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Created Experiments in Kato's inequality. Angle brackets don't render in my box. Those from blog do.
$$\langle \vert a\vert , H f\rangle \ge \langle (\sgn a)(Ka), f\rangle \quad \forall ..., f\ge0$$ Now me sleep, then work, then perhaps state an unsurprising theorem on stochastic semigroups, then sleep again, etc., then give proof of theorem. That's the plan.