Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily selfadjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{tA},e^{t(A+B)}$, $t>0$. I would like to ask for some literature on the consequences of the condition \begin{equation} e^{t(A+B)}e^{tA} \in S_{p}(\mathcal{H}) \end{equation} where $p\geq 1$ and $S_{p}(\mathcal{H})$ is the pSchatten class of operators on $\mathcal{H}$. In particular, I would like results crucially using that this difference is HilbertSchmidt, for example, and not just that it is compact as a result.
The problem is discussed in a more general setting (operator ideals in Banach spaces) for socalled analytic semigroups (parabolic problems) in
Blunck, S.; Weis, L., Operator theoretic properties of differences of semigroups in terms of their generators, Arch. Math. 79, No. 2, 109118 (2002). ZBL1006.47036.
The paper semms to be freely accessible. The idea is, if the differences (of some fractional powers) of the resolvents belong to the ideal and have a nice asymtptics, then the differences of the semigroups belong to the same ideal and have a nice decay.
The application they give is $e^{t(\Delta  V)}  e^{t\Delta}$ on the whole space, where the semigroups are not even compact, but the difference belongs to the Schatten classes you wish for.

$\begingroup$ Many thanks for your answer, it seems this fits and there are a few references in there I will study too. $\endgroup$ – folouer of kaklas Aug 3 at 0:48