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14) (0, t) = 0, w(π, t) = 0, t ∈ [0, ∞), ⎪ ⎪ ⎪ ∂x ⎪ ⎩ w(x, 0) = w0 (x), x ∈ (0, π). 6. Diagonalizable operators and semigroups 45 Let X = L2 [0, π] and let A be deﬁned by D(A) = dz (0) = z(π) = 0 dx z ∈ H2 (0, π) Az = d2 z dx2 , ∀ z ∈ D(A). It is easy to check the following properties. 14) iﬀ z is continuous with values in D(A) (endowed with the graph norm), continuously diﬀerentiable with values in X and it satisﬁes the equations ∀t z(t) ˙ = Az(t) 0 , z(0) = w0 . • The family of functions (ϕk )k∈N , deﬁned by ϕk (x) = 2 cos π k− 1 2 ∀ k ∈ N, x ∈ (0, π), x consists of eigenvectors of A, it is an orthonormal basis in X and the corresponding eigenvalues are λk = − k − 1 2 2 ∀ k ∈ N.

Suppose that |λ| > A . Then λI − A = λ I − A λ . 6. Hence, λ ∈ ρ(A). 26 Chapter 2. Operator Semigroups For any A ∈ L(X), the number r(A) = max |λ| λ∈σ(A) is called the spectral radius of A. It follows from the last proposition that r(A) A . A stronger statement will be proved at the end of this section. 11. If A ∈ L(X) and r > r(A), then there exists mr n A mr r 0 such that ∀ n ∈ N. n Proof. For every α, γ > 0 we denote Dα = {s ∈ C | |s| < α} , For α = 1 r(A) Cγ = {s ∈ C | |s| = γ} . we deﬁne the function f : Dα → L(X) by f (s) = (I − sA)−1 .

5, A is the generator of a strongly continuous semigroup T on X given by 1 2 e−(k− 2 ) Tt z = t z, ϕk ϕk ∀t 0, z ∈ X . 15) k∈N Note that this semigroup is also exponentially stable. 11. Everything we have said in this section remains valid if we replace N with another countable index set, such as Z. Sometimes it is more convenient to work with a diﬀerent index set, as the following example shows. 12. Let X = L2 [0, 1]. For α ∈ R we deﬁne A : D(A) → X by D(A) = z ∈ H1 (0, 1) | z(1) = eα z(0) , dz ∀ z ∈ D(A).