Pointwise convergence for semigroups in vector-valued L^p spaces

Suppose that {T_t : t ≥ 0} is a symmetric diffusion semigroup on L ²(X) and denote by T̃_t:t ≥ 0 its tensor product extension to the Bochner space L^p(X,B), where B belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf-Dunford-Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of T̃_t:t ≥ 0 on L ^p(X,B). As an application, we show that such continuations exhibit pointwise convergence.