Generators of detailed balance quantum Markov semigroups

For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state, we defne an adjoint T' with respect to the scalar product determined by the invariant state. In this paper, we solve the open problems of characterizing adjoints T' that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators in the Gorini-Kossakowski-Sudarshan-Lindblad representation of the generator of T .