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Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and $U'(t)+A(t)U(t)=0$.

Question: What condition on $A(t)$ guarantees that $U(t)$ is of trace class for all $t>0$?

More specifically, I am interested in situation when the following situation: Let $M$ be a compact manifold, $H=L^2(M)$, and let $A(t)$ be a smooth family of positive definite elliptic operators on $M$. If $A(t)$ is independent of $t$, then $U(t)=e^{-tA}$ is of trace class. So I hope that if $A'(t)$ is not very large, then $U(t)$ is still of trace class. What is the precise condition on $A'(t)$ which guarantees this?

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  • $\begingroup$ Don't you want $U(t)$ is identity plus trace class? $\endgroup$
    – Peter Michor
    Commented May 29, 2015 at 14:53
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    $\begingroup$ I don't think so. If $A$ is independent of $t$, then $U(t)$ is the heat operator and is of trace class. $\endgroup$
    – Maxim Braverman
    Commented May 29, 2015 at 19:18
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    $\begingroup$ In the case $A$ is independent of $t$ and is elliptic, you need more to ensure that $e^{-tA}$ is of trace class. This is false if $Af=af$ for some $a\in(0,+\infty)$, but it is true if $A$ has order $m>0$. Presumably, you had in mind the case where $A$ is differential of second order. $\endgroup$
    – Denis Serre
    Commented Oct 26, 2015 at 20:08

2 Answers 2

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You may want to take a look at this Ph.D. thesis, especially at Thm. 5.5.2, which as far as I know is the sharpest currently available result concerning regularity of nonautonomous problems. The theorem states that the solution to your problem is at most in $V$ for each $t$, under suitable assumptions. Now, if you are thinking of a time-dependent diffusion equation, $V=H^1(\Omega)$, which is never trace-class-embedded in $L^2(\Omega)$, even in 1-dimension. In the autonomous case this is not a problem, since the semigroup law still allows us to conclude that $e^{-tA}$ is of trace class for all $t>0$ iff $V\hookrightarrow H$ is of $p$-Schatten class for some $p<\infty$.

But in the autonomous case this smoothing property seems to fail, so in general you won't get any more than $p$-Schatten class for all $p>1$ in the case of a nonautonomoum problem associated with a differential operator of order 1.

As suggested by @DenisSerre, things change if you consider higher order operators: if $A(t)$ is for all $t$ a 2k-th order differential operator with domain $H^k(\Omega)$, then you do have trace class evolution operators as soon as $k>d$.

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Maybe, the following paper or references therein can help:

  • Josef Teichmann: Another approach to some rough and stochastic partial differential equations, arXiv/0908.2814, Stochastics and Dynamics 11 (2011), no. 2-3, 535-550.
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