Literature request: Schatten class difference of semigroups

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 $$$$e^{-t(A+B)}-e^{-tA} \in S_{p}(\mathcal{H})$$$$ where $$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.

1 Answer

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.

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