Generators of Detailed Balance Quantum Markov Semigroups

For a quantum Markov semigroup $\T$ on the algebra $\B$ with a faithful invariant state $ρ$, we can define an adjoint $\widetilde{\T}$ with respect to the scalar product determined by $ρ$. In this paper, we solve the open problems of characterising adjoints $\widetilde{\T}$ that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators $H,L_k$ in the Gorini Kossakowski Sudarshan Lindblad representation $\Ll(x)=i[H,x] - {1/2}\sum_k(L^*_kL_k x-2L^*_kxL_k + xL^*_kL_k)$ of the generator of $\T$. We study the adjoint semigroup with respect to both scalar products $<a,b> = \tr(ρa^* b)$ and $<a,b> = \tr(ρ^{1/2} a^* ρ^{1/2}b)$.

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