12 WILLIAM H. BARKER
Then F is said to be differentiable at \€ U if the following limit exists in the operator
topology of B(H):
F'(A) - lim[F(A+h)-F(A)]/h
h-*oo
Suppose {cn}^L1 is an orthonormal basis for H , and define
Fnm(A) = (F(A)eJeJ.
The following result will be needed later. The proof is straightforward, and hence is
omitted.
Proposition 2.1 Suppose F :U-*B(H) is such that
(i) F
n m
is analytic on U for all n,m;
(ii) For each \eU, sup | F ' ( A ) | (1+| n\) (1+| m|) oo;
n,m
0») 8UP I F n
m
( A )l ( l + | n | ) 2 ( l + | m | ) J o o
A6 U,n,m
Then F is differentiable on U, and (F ') = (F Rm) '•
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