SOME REMARKS ABOUT THE R-BOUNDEDNESS

Let X,Y be UMD-spaces that have property (α), 1< p< ∞ and let M be anR-bounded subset in L(X, Y). It is shown that {T(M_k)_(k∈z): M_k, k(M_(k+l)-M_k) ∈M for k∈Z} is an R-bounded subset of L(L^p (0,2π; X), L^p(0,2π; Y)), where T(M_m)_(k∈zdenotes the L^p-multiplier given by the sequence (M_k)_(k∈z), This generalizes a resultof Venni [10]. The author uses this result to study the strongly L^p-well-posedness ofevolution equations with periodic boundary condition. Analogous results for operator-valued L^p-multipliers on R are also given.