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9, f is analytic. 6, for |s| · A < 1 we have f (s) = I +sA+s2 A2 +s3 A3 +· · · . This, together with Cauchy’s formula from the theory of analytic functions, implies that for every γ > 0 such that γ · A < 1, An = 1 2πi f (s) Cγ ds sn+1 ∀ n ∈ N. 5) By Cauchy’s theorem the above formula remains valid for every γ ∈ (0, α). Decγ γ1n . Denoting r = γ1 and noting cγ = maxs∈Cγ f (s) , we obtain that An mr = cγ , we obtain the desired estimate. Let A : D(A) → X with D(A) ⊂ X. We define the space D(An ) recursively: D(An ) = {z ∈ D(A) | Az ∈ D(An−1 )} .

6. Diagonalizable operators and semigroups 41 shows that (sI − A)Rs z = z ∀ z ∈ X. This implies that s ∈ ρ(A) and (sI − A)−1 = Rs . 3. Let A : D(A) → X be diagonalizable. Let (φk ) be a Riesz basis consisting of eigenvectors of A. Let (φ˜k ) be the biorthogonal sequence to (φk ) and denote the eigenvalue corresponding to the eigenvector φk by λk . 4) k∈N λk z, φ˜k φk Az = ∀ z ∈ D(A). 5) k∈N Proof. Let s ∈ ρ(A). 18, (sI − A)−1 is a diagonal1 and the izable (and bounded) operator with the sequence of eigenvalues s−λ k −1 corresponding sequence of eigenvectors (φk ).

A, C) is observable if and only if Q > 0. Proof. If (A, C) is observable, then (as already mentioned) Qτ > 0 (for every τ > 0). Since Q Qτ , it follows that Q > 0. To prove the converse statement, suppose that (A, C) is not observable and take x ∈ Ker Ψτ , x = 0. Then CetA x = 0 for all t 0, hence Qx = 0, which contradicts Q > 0. The proof for R > 0 is similar, using the dual system. Chapter 2 Operator Semigroups In this chapter and the following one, we introduce the basics about strongly continuous semigroups of operators on Hilbert spaces, which are also called operator semigroups for short.

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