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Vladikavkaz Mathematical Journal 2009, Vol. 11, No 2, pp. 46–49

UDC 517.98

THE ORDER CONTINUOUS DUAL OF THE REGULAR

INTEGRAL OPERATORS ON Lp

Dedicated to Safak Alpay on the occasion of his sixtieth birthday

Anton R. Schep

In this paper we give two descriptions of the order continuous dual of the Banach lattics of regular integral operators on Lp. The ﬁrst description is in terms of a Calderon space, while the second one in terms of the ideal generated by the ﬁnite rank operators.

Mathematics Subject Classification (2000):47B65, 47B34.

Key words:Integral operators, order dual, Lp-spaces.

Introduction

LetI_r₍_Lp₎_{denote the collection of all regular integral operators on}_Lp ₍₁_{< p <}_∞₎_with

the regular norm k · kr. Then Ir(Lp) is a Banach function space on 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₍_Lp_{). Our first description follows from a result on} _I

r(Lp). In [3]

we showed that I_r₍_Lp₎ _{is equal to the Calderon space} ₍_L_∞_,₁₎p1′₍_Lt

∞,1)

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 Lp provided with an extension of the positive projective tensor norm is a Banach function space with the Fatou property, from which it follows that it is isometric to the order continuous dual ofI_r₍_Lp_).

The order continuous dual as Calderon space

Throughout this paper (X, µ) will denote a σ-finite measure space and p will be a fixed real number with1< p <∞. Recall the definition of a regular integral or kernel operator on

Lp-spaces. Let T(x, y)be a µ×µ-measurable function onX×X. ThenT(x, y) is the kernel of an integral operator T fromLp intoLp if

Z

X

|T(x, y)f(y)|dµ(y)<∞ a. e.

for allf ∈Lp _and

T f(x) =

Z

X

T(x, y)f(y)dµ(y)∈Lp

c

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The order continuous dual of the regular integral operators onLp 47 for all f ∈ Lp. If in addition |T(x, y)|is the kernel of an integral operator (denoted by |T|) fromLp _into_Lp_{, then}_T _{is called a regular (or order bounded) integral operator. By} _I

r(Lp)

we shall denote the collection of all such regular integral operators onLp. If we equip I_r₍_Lp₎

with the regular norm k · kr, i. e., the operator norm of the modulus operator |T|, then it is

well-known thatI_r₍_Lp₎_{becomes a Banach function space on}_X_×_X_{with the Fatou property.}

Many order-theoretic properties ofI_r₍_Lp₎_{are known, i. e.,}_I

r(Lp)equals the band generated

by Lp′ ⊗Lp in L_r₍_Lp₎ _{(see e. g. [4]). By} K_r₍_Lp₎ _{we shall denote the closure of} _Lp′ _⊗_Lp in L_r₍_Lp₎ _{with respect to the regular norm. It is a consequence of e. g. the Dodds-Fremlin}

theorem (see [4]) that K_r₍_Lp₎ _{is an ideal in} _I

r(Lp) and is equal to the ideal of elements of

order continuous norm inI_r₍_Lp_{). Finding a description of the order continous dual of}_I r(Lp)

is the same as finding a description of the norm dual ofK_r₍_Lp_{). For a measurable function}_F onX×X we define for 16p <∞ the normkFk∞,p as follows

kFk∞,p =

° ° ° ° ° µZ

|F(x, y)|p_dy

¶1 p ° ° ° ° ° ∞ . We denote

L∞,p=

©

F ∈L0(X×X) :kFk∞,p<∞

ª

.

Given F on X×X we define the transpose of F by Ft(x, y) =F(y, x). Then Lt_∞_,p will denote the collection of all F such that Ft_∈_L

∞,p and the norm onLt∞,p will be defined by kFt_k

∞,p. In[3] we proved the following theorem.

Theorem 1.1. Let 1 < p < ∞. Then L∞,p′ ·Lt_∞_,p is a product Banach function space isometrically equal to I_r₍_L_p₎ _{and for any} _T _∈ I_r₍_L_p₎ _{we have a factorization} _T₍_{x, y}_{) =}

T1(x, y)·T2(x,y) withkTkr =||T1k∞,p′kT₂tk_∞,p. In particular Ir(Lp) = (L∞,1)

1

p′₍_Lt

∞,1)

1

p_.As a consequence we get our first description of the order continuous dual of I_r₍_Lp_).

Theorem 1.2. Let 1 < p < ∞. Then the order continuous dual of I_r₍_Lp₎ _{is equal to}

(L1,∞)

1

p′₍_Lt

1,∞)

1

p =_L

p′_,_∞Lt_p,_∞.

