Weak convergence of semigroups implies strong convergence of weighted averages

Abstract

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