Generators of Detailed Balance Quantum Markov Semigroups

For a quantum Markov semigroup $\text{T}$ on the algebra $\text{B}$ with a faithful invariant state $\rho$, we can define an adjoint $\tilde{\text{T}}$ with respect to the scalar product determined by $\rho$. In this paper, we solve the open problems of characterising adjoints $\tilde{\text{T}}$ 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 $\text{Ll}(x)=i[H,x]-1\mathrm{/}2{\sum }_k(L_k^{\ast }{L}_{k}x-2L_k^{\ast }x{L}_k+x{L}_k^{\ast }{L}_k)$ of the generator of $\text{T}$. We study the adjoint semigroup with respect to both scalar products $<a,b>=\text{tr}(\rho a^{\ast }b)$ and $<a,b> = \tr(\rho^{1/2} a^* \rho^{1/2}b)$.