12 CONWAY, HERRERO, AND MORREL
Proof, (a) Let Hx H
d
be the components of
f_1(^)
n P+(A) . By
Proposition 1.2 , each Hj is an analytic Cauchy region with cl LL c P+(A) ,
f(Hj) = Q , f(3Hj) = d£l , and f is a strictly Pj-valent map of Hj onto Q for some
positive integer p,- . Also Theorem 1.1 implies that for X in Q , ±oo =
ind {X - T) = £ { ind (a - A) : a G f - 1 ( Q ) n P±(A) s u c h t h a t f^a) = x ) T h u s
for at least one j , 1 j d , there is a point a in Hj such that ind (a - A) =
± oo ; for this value of j , let H = Hj .
(b) Now assume that cl Q c P+(T) \ P±00(T) . Adopt the notation of the
preceding paragraph. By Proposition 1.2 all the properties from part (b)
hold, with the possible exception of (ii) and (iii) . Let n = ind (X - T) for X
in Q. and nij = ind (a - A) for a in Hj . Since f(Z(f')) is a countable set, there
is a X in Q such that X £ f(Z(f')). Thus part (f) of Theorem 1.1 implies n =
I j mj
P j
.
It remains to establish (iii). For this, it may be assumed that X £
f(Z(f')) since such points are dense in Q and so the general result will follow
from this case by results of spectral theory (see Theorem 1.13 (iii) in [20]).
Let { : 1 i pj } be the distinct points in Hj such that ffeJ = X . Note
that mj = nul (at. - A) - nul (a« - A) for each j . By Theorem 1.1(f) .
nul (X - T) = Xij
n u l (aij
-
A)
= Z j
m
j Pj + Si j
n u l (aij
"
A)*
= X {
m
j Pj
: m
j ° I
+
Z y {
n u l (aij
- A)* : mj 0 }
+ X ( [mj pj + X i n u l (aij " A)*l : mj 0 }
But each of these last two summands is positive. Hence (iii) holds.
1.4 Proposition If f is an analytic function (not assumed to be completely
non-constant) defined in a neighborhood of E and N is a normal operator,
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