Kato’s inequality occurs in several areas of mathematics and physics. …

Semigroup domination

Let $U_t = \exp(t H)$ be a positivity preserving semigroup on $L^p(M)$ for some fixed $p$. Let $V_t = \exp(t K)$ a strongly continuous semigroup on some vector valued $L^p(M,E)$, where $E$ is a Euclidian vector space with norm $\vert\cdot\vert$. For $a\in E$ set $(\sgn a\cdot v):=\langle a,v\rangle/\vert a\vert$. But you might as well take $E=\R$.

From Simon’s proof of the above theorem one easily sees that Kato’s inequality is necessary for any semigroup on $L^1$ to be positivity preserving.

Is it sufficient? – The other implication in Simon’s proof can easily be reduced to the criterion

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which looks nice but ultimately begs the question in the language of resolvents. (For practical verification of the criterion you need to know the answer.) Using a crown jewel of semigroup tricks (from Dodziuk 19xy) the question can be begged more literally,

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and this might perhaps turn out to guide the way to prove sufficiency: … tbc … Perhaps construct a positivity preserving semigroup dominated by $U_t$. Or perhaps just do some clever calculus, based perhaps on Egorov’s theorem…

References

B. Simon, Kato’s Inequality and the Comparison of Semigroups, J. Funct. Analysis 32 (1979), 92-101

I. Shigekawa, $L^p$ contraction semigroups for vector valued functions