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Additional resources for An Introduction to Semigroup Theory
For every f ∈ D(A), the mapping Tt f = u(t, f ) : R+ → X is diﬀerentiable and, therefore, continuous. The density of D(A) and continuity of every Tt imply that the semigroup (Tt )t≥0 is strongly continuous; and, since it satisﬁes T0 f = u(0, f ) = f for all f ∈ D(A), (Tt )t≥0 is a C0 -semigroup. 2 Let T = (Tt )t≥0 be a C0 -semigroup in a Banach space X. 1. 13) exists. Obviously, D(G) is a linear submanifold of X and G is a linear map from D(G) to X. The set of all θ ∈ C such that there exists a bounded operator Qθ : X → X satisfying Qθ ◦ (θ IX − G) = ID(G) & (θ IX − G) ◦ Qθ = IX is called the resolvent set of G; the operator Qθ is called the resolvent of G at the point θ, and is denoted by Rθ (G).
So T is quasi-constrictive, but it is not constrictive, since it has no non-trivial eigenvector. 8. Let X = c0 with the sup-norm · . Denote by ek the element of X, the k-th coordinate of which is equal to 1 and all other coordinates are zero. Fix α ∈ C and deﬁne the operator Sα : X → X, Sα (ek ) = e1 + α e2 ek+1 k=1 . else Set Tα := (I + Sα )/2. Obviously, Tα is power bounded (moreover, it is contractive if |α| ≤ 1). For k ≥ 2, we have n Tαn (ek ) = 2−n l=0 So Tαn (ek ) = 2−n n [n/2] n ek+l . l , where [q] is the integer part of q.
24) lim n Rn (G) x = x. 22) to deduce that n Rn (G) x − x Rn (G) ◦ G x = ≤ n−1 Gx (∀x ∈ D(G)). 24) for x ∈ D(G). 24) holds for all x ∈ X. Next we show that lim Gn x = Gx n→∞ (∀x ∈ D(G)). 24). 23), we obtain 2 etGn = e−nt en t Rn (G) = e−nt ∞ 0 (n2 t)m m Rn (G). m! 22), we deduce that each etGn is a contraction: etGn ≤ e−nt (n2 t)m 1 = e−nt ent = 1. m! 2. Elementary theory of C0 -semigroups 29 To estimate the diﬀerence of etGn and etGk , we use the fact that etGk and etGn commute with Gn and Gk : d (s−t)Gn [e ◦ etGk (x)] = e(s−t)Gn ◦ etGk ◦ (Gk − Gn ) (x) dt (∀x ∈ X).