Continuity of semigroups on $L^2$ and $L^1$: Is this simple proof correct?

This is more a series of rather trivial comments than an answer but for my own convenience I post them like this.

First of all, as I stated in my comment above, the argument looks fine to me. I now went through it line by line several times and I couldn't spot a mistake.

• I'm always a bit afraid of reductions to positive and negative parts, so I spelled them out (sorry if that's all trivial for you): If $f \in L^1 \cap L^2$ then we can write $f = f^{+} - f^{-}$ and thus $P_t \geq 0$ gives $P_t f^\pm \geq 0$, so $(P_t f)^{\pm} = P_t (f^{\pm})$ and hence your limit argument in the first paragraph yields $$\|P_{t}f\|_1 = \int P_{t} f^{+} + \int P_t f^- \leq \int f^+ + \int f^- = \|f\|_1$$ as you asserted.

• At the very end of the first paragraph you seem to have mixed up "right side" and "left side". Of course, you could also replace monotone by dominated convergence here.

• As $L^2$-convergence implies convergence in measure (Vitali), I think there is no need to pass to a subsequence of $t_{n} \downarrow 0$ at the beginning of the second paragraph. More precisely, you could argue using the Fatou lemma for convergence in measure, which would make the argument a bit cleaner in my opinion (even if Fatou's lemma in measure is usually obtained from the pointwise a.e. version by exactly the same reduction).

• Again, I have no objection to the decomposition into positive and negative parts and the extension to all of $L^1$ is clear from an $\varepsilon / 3$-argument.

Summing up, I like your argument and I think it's correct.

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Unfortunately, I'm not able to explain the equality $\mathcal{F}_0 \mathcal{F}_t f(X_0) = P_t (1/P_t 1) P_t f (X_0)$ in Silverstein.