AN ABSOLUTE CONTINUITY FOR POSITIVE OPERATORS ON BANACH LATTICES

For pos it ve operators on a Banach latt ice, abso Iute cont nu ity cond t ions are cons dered. An operator abso Iute y cont nuous with respect to T is compared to sums of compositions of T together with orthomorphisms or in special cases projections. Consequences For compact operators on functions spaces C(X) are considered.


I. INTRODUCTION
For positive operators S and T between real Banach lattices several types of "absolute continuity" have been defined. Here, we consider an absolute continuity which will be applicable to spaces which are not necessarily Dedekind complete. Several approximations of an operator absolutely continuous with respect to T are provided in terms of sums of operators of the form Qi0ToHi where Qi and H are orthomorphisms. These approximations are compared to results known for operators S less than T and For operators on Dedekind complete Banach lattices. We also examine the relationship between this and previous notions of absolute continuity. We begin by recalling the following definitions.
DEFINITION. (Luxemburg [!]). Let E, F be Riesz spaces with S, T positive operators From E to F. We say that S is absolutely continuous with respect to T if for each positive element F in E, we have that SF is in the band generated by TF.
DEFINITION. (Feldman [2]). Let E, F be Banach lattices with S, T positive operators From E to F. We say that S is -absolutely continuous with respect to T if for each positive element f in E, we have that SF is in the closure of the order ideal generated by TF.
We note that for linear Functionals on E C(X) (the continuous real-valued functions on a compact topological space X) when E is Dedekind complete, these two notions are equivalent to the usual definition of absolute continuity. The absolute continuity Introduced here will be shown to be equivalent to the usual notion for functionals on any C(X).
In what Follows we will refer to a decreasing sequence {fk} of positive elements of a Banach lattice E as a positive decreasinq sequence in E. We now introduce our version of absolute continuity.
DEFINITION. Let E, F be Banach Lattices and let S, T be positive operators from E to F. We say that S is sequentia11y absolutely continuious (s-absolutely continuous) with respect to T if For each positive decreasing sequence {Fk} in E and each positive linear functional on F, we have that lim(@(TFn))=O implies lim((Sfn))=O.
We will be concerned with Banach lattices with quasi-interior points. An element e of a Banach lattice E is a quasi-interior point if the order ideal generated by e is dense in E. Recall that the order ideal generated by e is the set of all elements whose absolute value is bounded by some multiple of e. If E is equal to the order ideal generated by an element e then e is an order unit. Recall that if E is a Banach lattice with quasi-interior point, the elements of E can be represented as extended real valued functions on a contract set X each Finite on a dense subset (see [3]). We shall call X a representation space For E. Further, this representation contains C(X) as a dense order ideal. IF E has an order unit, the representation is equal to C(X). We denote bY Tthe subset of the linear operators from E to F which consists of a11 those positive operators S, For which S is s-absolutely continous with respect to T. Further, we denote the order ideal generated by a positive operator T by <T> and the set of positive operators which are less than some multiple of T by <T> +. In what follows, we identify elements in a Banach lattice with a quasi-interior point with their representation as extended real valued functions. IF S and T are positive operators from a Banach lattice E with a quasi-interior point e to a Banach lattice F, the range of S and T is contained in the closure of the lattice generated by the supremum of Se and Te. X will denote the representation space for E and Y the representation space for the Banach lattice generated by the supremum of Se and Te.

ABSOLUTE CONTINUITY.
We begin with 2 elementary lemmas. LEMMA I. Let S and T be linear Functionals on C(X), the set of continuous real valued Functions on a compact HausdorFF space X. Then S is s-absolutely continous with respect to T if and only if the measure associated with S is absolutely continous with respect to the measure associated with T.
PROOF. We note that a linear Functional @ on R corresponds to multiplication, thus @(TF n) converges to 0 if and only if TF n converges to O.
It is an easy exercise to see that S is s-absolutely continous to T then the measure associated with S is absolutely continous with respect to the measure associated with T.
The converse is a simple application of the Radon-Nikodym Theorem. LEMHA 2. Let E be a Banach lattice with quasi-interior point e and let $ be a positive linear functional on E. Given a representation space X for E, there exists a measure u such that For each g in E, (g) /gdu@.

PROOF.
Since C(X) is dense in E and the sequence [gAne} converges in norm to g For g non-negative, the sequence [@(gAne)} converges to Q(g). It can be verified that the measure corresponding to the restriction of to C() represents $.
We now give a sufficient condition for s-absolute continuity.
PROPOSITION I. Let E be a Banach lattice with quasi-interior point e and F a Banach lattice with S, T positive operators from E to F. If for each positive decreasing sequence of functions {fn in E and for each y in the representation space Y, the convergence of Tfn(y) to 0 implies the convergence of Sfn(y) to 0 then S is s-absolutely continous with respect to T. 0 and ySfndu SASfndU. Since h(y) 0 on A and {Tfn(y)} is decreasing, we have that Tfn(y) converges to 0 on A and thus Sfn(y) converges to 0 by hypothesis. Since Sfn(y) ! Sfl(y) and Sfn(y) converges to 0 on A, the Monotone Convergence theorem ilies im SySfndU lira /ASFndU 0 Thus we have that $(Sf n) converges to O, that is that S is s-absolutely continous with respect to T.

