Table Of ContentInverses of Disjointness
Preserving Operators
MEMOIRS
of the
American Mathematical Society
Number 679
Inverses of Disjointness
Preserving Operators
Y. A. Abramovich
A. K. Kitover
January 2000 0 Volume 143 0 Number 679 (first of4 numbers) 0 ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification.
Primary47B60, 47865, 47B38, 46A40, 46B40, 46B42; Secondary 54G05.
Library ofCongress Cataloging-in-Publication Data
Abramovich, Y. A. (Yuri A.)
Inverses ofdisjointness preserving operators / Y. A. Abramovich, A. K. Kitover.
p. cm. —- (MemoirsoftheAmerican Mathematical Society, ISSN 0065-9266 ; no. 679)
“January 2000, volume 143, number 679 (first of4 numbers)."
Includes bibliographical references
ISBN 0-8218-1397-8 (alk. paper)
1. Banach modules. 2. Operatortheory. 3. Banach lattices. I. Kitover, A. K. (Arkady K.).
II. Title. III. Series
QA3.A57 no. 679
[QA326]
510s—dc21
[512’.55] 99-054530
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Table of Contents
. Setting forth the problems
. Some history
. Synopsis of the main results 10
. Preliminaries 17
. The McPolin—Wickstead and Huijsmans—de Pagter—Koldunov
Theorems revisited 26
d-bases 32
. Band preserving operators and band-projections 47
. Central operators and Problems A and B 57
9. Range—domain exchange in the Huijsmans—de Pagter—Koldunov
Theorem 72
10. d—splitting number of disjointness preserving operators . 78
11. Essentially one—dimensional and discrete vector lattices 85
12. Essentially constant functions and operators on C[0, 1] . . 91
13. Counterexamples 104
14. Dedekind complete vector lattices and Problems A and B 126
15. Generalizations to (rd-complete vector lattices 143
16. Open problems 155
References 158
Index 163
vii
Abstract
A linear operator T : X —> Y between vector lattices is said to be disjointness
preserving if T sends disjoint elements in X to disjoint elements in Y. The bijective
disjointness preserving operators are the central object of this work. The following
three types of results are obtained.
1) For a bijective disjointness preserving operator T : X —> Y a number of re-
sults are proved demonstrating that (under some mild additional conditions on the
vector lattices) the inverse operator T"1 is also disjointness preserving, and further-
more the vector lattices X and Y are order isomorphic. Moreover, for a Dedekind
complete vector lattice X necessary and sufficient conditions are found under which
any bijective disjointness preserving operator on X has the disjointness preserving
inverse. We prove also that any d—isomorphic Dedekind complete vector lattices are
order isomorphic.
2) A general method is presented for producing bijective disjointness preserving
operators T : X —» Y (with various conditions on X and Y) for which T‘1 fails to
preserve disjointness.
3) A general methodis presentedforproducingbijectiveoperators T 2 X —» Yfor
which both T and T—1 preserve disjointness but X and Y are not order isomorphic.
It should be pointed out that the results referred to in 1)—3) answer several well
known open problems concerning operators on vector lattices.
1991 Mathematics Subject Classification: 47B60, 47B65, 47B38, 46A40, 46B40,
46B42; 54G05.
Key words and phrases: Disjointness preserving operators, band preserving op—
erators, invertible operators, order isomorphism, vector lattice, Dedekind complete
vector lattice.
Received by the editor January 22, 1998
viii
1. Setting forth the problems
In this section we set forth the main problems that will be addressed in our work.
Recall that a (linear) operator T : X —» Y between vector lattices is said to be
disjointness preserving if T sends disjoint elements in X to disjoint elements
in Y. Now, let T : X —> Y be a one-to-one disjointness preserving operator
from a vector lattice X onto a vector lattice Y. Therefore, the inverse operator
T‘1 : Y —» X exists. The following problem will be our main topic in this work.
Problem A: Is T‘1 also a disjointness preserving operator?
It is worth mentioning right here that in the majority of cases when we are able
to resolve Problem A in the affirmative, we are also able to prove one more surpris—
ing fact that the vector lattices (between which a bijective disjointness preserving
operator acts) are order isomorphic. This leads us naturally to the next problem
that will be our second main topic in this work.
Problem B: Let T : X —> Y be a bijective disjointness preserving operator such
that T‘1 is disjointness preserving. Are then X and Y order isomorphic?
Postponing a detailed description of the content of the present work until after
some history ofthe problems is discussed in Section 2, here we mention only that, as
we will demonstrate, in the absenceofsomeadditional hypotheses eachofthe above
problems has a negative answer. The majority of corresponding counterexamples
will be constructed in Section 13.
The existence of such examples has certainly led us to a change of emphasis
regarding the problems under consideration; namely, now we are interested in a
search for some additional (ultimately necessary and sufficient) conditions on vector
lattices or/and operator that would guarantee the affirmative solutions to these
problems. To a large extent we understand now what these conditions should be
and how to construct counterexamples in their absence. A considerable part of our
work (Sections 8-12 and 14) is devoted to these more general problems. Some of
our results in this direction have been announced without proofs in [AK2].
2. Some history
In someformorother disjointness preserving operators appeared in the literaturefor
the first time in 40’s (see, for example, [Vl,2]), but only during the last 15—20 years
have they become the object of systematic study. We mention here only several
monographs [AAK], [AB], [BGK], [L], [MN], [S], [Z], and the survey [H] in which
these operators occupy a prominent role. (Neither in this section nor anywhere
else in this work we intend to touch a vast literature on the spectral properties of
disjointness preserving operators.)
