G.W. JOHNSON and M.L. LAPIDUS
Assume that for all X in T, g(X,y) is_a strongly measurable function of
y in A. Suppose further that there exists h in L (A,y) such that
I I g(*y) I I 1 My) for y-a.e. y in A and all X in T. Set
G(X) = / g(X,y)dy(y), for X in T.
(Note that G is_well defined by the basic Bochner integrability criterion
[15, Theorem 3.7.4, p. 80].)
1°) Assume that for y-a.e. y in A, g(X,y) is_a strongly continuous
function of: X in T. Then G is_ strongly continuous in T.
2°) Assume that T is an open subset of C and that for y-a.e. y in A,
g(X,y) i£ an analytic function of X in T. Then G is_analytic in T.
PROOF. If g(X,y) is operator-valued, we consider the vector-valued
function g(A,y)i| for fixed ^ in E; so that we may assume that g is
1° is a consequence of the dominated convergence theorem for
Bochner integrals [15, Theorem 3.7.9, p.83].
Let E denote the dual space of E and
the duality bracket
between E and E . Fix ty in E . Recalling our earlier remark in Sec-
tion 0.2.E about the equivalence of all the natural notions of analytic-
ity, we see that 2 will follow if we show that G,(X) := * , G(X) is
analytic in T. Let g-.(X,y) := ij / , g(X,y). Clearly, under our assump-
tions, g- i (X,y) is an analytic function of X for y-a.e. y e A. Further,
by [15, Eq. (3.7.5), p. 80],G,(X) = / g-(X,y)dy(y). Moreover, for
1 A L
y-a.e. y, we have:
(0.17) |gl(x,y)| _ II**H^ I I g(A,y)||£ 1 IU*HE*h(y).
By hypothesis, the dominating function in (0.17) lies in L (A,y). 2°
now follows from the corresponding result for Lebesgue integrals of
scalar-valued functions depending on a parameter. •
When we apply Lemma 0.2 in this paper, we shall always choose T • C+
in case 1° and T « C, in case 2°.