2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/155733Suppose that T_t is a symmetric diffusion semigroup on L^2(X) and consider 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 the semigroup's extension to L^p(X,B). As an application, we show that such continuations exhibit pointwise convergence.In version2 we correct the error present in version 1 as well as removing one of the hypotheses of the main theorem. Section 2 is also rewrittenFunctional AnalysisSpectral Theory47D03text