A generalization of Sen-Brinon's theory

Let K be a complete discrete valuation field of mixed characteristic and k be its residue field of prime characteristic p > 0. We assume that [k: k^p] = p^h < ∞. Let G_K be the absolute Galois group of K and R be a Banach algebra over C_p:=K̄̂ with a continuous action of G_K. When k is perfect (i.e. h = 0), Sen studied the Galois cohomology H¹(G_K, R*) and Sen's operator associated to each class (Sen Ann Math 127:647-661, 1988). In this paper we generalize Sen's theory to the case h ≥ 0 by using Brinon's theory (Brinon Math Ann 327:793-813, 2003). We also give another formulation of Brinon's theorem (à la Colmez).