E. M. BATOR AND D. R. SLAVENS

with Ix = (, (  xaI P)1/P, and
<}Xa     = x=(x,)E     1Xa:supx< oo},            (2)
with IxI = sup, xII.
THEOREM 1. Let X = ( a(d Xa)c. If Z is a Banach space, then a bounded
linear operator T: X - Z is DPI if and only if it is completely continuous.
PROOF. Suppose that T: X - Z is DPI but not completely continuous. Let
1 E A, Yi1= Xg, and Y2 = (i$E X). Thus X = Y1 E. Y2. Since T is not com-
pletely continuous, there exists a normalized weakly null sequence (x,yn)
in Y1 (, Y2 and c > 0 such that IT((x,y,)) I > c for each n E N. Define
TI : Yi - Z and T2 : Y2 -> Z by T(x) = T((x,0)) and T2(y) = T((0,y)). It
is clear that T((x,y)) = Ti(x) + T2(y). Thus, by passing to a subsequence if
needed, we may assume that either (a) IT1(x) I > E/2 or (b) IT2(y) > c/2,
for each n E N. Without loss of generality, we assume that (a) holds. Let y E Y2
such that Iy II = 1. As (x) is weakly null in X1, the sequence ((x, y))"I con-
verges weakly to (0,y), and each (xa,y) and (0,y) belong to Sx1 x2. As T is
DP1, T((x, y)) - T((0, y)) in norm. Therefore, T1(x) - 0 in norm, which is
a contradiction.                                                 Q
COROLLARY 2. Let X = (aE 9X). Then X has the DP property if and
only if it has the DPP. Likewise, X has the KKP if and only if X is a Schur space.
Observe that if (aE X,)c has the DPI property (KKP) and hence the DPP
(Schur property), then each X, has the DPP (Schur property). If s is finite, then
the converse also holds. However, the converse need not be true for infinite
-4. For instance, if X = (,EN 1 , then X contains a complemented copy of
42 (see [6, page 61]) and hence X does not have the DPP (Schur property), even
though each -z has the DPP (Schur property).
We will now consider DPI operators from C(H,X) to Y, where X and Y
are Banach spaces, H is a compact Hausdorff space with Borel subsets 1,
and C(H,X) is the Banach space (sup-norm) of X-valued continuous func-
tions on H. If T : C(H,X) - Y is a bounded linear map, then there exists a
unique weakly regular set function m : I - L (X, Y**) such that rig(H) < 00
and T(f) = ffdm  for each f E C(H,X). This is denoted by mF- T. (Note
that f(A) = supTE(A) {IAjET m(Ai)xi :l  x Ix i < 1}, where H(A) denotes the
collection of finite Y partions of A. See [2] or [4, Chapter 3] for a discussion of
the Riesz representation theorem in this setting.) The operator m  T is said
to be strongly bounded if f(Ai) - 0 for each pairwise disjoint sequence (A)
of members of Y.
LEMMA 3. If m - T : C(H,X) - Y is a DPI operator, then m is strongly
bounded.

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