2 SPECTRUM OF AN ELEMENT OF A NORMED ALGEBRA 313

cannot be defined and analytic in the region |(| > r, because the function
£H-»(£? - £x)~\ which is analytic and equal to the sum of the series ]T ({jc)n

n

for all sufficiently small |£|, would then be equal to the sum of this series for
|£| <r~1 by virtue of (9.9.1) and (9.9.2). But this is absurd, because if
p(x)-1 <r'~1 <r~\ we have \\(&)n\\ 'Z (\t\rj for all sufficiently large n,
and (|£kT tends to +00 as n tends to +00 when |£| > r'"1.

The number p(x) is called the spectral radius of x. We have

(15.2.7.1) P(x)£ \\x\\,

(15.2.7.2) p(xk) = (p(x))k

for all integers A: ^ 0, by virtue of (15.2.4(iii)) and (15.2.3.1).

(15.2.8) (i) Let A, B be two Banach algebras with nonzero unit elements
e, ef respectively, and let u : A-»B be an algebra homomorphism such that
u(e) = e'. Then SpB(w(*)) cz SpA(x)for all x e A.

(ii) In particular, suppose that A is a closed subalgebra of B, having the
same unit element. Then SpB(x) c SpA(x), and every frontier point of SpA(jt)
belongs to SpB(x). Hence, z/SpA(,x) has empty interior, we have SpB(jc) = SpA(jc).

(i) If £ e C is such that x < £e is invertible in A, then u(x — £e) =
u(x) — £e' is invertible in B, whence the result.

(ii) The first assertion is a particular case of (i). To prove the second
assertion, it is enough to show that if A0 is a frontier point of SpA(x), then
x — A0 e is not invertible in B. Now, by hypothesis there exists a sequence
(AM) of regular values for x (in A) tending to A0 . For each n ^ 1, the inverse
(x — kne)~l of x — kne in A exists, and is therefore also the inverse of x — "kne
in B. If A0 <£ SpB(x), the sequence ((x — Xne)~l) would tend in B to the inverse
y ofx — A0e in B (15.2.4(ii)). But since A is closed in B, we should havey e A,
so that y would also be the inverse of x — 10 e in A, which is absurd.

PROBLEMS

1. Let A be a normed algebra with no unit element. Then A can be considered as a
closed two-sided ideal in the normed algebra A defined in Problem 5 of Section 15.1.
For each x 6 A, the spectrum of x in A (denoted by Sp(*) or SpA(*)) is defined to be
the spectrum of x in A, and p(x) is the spectral radius of x considered as an element
of A.

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"ITTSBIJRGH, PEE'SYLVAIIA 152T3