Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by anon-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad [14], Stinespring [19], and Kraus [13] imply that the generator of the semigroup has the formwhere Vn and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring.We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains. Motivation and Overview of our ResultsIn this section we motivate and overview our results while precise definitions appear in section 2. In the early seventies, R.S. Ingarden and A. Kossakowski (see [11] and [12]) postulated that the time evolution of a statistically open system, in the Schrodinger picture, be given by a one-parameter semigroup of linear operators acting on the trace-class operators of a separable Hilbert space H satisfying certain conditions. In the Heisenberg picture the situation translates to a one-parameter semigroup (T t ) t≥0 acting on B(H) (the bounded operators on a Hilbert space H) where each T t is positive and σ-weakly continuous, satisfying T t (1) = 1 for all t ≥ 0, and where the map t → T t A is σ-weakly continuous for eachThe article is part of the second author's Ph.D. thesis which is prepared at the University of South Carolina under the supervision of the first author. 1 2 GEORGE ANDROULAKIS AND MATTHEW ZIEMKE A ∈ B(H). In 1976, G. Lindblad [14] added to the formulation the condition that each T t be completely positive rather than simply positive, a condition which he justified physically. Results of Stinespring [19, Theorem 4] and Arveson [1, Proposition 1.2.2] further justify this condition by proving that if an operator has a commutative domain or target space then positivity and complete positivity are equivalent. Further, under the assumption that the map t → T t is uniformly continuous, the semigroup is called a uniformly continuous QMS, the generator L of the semigroup is bounded, and Lindblad was able to write L in the form L(A) = φ(A) + G * A + AG where φ is completely positive and G ∈ B(H). Using an earlier theorem of Stinespring [19] we can then write φ in the form φ(A) = V * φ(A)V where V : H → K for some Hilbert space K and π : B(H) → B(K) is a normal representation. Further, a theorem due to Kraus [13] lets us write π in the form π(A) = ∞ n=1 W * n AW n where W n : K → H is a bounded linear operator. When we combine Stinespring's and Kraus' results we are then able to write φ in th...

Abstract. We study semigroups of bounded operators on a Banach space such that the members of the semigroup are continuous with respect to various weak topologies and we give sufficient conditions for the generator of the semigroup to be closed with respect to the topologies involved. The proofs of these results use the Laplace transforms of the semigroup. Thus we first give sufficient conditions for Pettis integrability of vector valued functions with respect to scalar measures. MotivationThe motivation for this article comes from the area of quantum Markov semigroups (QMSs) and the well known problem of describing the form of the generator of a general QMS. A semigroup on a Banach space X is a family (T t ) t≥0 of bounded operators on X satisfying T 0 = 1 and T t T s = T t+s for t, s ≥ 0.A QMS on a von Neumann algebra M is a semigroup (T t ) t≥0 on M such that each T t is completely positive, σ-weakly-σ-weakly continuous, and satisfies T t (1) = 1, and the map t → T t A is σ-weakly continuous for each A ∈ M. This definition was finalized in 1976 by Lindblad [17] who was able to describe the form of the generator of the semigroup in the case that the von Neumann algebra M is semifinite and the map t → T t is norm continuous.

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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, or under the assumption that the generator of the minimal unitary dilation of the associated semigroup of contractions is compact.