COMPLETING THE FUNCTIONAL CALCULUS 17
G(S) = G(T) u c l A ,
olre(S) = cl A ,
a0(S) = a0(T) \ A .
Note that this implies that o0(S) is finite and o(S) has only a finite number of
components. Moreover S can be chosen so that #"(S; X) = 9{(T; X) and
S | !H(T; X) = T| H(T\ X) for X in G
0
(S) . Let ax , . . . , a
n
be the components of
a(S) \ a0(S) . Using the Riesz-Dunford functional calculus, S is similar to Sl
0 S
n
0 F , where each Sj = s | 9{{S\ Gj) and F = s l #"(T;
G
0
(S) )
. It is
routine to check tha t the properties listed in the statement of the
proposition are verified.
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