3 - Semigroups of class AK

Summary

A semigroup {V_t, t ∈ R⁺, X} belongs to the class AK (or it is asymptotically compact) if it possesses the following property: for every B ∈ B such that γ⁺(B) ∈ B, each sequence of the form {V_tk(x_k)}^∞_k=1, where x_k ∈ B and t_k +∞, is precompact.

Here we restrict ourselves to the case of continuous semigroups of class AK. We begin with two elementary propositions concerning continuous semigroups.

Proposition 3.1

For every compact set K and t ∈ R⁺the set γ⁺[0,t](K) is compact.

The proof is evident.

Proposition 3.2

If K is compact and γ⁺(K) is precompact then w(K) is a non-empty compact invariant set attracting K.

Proof In fact, this proposition was proved in Chapter 2. Denoting K₁ := [γ⁺(K)]x we know this is compact and V_t(Ki) ⊂ K₁. So we have a semigroup {V_t, t ∈ R⁺, K₁) of continuous operators V_t acting on a metric space K₁. Hence, this semigroup is of class K, and we may apply Theorem 2.1 to obtain the desired result.

Now we pass to the semigroups of class AK.

Proposition 3.3

Let {V_t, t ∈ R⁺, X} be a continuous semigroup of class AK. Suppose that K is a compact set such that the γ⁺ (K) is bounded. Then γ⁺(K) is precompact and thus the statement of Proposition 3.2 is true.

Proof Let y_nn = 1,2,…, be an arbitrary sequence of points from γ⁺(K), i.e. y_n = V_tn(x_n) for some x_n ∈ K and t_n ∈ R⁺.