The order continuous dual of the regular integral operators on Lp

Let I r ( L p) denote the collection of all regular integral operators on L ^p (1 < p <  to ) with the regular norm || • ||_r . Then I r ( L p) is a Banach function space on X x X with the Fatou property. In this paper we shall give two distinct descriptions of the order continuous dual, or associate space, of I r ( L ^p ) . Our first description follows from a result on I r ( L ^p ) . In [3] 11

we showed that I r ( L p) is equal to the Calderon space ( Ь _од) p⁰ ( L oo 1 ) ^p • As a consequence we derive our first description via Lozanovskii’s duality theorem. Then we will present a different description of this space. We will prove that the order ideal generated by the finite rank operators on L ^p provided with an extension of the positive pro jective tensor norm is a Banach function space with the Fatou property, from which it follows that it is isometric to the order continuous dual of I r ( Lp).

The order continuous dual as Calderon space

Throughout this paper (X, ^) will denote a ст-finite measure space and p will be a fixed real number with 1 < p < to. Recall the definition of a regular integral or kernel operator on bp-spaces. Let T(x,y) be a ^ x ^-measurable function on X x X. Then T(x,y) is the kernel of an integral operator T from Lp into Lp if j |T(x,y)f (y)|d^(y) < to a. e.

X for all f G Lp and

Tf ( x ) = j T ( x,y ) f ( y)dy ( y) G b ^p

X

p

.

∞

We denote

L ^ ,p = {F E L o ( X x X ) : | F U,p <  ГО .

Given F on X x X we define the transpose of F by F t ( x,y ) = F ( y,x). Then L^ p will denote the collection of all F such that F ^t ∈ L _{∞ ,p} and the norm on L ^t _{∞ ,p} will be defined by k F ^t k _{∞ ,p} . In[3] we proved the following theorem.

Theorem 1.1. Let 1 < p <  ro . Then L _x ,p • L^ p is a product Banach function space isometrically equal to I r ( L p ) and for any T E I r ( L p ) we have a factorization T ( x, y ) = 11

T i ( x,y ) • T 2( x,y ) with k T k r = || T i k ^ ,p ' ||T tU,p . In particular I r( L p) = ( L ^ , i ) ^p' ( L^ ) ^p . As a consequence we get our first description of the order continuous dual of I r ( Lp).

Theorem 1.2. Let 1 < p <  ro . Then the order continuous dual of I r ( Lp) is equal to ¹ t ¹                   t

^{( L} 1 ,^) ^{p ( L} i ,^) ^p = ^L p ' , ^ ^L p, ^-

C Recall first Lozanovskii’s theorem. If E and F are Banach function spaces with the Fatou 1      1 ⁰                  1           1

property and 1 < p < ro, then IEp Fp' I = (E 0)p (F 0)p'. Applying this to the situation at hand we get (Ir(U))’ = (L'^ 1)p' ((L^ 1 )0)p . Now it is easy to see that L0^ 1 = Lp^ and similarly (L^ 1)0 = L* ^. Hence the result follows. B

The order continuous dual as the ideal generated by the finite rank operators

We now recall from [2] some facts about the positive projective tensor product of Banach lattices, applied to our situation. The Riesz space tensor product Lp0 ⊗Lp of Lp0 ⊗ Lp can be identified with the Riesz subspace generated by Lp 0 Lp in Ir (Lp). On Lp 0Lp we can define the positive pro jective tensor norm k · k|π| , which by Theorems 2.1 and 2.2 of [2] is equal to kf k|n| = inf{kgkp'khkp : |f | 6 g 0 h, 0 6 g E Lp', 0 6 h E Lp}.

Note, in general we can not restrict ourselves to majorants consisting of single tensors, but we can in this particular case. Moreover the positive projective tensor product Lp0 ⊗|π|Lp of Lp0 with Lp is now the completion of Lp0 ⊗Lp with respect to this norm. In Theorem 2.1 of [2] it was also observed that Lp00|n|Lp is contained in the ideal generated by Lp0 0 Lp in Ir(Lp). It was proved by Fremlin [1]that L2⊗e |π|L2 is not Dedekind complete in general and similar arguments show that the same is true for Lp0 ⊗e |π|Lp. We consider therefore the extension of the positive pro jective norm to the ideal generated by the finite rank operators. More precisely, denote by F(Lp) the ideal in Ir(Lp) generated by the finite rank operators and define for f G F(Lp)

k f ||_N =inf {k g k p o k h k p : | f | 6 g 0 h, 0 6 g G L ^p 0 , 0 6 h G L p } .

