LOZANOVSKIǏ’S PROOF OF DUNFORD’S THEOREM
Anton R. Schep
University of South Carolina
Let (X, µ) and (Y, ν) be σ-finite measure spaces. We denote by Lr (Ls ) the
Banach lattice {f (x, y) ∈ L0 (X × Y, µ × ν) : k kfy ks kr < ∞}, where fy denotes
the function x 7→ f (x, y). We denote by kf kr,s the norm k kfy ks kr < ∞ of f . It
is well known, see [B], that for 1 ≤ r, s ≤ ∞ the order continuous dual of Lr (Ls )
0
0
equals Lr (Ls ). In particular for 1 < p ≤ ∞ the space L∞ (Lp ) is the dual space
0
of L1 (Lp ), so that the unit ball of L∞ (Lp ) is weak*-compact with respect to this
duality. For k ∈ L∞ (Lp ) we denote by Tk the integral operator with kernel k, i.e.
Z
Tk (f ) =
k(x, y)f (y)dν(y) a.e.
Y
It is an exercise (by using adjoints) to show that Tk is a bounded linear operator
from L1 into Lp with kTk k = kkk∞,p .
Theorem (Dunford’s theorem [D]). Let T be a bounded linear operator from
L1 (Y, ν) into Lp (X, µ). Then there exists k ∈ L∞ (Lp ) such that T = Tk .
Proof (Lozanovskiǐ [L]). We may assume that T ≥ 0 and kT k ≤ 1. Define M =
{U : U = (χA1 , . . . , χAn ) : Ai ∩ Aj = ∅ for i 6= j, 0 < µ(Ai ) < ∞}. Then for U ∈ M
we denote
R(U ) = {k ∈ L∞ (Lp ) : k ≥ 0 a.e. , kkk∞,p ≤ 1
and Tk (χAi ) = T (χAi ) for i = 1, . . . , n}.
We shall first show that R(U ) 6= ∅ for all U ∈ M. Let U = (χA1 , . . . , χAn ) ∈ M.
Define
n
X
T (χAi )(x)χAi (y)
k(x, y) =
.
ν(A
)
i
j=1
Now clearly k ≥ 0, Tk (χAi ) = T (χAi ) and
n
X T (χ )(x)χ (y) Ai
Ai
kkk∞,p = ess sup ν(Ai )
y
i=1
≤ ess sup
y
n
X
p
χAi (y) = 1,
i=1
so k ∈ R(U ). Hence R(U ) =
6 ∅ for all U ∈ M. Next we show that R(U ) is weak*∞
p
closed in L (L ). Let kτ ∈ R(U ) such that kτ → k weak*. Then it easy to see
1
2
ANTON R. SCHEP
0
that k ≥ 0 and kkk ≤ 1. Let U = (χAi ) and let g ∈ Lp (X, µ). Then χAi (y)g(x) ∈
0
L1 (Lp ), so hkτ , g χAi i → hk, g χAi i. But by Tonelli’s theorem we have
Z
hkτ , g χAi i =
Z
g(x)Tkτ (χAi )(x)dµ(x) =
Z
Z
and
hk, g χAi i =
g(x)
g(x)T (χAi )(x)dµ(x)
k(x, y)χAi (y)dν(y) dµ(x).
Hence we have that
Z
Z
Z
χ
χ
g(x)T ( Ai )(x)dµ(x) = g(x)
k(x, y) Ai (y)dν(y) dµ(x)
0
for all g ∈ Lp . It follows that T (χAi ) = Tk (χAi ) for all i. Hence Tk ∈ R(U ).
We conclude from this that R(U ) is a weak*-compact subset of L∞ (Lp ). Next
we show that {R(U ) : U ∈ M} has the finite intersection property. Let U1 and
U2 ∈ M. Then let U denote the common refinement of U1 and U2 , where sets of
measure zero are omitted. Then U ∈ M and R(U ) ⊂ R(U1 ) ∩ R(U2 ) by linearity
of T and Tk . Hence {R(U ) : U ∈ M} has the finite intersection property. Weak*compactness of the unit ball of L∞ (Lp ) implies that there exists k ∈ L∞ (Lp ) such
that k ∈ ∩ {R(U ) : U ∈ M}. This implies that T (χA ) = Tk (χA ) for all measurable
sets A ⊂ Y with 0 < ν(A) < ∞. It follows from this that Tk = T and the proof is
complete.
Generalizations and Applications
One can generalize the above proof and obtain proofs of the author’s results
concerning characterizations of Carleman and Hille-Tamarkin integral operators
(see [S2]) and also Schachermayer’s characterizations of integral operators ([S1]).
References
A. Benedek and R. Panzone, The spaces Lp with mixed norm, Duke Math. J. 28 (1961),
301–304.
[D] N. Dunford, Integration and linear operations, Trans. Amer. Math. Soc. 40 (1936), 474–494.
[L] G. Ya. Lozanovskiı̌, N. Dunford’s theorem, Izv. Vysš Učebn. Zaved. Matem. 8 (1974), 58–59.
[S1] W. Schachermayer, Integral operators on Lp –spaces, Indiana Univ. Math. J. 30 (1981),
123–140.
[S2] A.R. Schep, Generalized Carleman operators, Proc. Kon. Ned. Akad. v. Wet. Series A 83(1)
(1980), 49–59.
[B]
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208