Perturbation Theory for the Logarithm of a Positive Operator

In various contexts in mathematical physics one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, that we denote by $\Delta$, can be related by a perturbative series to another operator $\Delta_0$, whose logarithm is known. We set up a perturbation theory for the logarithm $\log \Delta$. It turns out that the terms in the series possess remarkable algebraic structure, which enable us to write them in the form of nested commutators plus some "contact terms."