# The induced semigroup of Schwarz maps to the space of Hilbert-Schmidt operators

## TL;DR

This paper establishes a connection between semigroups of Schwarz maps on operator algebras and semigroups on Hilbert-Schmidt operators, providing continuity properties and a description of generators in quantum Markov semigroups.

## Contribution

It introduces a framework linking Schwarz semigroups to Hilbert-Schmidt operator semigroups and characterizes their generators under certain conditions.

## Key findings

- Existence of associated semigroups of contractions on Hilbert-Schmidt space.
- Weak* continuity implies strong continuity of the associated semigroup.
- Characterization of the generator of quantum Markov semigroups with compact resolvent.

## Abstract

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.