Let T = {T(t)}t[ges ]0 be a
C0-semigroup on a Banach space X. The following results are proved.
(i) If X is separable, there exist separable Hilbert spaces X0
and X1, continuous dense embeddings
j0[ratio ]X0 → X and
j1[ratio ]X → X1, and
C0-semigroups T0 and T1 on
X0 and X1 respectively, such that
j0 ∘ T0(t)
= T(t) ∘ j0 and
T1(t) ∘ j1
= j1 ∘ T(t) for all t [ges ] 0.
(ii) If T is [odot ]-reflexive, there exist reflexive Banach spaces
X0 and X1 , continuous dense embeddings
j[ratio ]D(A2) → X0,
j0[ratio ]X0 → X,
j1[ratio ]X → X1, and
C0-semigroups T0 and T1 on
X0 and X1 respectively, such that
T0(t) ∘ j = j ∘ T(t),
j0 ∘ T0(t)
= T(t) ∘ j0 and
T(t) ∘ j1
= j1 ∘ T(t) for all t [ges ] 0,
and such that σ(A0) = σ(A)
= σ(A1), where Ak is
the generator of Tk, k = 0, [emptyv ], 1.