@AlessandroCodenotti What does this mean in practice? I guess that there is a continuous function $c(t)$ vanishing at 0 so that $|T(t)f - f| \leq c(t) |f|$, uniformly in $f$, using $L^p$ norms on both sides. Why should I get a contradiction here?
Well in particular if $|f_n| = 1$ for all $n$ but $f$ does not converge --- eg, take the $f_n$ to be the indicator function of the "even length $1/2^n$ intervals in the unit interval", so 1 on [0,1/2^n], [2/2^n, 3/2^n] and so on. Then we should have $T(1/2^{n+1})f_n - f_n \to 0$ in $L^p$. But this translation is precisely the indicator function of…