By Wilfred Kaplan
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Extra info for Introduction to Analytic Functions (Addison-Wesley Series in Mathematics)
In particular, A[Sω ] is an algebra of functions containing every rational function with poles outside of Sω . Proof. (i)⇒(ii). If g(z) := f (z)(1 + z)−n ∈ E, then clearly condition 1) is satisﬁed. Because g has a ﬁnite polynomial limit at 0, also f = (1 + z)n g has a ﬁnite polynomial limit at 0. (ii)⇒(iii). Choose α as in (ii), and let n > α. Then (f (z) − f (0))(1 + z)−n ∈ H0∞ . (iii)⇒(i). This is trivial. 11. Let A ∈ Sect(ω). Then A[Sω ] ⊂ MA . For f ∈ A[Sω ] the following assertions hold. a) If A is bounded, so is f (A).
More precisely, we require the following: 1) A ∈ Sect(ω). 2) g ∈ M[Sω ]A and g(A) ∈ Sect(ω ). 3) For every ϕ ∈ (ω , π) there is ϕ ∈ (ω, π) with g ∈ M(Sϕ ) and g(Sϕ ) ⊂ Sϕ . Under these requirements obviously g(Sω ) ⊂ Sω . 2. (Composition Rule) Let the operator A and the function g satisfy the conditions 1), 2), and 3) above. 9) for every f ∈ M[Sω ]g(A) . Let us ﬁrst discuss the case that g = c is a constant. Then g(A) = c, and if c = 0, everything is easy by Cauchy’s theorem. 8). So f ◦ g is in 42 Chapter 2.
If (An )n is a sectorial approximation for A on Sω , we write An → A (Sω ) and speak of sectorial convergence. 3. a) If An → A (Sω ) and all An as well as A are injective, then A−1 → A−1 (Sω ). b) If An → A (Sω ) and A ∈ L(X), then An ∈ L(X) for large n ∈ N, and An → A in norm. c) If An → A (Sω ) and 0 ∈ (A), then 0 ∈ (An ) for large n. d) If (An )n ⊂ L(X) is uniformly sectorial of angle ω, and if An → A in norm, then An → A (Sω ). e) If A ∈ Sect(Sω ), then (A + ε)ε>0 is a sectorial approximation for A on Sω .
Introduction to Analytic Functions (Addison-Wesley Series in Mathematics) by Wilfred Kaplan