 By A.M. Fink

ISBN-10: 3540067299

ISBN-13: 9783540067290

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Extra info for Almost Periodic Differential Equations

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Exercises. 11. SPECTRAL PERMANENCE THEOREM 31 by a convergent power series of the form ∞ f (z) = an z n , z ∈ ∆, n=0 for some sequence a0 , a1 , a2 , . . in C satisfying n |an | < ∞. 6. If f ∈ B satisﬁes f (z) = 0 for every z ∈ ∆, then g = 1/f belongs to B. In the following exercise, Z+ denotes the additive semigroup of all nonnegative integers. (2) Let T be the isometric shift operator that acts on 1 (Z+ ) by T (x0 , x1 , x2 , . . ) = (0, x0 , x1 , x2 , . . ), and let a = (a0 , a1 , a2 , . .

Let A be a Banach algebra with normalized unit 1 and let I be a proper ideal in A. Then for every z ∈ I we have 1 + z ≥ 1. In particular, the closure of a proper ideal is a proper ideal. Proof. 2 z must be invertible in A; hence 1 = z −1 z ∈ I, which implies that I cannot be a proper ideal. The second assertion follows from the continuity of the norm; if 1 + z ≥ 1 for all z ∈ I, then 1 + z ≥ 1 persists for all z in the closure of I. 3. If I is a proper closed ideal in a Banach algebra A with normalized unit 1, then the unit of A/I satisﬁes 1˙ = inf 1 + z = 1; z∈I hence the unit of A/I is also normalized.

Deduce that every linear functional f : A → C satisfying f (xy) = f (x)f (y), x, y ∈ A, is continuous. 9. Commutative Banach Algebras We now work out Gelfand’s generalization of the Fourier transform. Let A be a commutative Banach algebra with unit 1 satisfying 1 = 1. We consider the set hom(A, C) of all homomorphisms ω : A → C. An element ω ∈ hom(A, C) is a complex linear functional satisfying ω(xy) = ω(x)ω(y) for all x, y ∈ A; notice that we do not assume that ω is continuous, but as we will see momentarily, that will be the case.