Norm continuity of $c_0$-semigroups
Volume 134 / 1999
Studia Mathematica 134 (1999), 169-178
DOI: 10.4064/sm-134-2-169-178
Abstract
We show that a positive semigroup $T_t$ on $L_p(Ω,ν)$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of "smoothing properties" of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.
