Extrapolation Spaces for Semigroups

To a strongly continuous semigroup ? T(t) t0 on a Banach space X we will associate semigroups ? T n (t) t0 on new Banach spaces X n for each n 2 Z. This construction is inspired by the classical Sobolev spaces and, due to its simplicity, of great help in understanding abstract and concrete semigroups. We start with a strongly continuous semigroup ? T(t) t0 on a Banach space X for which we assume that its growth bound ! 0 is negative. Therefore, the generator ? A; D(A) is invertible and A ?1 2 L(X): In addition, we assume, after renorming X if necessary, that kR(; A)k 1 for all > 0. On the domains D(A n) of A n , n 2 N, we now introduce new norms k k n. 1.1 Deenition. For each n 2 N and x 2 D(A n) we deene the n-norm kxk n := kA n xk and call X n := (D(A n); k k n) the n-th Sobolev space associated to ? T(t) t0. The operators T(t) restricted to X n will be denoted by T n (t) := T(t) j X n : It turns out that the restrictions T n (t) behave surprisingly well on X n. 1.2 Proposition. With the above deenitions the following holds. (i) Each X n is a Banach space. (ii) The operators T n (t) form a strongly continuous semigroup ? T n (t) t0 on X n. (iii) The generator A n of ? T n (t)