PROOF. Let
We note that in the case when F C(Y), the converse of the proposition is also true since y0T defines a positive linear functional on F.
It is obvious that if S < T then S is s-absolutely continous with respect to T, i.e. contains <T> +. it is an easy exercise to show that 7 is closed and thus contains even the closure of <T> +. PROPOSITION 2. ]vis a closed subset of L(E, F), the linear operators from E to F with respect to the operator norm. In particular, contains the closure of <T> + We now compare and contrast these notions oF order and absolute conti nuity when the range s an M-space. THEOREM I. Let E be a Banach lattice with quasi-interior point and S, T be positive operators from E to C(Y). Consider the conditions. i) S is -absolutely continous wlth respect to T (in the sense of Luxemburg) ii) S is -absolutely continous with respect to T iii) S is s-absolutely continous with respect to T iv) S is in the closure of <T> +. to T and f Z O, SF(y) 0 implies Tf(y) > O. For a given > 0 and g such that 0 < g SF, let A be the set [yl (g e)(y) Z 0}. Then A is compact and hence there exists a > 0 such that TF Z (g e). Therefore we have that Tf (SF e) and thus (5F Be)VO is in the order ideal generated by TF and therefore SF is In the uniform closure of the ideal generated by Tf. Thus we have (iii) implies (ii). That (Ii) implies (i) Follows from the Fact that the closure of the ideal generated by TF is contained in the band generated by Tf.
That no other implications hold is shown by the following examples. We will assume that C(X) is endowed with the sup-norm topology. EXAMPLE I. We give here operators S and T such that S is absolutely continuous with respect to T, but $ is not -absolutely continuous with respect to T. Define  0 then SF is not in the closure of the order ideal generated by TF and thus S is not -absolutely continuous with respect to T. EXAMPLE 2. Here we give operators S and T such that S is -absolutely continous with respect to T, but S is not s-absolutely continous with respect to T. Let N" denote the one point compactiFication of N. Define operators from C(N') to C(N') as follows.
Sf(x) Fix) Tf(x) {I: [fin)l/In2)]1 (1 denotes Fix) 11 n=! Since TF is constant, 5f is less than some multiple of TF, i.e. in the order ideal generated by TF0 Thus S is -absolutely continous with respect to T.

IIS-T'II 2_ IlSg-T'gil Z ISg(m) T'g(m) > l/2
and thus S can not be In the closure of the order ideal generated by T. It is routine to check using Proposition that S Is s-absolutely continous wlth respect to T. 3

PROOF. If e is a quasi-lnterior point of E then For a given f in E +, we
have that e + f is also a quasl-interior point. Thus we can choose the representation space X so that IF F is In C(X) and SF and TF are in C(Y). Now, assume that S is s-absolutely continous with respect to T. For each fixed y in Y, let Uy and Vy be the measures corresponding to the functionals (y0T) and (y0S), respectively. By the Riesz Representation Theorem, Sfgyduy.
Since (y0S) is continuous, we have that S(1)y < (R), and therefore gy is in LI(X, Uy). Given 6 > 0 and f in C(X), there is an hy in C(X) such that lgy hyll < Elllfll. (see [5], Thm 25.10). For each y in Y, choose such a neighborhood. Since Y is compact there is a finite nun3er of these neighborhoods which cover Y. We label the neighborhoods N for 1,2 n. Further, functions qi in C(Y) can be chosen (a partition of unity, see e.g. [6], p. 63) such that n 0 qi 'l'ilqi I We define orthomorph sms QI on C (Y) and H on C(X) by Thus we have by extending to E and F (e.g., see [2] The next result, motivated by results which were established for operators S which are in the ideal generated by T (on M-spaces by AIiprantis and Burkinshaw [6] and on Banach lattices with quasi-interior points by Haid [9]), is a direct corollary of Theorem 2. COROLLARY 2. Let E and F be Banach lattices with quasi-interior points and let S, T be positive operators from E to F such that S is s-absolutely continous with respect to T. Then where hy is the continuous function as described in the proof of Theorem 2.
Letting HM be the orthomorphism on C(X) defined by multiplication by by, we have that G is compact, since both S and THy are compact. Further, as in the proof of Theorem 2, we have 16f(y) < 6.
We will show that there is a neighborhood Ny of y such that GF(Ny) is contained in (-3&,36) for all F in C(X) such that Ilfil, 1.
Assume that this is not the case. Then there exists a net lye} in Y such that Ye converges to y, and there exist functions fe in C(X) with ilFall and satisfying both of the following i) Gfa(Ya > 3&, For all a. 11) GFa(y) < 6.
Since G is compact, there exists a subnet of {GFo} converging to some Function g" in C(Y We Further note that the approximation given in Theorem 2 is not, in general, uniform. Let E F C(N') and define SF(x) f(x) and TF(x) (n(n)/n2=i + f((R)))l. As stated earlier S is s-absolutely continous with respect to T. T is compact (it has rank I), but S is not compact. IF the approximation given in Theorem 2 were uniform, then Theorem 4 would imply S is compact.