It is clear that this defines a norm on F ( L p ) if we realize that the unit ball B f = { f G F ( L p ) : k f |||_n| 6 1 } is the solid hull of the unit ball of L ^p 0 0 L ^p (and in fact also of the unit ball of L ^{p 0} ⊗ e | _π | L ^p ). The main result of this section is now the following theorem.

Theorem 2.1. The normed Kothe space (F( L p ) , || • k | _n | ) has the Fatou property. In particular (F( L p ) , || • k | _n | ) is complete.

C It suffices to prove that the unit ball B f = { f G F( L p ) : k f | | _n | 6 1 } is closed in measure in L 0 ( X x X, ^ x ^ ) . Let 0 6 f _n ( x,y ) G B f and assume f _n ( x,y ) ^ f ( x,y ) a. e. on X x X . Without loss of generality we can assume that | f _n | | _n | <  1 for all n . Then there exist g _n and h _n with | g _n | _p o 6 1 , | h n k _p 6 1 such that f _n 6 g _n 0 h _n . By using Komlos’ Theorem (see Theorem 3.1 of [3]) we can find subsequences g n k and h n k such that g n k ( y ) Cesaro converges a. e. on X to g ( y ) and h n k ( x ) Cesaro converges a. e to h ( x ) on X . This implies that k g k p o 6 1 and ||h k p 6 1 . Now g n k 0 X X Cesaro converges a. e. to g 0 x x and x x 0 h n k Cesaro converges a. e. to x x 0 h on X x X . From 0 6 f n k 6 g n k 0 X X • X X 0 h n k it follows now from Theorem 2.3 of [3] that f ( x, y ) 6 g 0 X X • X x 0 h = g 0 h , which shows that f G B f . This shows that B f is closed in measure and thus the normed Kothe space (F( L p ) , k • k | _n | ) has the Fatou property. B

As a consequence of the above theorem we get our second description of the order continuous dual of I r ( L p ) .

Theorem 2.2. The order continuous dual of the space (F ( L p ) , k^k | _n | ) is I r ( L p ) - Therefore the norm dual of K r ( L p ) and the order continuous dual of I r ( L p ) is equal to (F( L p ) , k • k | _n | ) -

C Let k(x,y) define an order continuous functional T of norm less or equal to one on (F(Lp), k • k|n|). Then h|k|, |g| 0 |h|i < ro for all g 0 h G Lp 0 Lp implies that k defines an integral operator Tk G Ir(Lp). Moreover kTkkr = sup(h|k|, |g| 0 |h|i : kg 0 hk|n| 6 1) = sup((h|k|, |f |) : kf kH 6 1) = IMf0•

Hence the map k ^ T k is an isometric lattice isomorphism from (F( L p ) , k • k | _n | ) into I r ( L p ) . Now it is straightforward to verify that this mapping is onto, as the kernel of a positive kernel operator in I r ( L p ) of norm less or equal to one defines a positive linear functional on (F( L p ) , k • k | _n | ) of norm less or equal to one and the proof of the first assertion follows. The remaining statements are now an immediate consequence from the Fatou property of the norm on (F( L p ) , k • k | _n | ) . B

Remark. (1) One can ask why the above results are restricted to L ^p -spaces and whether the results can’t be extended to more general Banach lattices. For the first part of the last theorem that seems possible, but the second part is not clear as it depends on the Fatou property of (F( L p ) , k • k | _n | ) . The proof of this fact in Theorem 2.1 depended essentially on the fact that the positive projective norm on L ^{p 0} ⊗ | _π | L ^p can be obtained using only single tensors instead of finite sums of tensors. This property is known only for E ⁰ ⊗ E when E is isomorphic to an L ^p -space (see [2]).

(2) We observe that the second part of the ab ove theorem can be considered as an order continuous analogue of the classical duality of compact operators, trace class operators and all bounded operators.