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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.

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The problem is discussed in a more general setting (operator ideals in Banach spaces) for so-called 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, 109-118 (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.

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  • $\begingroup$ 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

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