We prove that for every semigroup of Schwarz maps on the von Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak ∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe the form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear operators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent.

中文翻译：

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Schwarz 映射到 Hilbert-Schmidt 算子空间的诱导半群

我们证明，对于 Hilbert 空间上所有有界线性算子的冯诺依曼代数上的 Schwarz 映射的每个半群，该空间具有一个次不变的忠实正常状态，在 Hilbert 空间的 Hilbert-Schmidt 算子的空间上存在一个相关的收缩半群。此外，我们证明如果原始半群是弱 ∗ 连续的，那么关联的半群是强连续的。我们在 Hilbert 空间的有界算子上引入了半群的扩展生成器的概念，关于 Hilbert 空间的正交基。