005020x.
@article{treschev2020norm,
title = {{$$\mu$$-Norm of an Operator}},
author = {Treschev, Dmitrii Valer'evich},
journal = {Proceedings of the Steklov Institute of Mathematics},
volume = {310},
number = {1},
pages = {262--290},
year = {2020},
doi = {10.1134/s008154382005020x}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.