One of the external reasons for the recent interest in disjointness preserving
operators is the fact that precisely the regular disjointness preserving operators
allow a multiplicative representation as weighted composition operators; thus the
disjointness preservingoperators providean abstract framework for averyimportant
class ofoperators in analysis. Werefer to [AAK] and [A] for results in this direction.
All vector lattices in this work are assumed to be Archimedean and over the
reals. At the same time it should be stressed that all our results remain true for the
complex scalars as well.
Two elements 221,12 of a vector lattice are disjoint (in symbols: :01 J. :02) if
[.21] A [1:2] = 0. Following [AVK2] we say that a one—to-one and onto operator
T : X —+ Y between vector lattices is a d-isomorphism ifboth T and T'1 preserve
disjointness,l that is, I] i at; in X ifand only if T31 L T2; in Y.
In terms ofd-isomorphisms Problem B simply asks ifd-isomorphic vector lattices
are order isomorphic. It was proved in [AVK1,2] that for Banach lattices the answer
is “yes.”
We will discuss below some of the known cases when Problem A has an affir-
mative solution. The strongest result that has been obtained so far was established
by Huijsmans—de Pagter [HP] and independently by Koldunov [K3]. T0 formulate
this result we need to remind the reader the definition of an (rd-complete (rela-
tively uniformly complete) vector lattice. Among various equivalent definitions we
1This definition ofa d-isomorphism is more restrictive than that in [AVKl], where it was not
required that T be onto.
INVERSES OF DISJOINTNESS PRESERVING OPERATORS 3
choose the one that is most convenient for us. Namely, we say that a vector lattice
X is (rd—complete if for each x E X the principal ideal X(.1:) generated by a:
is order isomorphic to the space of continuous functions C(K) on an appropriate
compact Hausdorff space. We refer to [V3] or [Z] for an internal definition of (ru)—
completeness. The following theorem explains the role played by (rd-completeness
in the problems under discussion.
Theorem 2.1 ([A], Theorem A) Let X,Y be two arbitrary vector lattices and
T : X a Y be a disjointness preserving operator. Then the following conditions are
equivalent:
1) T is regular,
2) T is sequentially (ru)-continuous,
3) T is sequentially (r1, — o)—continu0us,
4) T is order bounded.
5) T satisfies condition (R), that is, infn(|Tz;,| + |T$Z|) = 0, for any two se—
quences (mg) and (wit) that (rd-converge to zero in X.
In actuality [A, Theorem A] contains one more equivalent property which relates
theaboveconditions with themultiplicativerepresentationofdisjointness preserving
operators. It is obvious that condition (R), introduced in 5), is the weakest of all
five conditions. There were numerous attempts to replace two sequences in (R) by
just one. This was accomplished by McPolin and Wickstead in [MW] who gave a
very technical proof of this result. In Section 5 we will present a very simple and
elementary proofof this important fact. Moreover, as we will show, it is enough to
consider only monotone (rd-convergent sequences in (R).
Now we are ready to formulate the Huijsmans—de Pagter—Koldunov Theorem
(see [HP, Theorems 2.1, 2.3 and Corollary 2.2], and [K3, Theorem 3.6]). For brevity,
we will often refer to this theorem as the HPK—Theorem.
Theorem 2.2 (Huijsmans—de Pagter—Koldunov) Let T be a one-to-one dis-
jointness preserving operatorfrom an (rd—complete vector lattice X into a normed
4 Y. ABRAMOVICH AND A. KITOVER
lattice Y. Then T321 _L T32 in Y implies that 9:1 J. :52. Moreover, if the operator T
is additionally onto, then T is regular.
In other words, for such vector lattices as in Theorem 2.2, Problem A has an
affirmative solution. It will be easy to show (see Corollary 4.13) that under the
same conditions Problem B also has an affirmative solution. It is important to note
that the surjectivity hypothesis in the HPK—Theorem is essential for the regularity
of T (even if X and Y are Banach lattices). This follows from Example 1 in [A].
Nevertheless, we will demonstrate later that both hypotheses on T in the HPK—
Theorem as well as the normability assumption on Y can be considerably relaxed.
This (together with an independent proof of the HPK—Theorem) will be done in
Lemma 5.2 and Theorem 5.3.
It was proved in [AVK2, Theorem 4.3] (see also [AVK1, Theorems 4 and 5]) that
each d—isomorphism between Banach lattices is continuous. The previous theorem
allows us to improve this result on automatic continuity by relaxing the assumption
that T : X ——i Y is a d—isomorphism to the preservation of disjointness only. The
prooffollows immediatelyfromTheorem2.2 sinceeachregularoperatoron a Banach
lattice is continuous.
Corollary 2.3 IfT : X —» Y is a bijective disjointness preserving operator, where
X is a Banach lattice and Y is a normed lattice, then T is continuous.
Another theorem of Huijsmans and de Pagter, in which X is a discrete vector
lattice, reads as follows.
Theorem 2.4 ([HP], Theorem 2.6) IfT is a one-to-one disjointness preserving
operatorfrom a discrete vector lattice X into a vector lattice Y, then the operator
T‘1 : T(X) —> Y is also disjointness preserving.
We will obtain Theorem 2.4 as a special case ofTheorem 11.2 in which a similar
result will be proved for a more general class ofvector lattices. Note that under the
conditions ofTheorem 2.4 the vector lattice Y does not need to be discrete. Indeed,
