Weak convergence of semigroups implies strong convergence of weighted averages

Abstract:

For a fixed $p,1 \leqslant p < \infty$, let $\{ {T_t}:t > 0\}$ be a strongly continuous semigroup of positive contractions on ${L_p}$ of a $\sigma$-finite measure space. We show that weak convergence of $\{ {T_t}:t > 0\}$ in ${L_p}$ is equivalent with the strong convergence of the weighted averages $\int _0^\infty {{T_t}f{\mu _n}(dt)(n \to \infty )}$ for every $f \in {L_p}$ and every sequence $({\mu _n})$ of signed measures on $(0,\infty )$, satisfying ${\sup _n}||{\mu _n}|| < \infty ;{\lim _n}{\mu _n}(0,\infty ) = 1$; and for each $d > 0,{\lim _n}{\sup _{c \geqslant 0}}|{\mu _n}|(c,c + d] = 0$. The positivity assumption is not needed if $p = 1$ or 2. We show that such a result can be deduced-not only in ${L_p}$, but in general Banach spaces-from the corresponding discrete parameter version of the theorem.