Long time growth bounds

In most applications of semigroup theory one is given the generator Z explicitly, and has to infer properties of the solutions of the evolution equation ƒ′ (t) = Zƒ(t), i.e. of the semigroup Tt. This is not an easy task, and much of the analysis depends on obtaining bounds on the resolvent norms. We devote this section to establishing a connection between the spectrum of Z and the long time asymptotics of Tt. Before starting it may be useful to summarize some of the results already obtained. These include

1. (i) If ∥Tt∥ ≤ Meat for all t ≥ 0 then Spec(Z) ⊆ {z : Re(z) ≤ a} and ∥Rz∥ ≤ M/(Re(z) − a) for all z such that Re(z) > a. (Theorem 8.2.1)

2. (ii) If Rz is the resolvent of a densely defined operator Z and ∥Rx∥ ≤ 1/x for all x > 0, then Z is the generator of a one-parameter contraction semigroup, and conversely. (Theorem 8.3.2)

3. (iii) For every ε > 0 there exists a densely defined operator Z acting in a reflexive Banach space such that ∥Rx∥ ≤ (1+ ε)/x for all x > 0, but Z is not the generator of a one-parameter semigroup. (Theorem 8.3.10)

4. (iv) Tt is a bounded holomorphic semigroup if and only if there exist α > 0 and N < ∞ such that the associated resolvents satisfy ∥Rz∥ ≤ N|z|−1 for all z such that |Arg(z)| ≤ α + π/2. (Theorems 8.4.1 and 8.4.2)

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