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This semigroup is called exponentially stable if ω0 (T) < 0. 4. The linear operator A : D(A) → X, deﬁned by Tt z − z exists , t → 0, t>0 t Tt z − z Az = lim ∀ z ∈ D(A), t → 0, t>0 t D(A) = z∈X lim is called the inﬁnitesimal generator (or just the generator) of the semigroup T. 1. 1), then its generator is A. 5. Let T be a strongly continuous semigroup on X, with generator A. Then for every z ∈ D(A) and t 0 we have that Tt z ∈ D(A) and d Tt z = ATt z = Tt Az . dt Proof. 5) 0 and τ > 0, then Tτ − I Tτ − I Tt z = Tt z → Tt Az, as τ → 0 .

If A : D(A) → X, where D(A) ⊂ X, then λ ∈ C is called an eigenvalue of A if there exists a zλ ∈ D(A), zλ = 0, such that Azλ = λzλ . In this case, zλ is called an eigenvector of A corresponding to λ. The set of all the eigenvalues of A is called the point spectrum of A, and it is denoted by σp (A). The following proposition is an elementary spectral mapping theorem for the point spectrum of an operator. 28 Chapter 2. 18. Suppose that A : D(A) → X, where D(A) ⊂ X and λ, s ∈ C, λ = s, s ∈ ρ(A). Then the following statements are equivalent: (1) λ ∈ σp (A).

5. 3, A is the generator of a strongly continuous semigroup T on X if and only if sup Re λk < ∞. 8) k∈N If this is the case, then k∈N and for every t 0, eλk t z, φ˜k φk Tt z = k∈N ∀ z ∈ X. 9) 42 Chapter 2. Operator Semigroups A semigroup as in the last proposition is called diagonalizable. Proof. 7) holds. 9) deﬁnes a bounded operator Tt on X. It is easy to see that this family of operators satisﬁes the semigroup property and it is uniformly bounded for t ∈ [0, 1]. It is clear that the function t → Tt z is continuous if z is a ﬁnite linear combination of the eigenvectors φk .