Let A be a Banach algebra

All the spaces, algebras in this notes are assumed to be over C.

1. Banach Algebra

Let (A, k · k) be a complete normed algebra over C. We say that A is a Banach algebra if kabk ≤ kakkbk,

for all a, b ∈ A. A unital Banach algebra is a Banach algebra with (multiplicative) identity 1.

Proposition 1.1. Let A be a Banach algebra. If a ∈ A with kak < 1, then 1−a is invertible and (1 − a)⁻¹ =P∞

n=0aⁿ. Moreover k(1 − a)⁻¹k ≤ (1 − kak)⁻¹. Proof. Let sn=Pn

k=0a^k. It is easy to show that (sn) is a Cauchy sequence in A : for n ≥ m, sn− s_m=P

k=m+1a^k. For n ≥ m, by triangle inequality, ks_n− s_mk ≤

n

X

k=m+1

kak^k< kak^m+1 1 − kak.

Since A is complete, we denote the limit of (sn) by b. Since (1 − a)sn= sn(1 − a) = 1 − aⁿ and lim_n→∞aⁿ= 0, we see that

(1 − a)b = b(1 − a) = 1.

Hence b = (1 − a)⁻¹. By definition, b = P∞

n=0aⁿ. Using triangle inequality and the norm inequality, we also have

kbk ≤

∞

X

n=0

kak^k= 1 1 − kak.



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