# Forest above Ground Biomass Inversion by Fusing GLAS with Optical Remote Sensing Data Page: 2,376

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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.2376

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Bator, Elizabeth M. & Slavens, Dawn R. Forest above Ground Biomass Inversion by Fusing GLAS with Optical Remote Sensing Data, article, February 11, 2003; Cairo, Egypt. (digital.library.unt.edu/ark:/67531/metadc990996/m1/2/: accessed February 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.