⊳Recall first Lozanovskii’s theorem. IfEandF are Banach function spaces with the Fatou

property and 1 < p < ∞, then ³E1p_F

1

p′´′ _{= (}_E′₎1p(_F′)

1

p′_{. Applying this to the situation at}

hand we get (I_r₍_Lp₎₎′ _{= (}_L′ ∞,1)

1

p′₍₍_Lt

∞,1)′)

1

p_{. Now it is easy to see that} _L′

∞,1 = L1,∞ and

similarly(Lt_∞_,₁)′ ₌_Lt

1,∞. Hence the result follows.⊲

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 productLp′⊗Lp _of_Lp′

⊗Lp can be identified with the Riesz subspace generated byLp′

⊗Lp_in_I

r(Lp). OnLp ′

⊗Lp _{we can define}

the positive projective tensor norm k · k_|π|, which by Theorems 2.1 and 2.2 of [2] is equal to

kfk_|_π_|= inf{kgkp′khk_p : |f|6g⊗h, 06g∈Lp ′

, 06h∈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 productLp′⊗e|π|Lp ofLp

′

withLp _{is now the completion of}_Lp′

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48 Anton R. Schep

was also observed thatLp′⊗e_|_π_|Lp is contained in the ideal generated by Lp′⊗Lp inI_r₍_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 forLp′⊗e_|_π_|Lp. We consider therefore the extension of the positive projective norm to the ideal generated by the finite rank operators. More precisely, denote by F₍_Lp₎ _{the ideal in} _I

r(Lp) generated by the finite rank operators and define for f ∈F₍_Lp₎

kfk_|_π_|= inf{kgkp′khk_p: |f|6g⊗h, 06g∈Lp ′

, 06h∈Lp}.

It is clear that this defines a norm on F₍_Lp₎ _{if we realize that the unit ball} _B_F ₌ _{f _∈ F₍_Lp_{) :}_kf_k

|π| 61} is the solid hull of the unit ball of Lp

′

⊗Lp _{(and in fact also of the unit}

ball ofLp′

e

⊗|π|Lp). The main result of this section is now the following theorem.

Theorem 2.1. The normed K¨othe space (F₍_Lp₎_,_{k · k}

|π|) has the Fatou property. In

particular(F₍_Lp₎_,_{k · k}

|π|) is complete.

⊳ It suffices to prove that the unit ball BF = {f ∈ F(Lp) : kfk_|_π_| 6 1} is closed in measure inL0(X×X, µ×µ). Let 06fn(x, y)∈BF and assume fn(x, y)→f(x, y) a. e. on X×X. Without loss of generality we can assume thatkfnk_|π|<1 for alln. Then there exist

gnandhnwithkgnkp′ 61,kh_nk_p 61such thatf_n6g_n⊗h_n. By using Komlos’ Theorem (see

Theorem 3.1 of [3]) we can find subsequencesgnk andhnk such thatgnk(y) Cesaro converges a. e. onXto g(y)andhn_k(x)Cesaro converges a. e toh(x)onX. This implies thatkgkp′ 61

andkhkp61. Nowgnk⊗χX Cesaro converges a. e. tog⊗χX andχX⊗hnk Cesaro converges a. e. toχX⊗hon X×X. From06fnk 6gnk⊗χX·χX⊗hnk it follows now from Theorem 2.3 of [3] thatf(x, y)6g⊗χX ·χX⊗h=g⊗h, which shows thatf ∈BF. This shows that

BF is closed in measure and thus the normed K¨othe space (F(Lp),k · k_|_π_|) has the Fatou property.⊲

As a consequence of the above theorem we get our second description of the order continuous dual ofI_r₍_Lp_).

Theorem 2.2.The order continuous dual of the space(F₍_Lp₎_,_k·k_|_π_|₎_isI_r₍_Lp₎_{. Therefore}

the norm dual ofK_r₍_Lp₎_{and the order continuous dual of}_I

r(Lp)is equal to(F(Lp),k · k|π|).

⊳ Let k(x, y) define an order continuous functional T of norm less or equal to one on (F₍_Lp₎_,_{k · k}

|π|). Then h|k|,|g| ⊗ |h|i <∞ for all g⊗h ∈Lp

′

⊗Lp implies that k defines an integral operator Tk ∈Ir(Lp). Moreover

kTkkr = sup(h|k|,|g| ⊗ |h|i:kg⊗hk|π|61) = sup((h|k|,|f|i:kfk|π|61) =kkkF′.

Hence the mapk7→Tk is an isometric lattice isomorphism from(F(Lp),k · k|π|)intoIr(Lp).

Now it is straightforward to verify that this mapping is onto, as the kernel of a positive kernel operator in I_r₍_Lp₎ _{of norm less or equal to one defines a positive linear functional} on (F₍_Lp₎_,_{k · k}_|_π_|₎ _{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₍_Lp₎_,_{k · k}

|π|).⊲

Remark. _{(1) One can ask why the above results are restricted to}_Lp_{-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₍_Lp₎_,_{k · k}

|π|). The proof of this fact in Theorem 2.1 depended essentially on

the fact that the positive projective norm on Lp′ ⊗_|π|Lp can be obtained using only single

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The order continuous dual of the regular integral operators onLp 49 (2) We observe that the second part of the above theorem can be considered as an order continuous analogue of the classical duality of compact operators, trace class operators and all bounded operators.

References

1. Fremlin D. H.Tensor products of Banach lattices // Math. Ann.—1974.—Vol. 211.—P. 87–106. 2. Schep A. R.Factorization of positive multilinear maps // Illinois J. Math.—1984.—Vol. 28, №. 4.—

P. 579–591.

3. Schep A. R.Products of Cesaro convergent sequences with applications to convex solid sets and integral operators // Proc. Amer. Math. Soc.— 2009.—Vol. 137, №. 2.—P. 579–584.

4. Zaanen A. C.Riesz spaces, II (North-Holland Mathematical Library).— Amsterdam: North-Holland Publishing Co., 1983.—Vol. 30.—720 p.

Received November 11, 2008.

Anton R. Schep

Department of Mathematics University of South Carolina,Prof. USA, Columbia, SC 29208