Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) 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 p-Schatten class of operators on $\mathcal{H}$. In particular, I would like results crucially using that this difference is Hilbert-Schmidt, for example, and not just that it is compact as a result.



Literature request: Schatten class difference of semigroups
The problem is discussed in a more general setting (operator ideals in Banach spaces) for the so-called analytic semigroups (parabolic problems) in


*Blunck, S.; Weis, L.*, [**Operator theoretic properties of differences of semigroups in terms of their generators**](http://dx.doi.org/10.1007/s00013-002-8292-3), Arch. Math. 79, No. 2, 109-118 (2002). [ZBL1006.47036](https://zbmath.org/?q=an:1006.47036).


The paper seems 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 asymptotics, 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 classe you wish for.


