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3). The set M(Sϕ )A M[Sω ]A := ϕ∈(ω,π) is called the domain of this calculus. As in Chapter 1 we write H(A) := {f ∈ M[Sω ]A | f (A) ∈ L(X)} as an abreviation. For simplicity, we frequently omit explicit reference to the sector and simply write ‘MA ’ instead of ‘M[Sω ]A ’, provided that no ambiguities occur. Also we use the abbreviation O[Sω ]A := O[Sω ] ∩ MA and similar ones for other function classes. 7. 1) One should keep in mind that a function f belonging to E[Sω ] or M[Sω ]A , is actually defined on a larger sector Sϕ for some ϕ ∈ (ω, π].

Proof. a) It is immediate that A[Sω ] ⊂ MA . Let f ∈ A, and choose n such that F := f (1 + z)−n ∈ E. If A is bounded, then f (A) = (1 + A)n F (A) ∈ L(X). 1 we have ((z − µ)f (z))(A) = (1 + A)n+1 f (z)(z − µ) (1 + z)n+1 (A) = (1 + A)n+1 [(A − µ)(1 + A)−1 ]F (A) = (A − µ)(1 + A)n+1 (1 + A)−1 F (A) = (A − µ)(1 + A)n F (A) = (A − µ)f (A). c) Note that the operator F (A) commutes with (1 + A)−n , whence D(An ) is F (A)-invariant. This gives D(An ) ⊂ D(f (A)). For arbitrary x ∈ D(f (A)) we have Tt (x) := (t(t + A)−1 )n x → x as t → ∞.

6) is a homomorphism of algebras. Moreover, it has the following properties: a) z(1 + z)−1 (A) = A(1 + A)−1 . 3. The Natural Functional Calculus Γ = Γω f (A) = 1 2πi 33 ,δ Z Γ f (z)R(z, A) dz σ(A) ∂Sω Figure 4: The contour of integration avoids 0 with f being analytic there. b) If B is a closed operator commuting with the resolvents of A, then B also commutes with f (A) for each f ∈ E(Sϕ ). In particular, each f (A) commutes with A. c) If x ∈ N(A) and f ∈ E(Sϕ ), then f (A)x = f (0)x. d) Let B denote the injective part of A.

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