On dilatation of contracting cocycles and perturbations of the group of shifts on the line by cocycles, II

摘要

Let S = (St)t∈R be the group of translations in the Hilbert space H = L~2(R) with the action given by (Stf)(x) = f(x + t) , f ∈ H. To this action, a natural filtration of H by the family of spaces Ht ∈ H consisting of functions with the support on the interval [-t, +∞) is associated. Note that S_tH_0 = H_t , t ∈R. A strongly continuous family of bounded operators W = (W_t)_(t ∈R+) is called a cocycle of S if the condition W_(t+s) = W_tS_tW_sS_(-t), s,t≥ 0, W_0 = I is satisfied (here I denotes the identity operator); this condition guarantees that the family of bounded operators u0001S = {S_t = W_tS_t, t ∈R+} forms a C_0-semigroup of bounded operators.