Geodesic
Entropy of a unitary operator on $L^2(\pmb{\mathbb{T}}^n)$
Sbornik. Mathematics, Tome 213 (2022) no. 7, pp. 925-980 Cet article a éte moissonné depuis la source Math-Net.Ru

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The concept of the $\mu$-norm of an operator, introduced in [28], is investigated. The focus is on operators on $L^2(\mathbb{T}^n)$, where $\mathbb{T}^n$ is the $n$-torus (the case when $n=1$ was previously considered in [29]). The main source of motivation for the study was the role of the $\mu$-norm as a key tool in constructing a quantum analogue of metric entropy, namely, the entropy of a unitary operator on $L^2(\mathcal X,\mu)$, where $(\mathcal X,\mu)$ is a probability space. The properties of the $\mu$-norm are presented and some ways to calculate it for various classes of operators on $L^2(\mathbb{T}^n)$ are described. The construction of entropy proposed in [28] is modified to make it subadditive and monotone with respect to partitions of $\mathcal X$. Examples of the calculation of entropy are presented for some classes of operators on $L^2(\mathbb{T}^n)$. Bibliography: 29 titles.
Keywords: Hilbert space, $\mu$-norm of an operator, metric entropy, Schrödinger propagator, operator theory. 
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K. A. Afonin; D. V. Treschev. Entropy of a unitary operator on $L^2(\pmb{\mathbb{T}}^n)$. Sbornik. Mathematics, Tome 213 (2022) no. 7, pp. 925-980. http://geodesic.mathdoc.fr/item/SM_2022_213_7_a1/

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