Table Of ContentLecture Notes ni
Mathematics
Edited by A. Dold and B. Eckmann
Series: Mathematisches Institut der Universit~t Erlangen-N0rnberg
Advisers: .H Bauer dna .K Jacobs
904
Klaus Donner
noisnetxE fo Positive Operators
dna Korovkin Theorems
IIIII 11! ETHICS MIIIIII ETH-BIB !
O0100000802840
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Berlin Heidelberg New York 1982
Authors
Klaus Donner
Mathematisches Institut, Universit~t Erlangen-N~rnberg
BismarckstraBe 1 1/2, 8520 Erlangen
AMS Subject Classifications (1980) 40-A-05, 46-A-22, 46-B-30
ISBN 3-540-11183-2 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38?-11183-2 Springer-Verlag New York Heidelberg Berlin
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Section I: Cone imbeddings for vector lattices . . . . . . . . 2
Section 2: A vector-valued Hahn-Banach theorem . . . . . . . . 12
Section 3: Bisublinear and subbilinear functionals . . . . . . 30
Section 4: Extensions of L1-valued positive operators ..... 68
Section 5: Extension of positive operators in LP-spaces. 84
Section 6: The Korovkin closure for equicontinuous
nets of positive operators. . . . . . . . . . . . . 105
Section 7: Korovkin theorems for the identity mapping
on classical Banach lattices . . . . . . . . . . . . 127
Section 8: Convergence to vector lattice homomorphisms
and essential sets . . . . . . . . . . . . . . . . . 162
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . 174
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Introduction
Examining the various branches of functional analysis, it will be dif-
ficult to find a section in which extension problems of linear opera-
tors are completely absent. Usually, there are certain properties such
as continuity, positivity, compactness, contractivity, or commutativ-
ity with given operators that have to be preserved in connection with
the extension of the linear operator in question. For linear func-
tionals, extension problems can often be satisfactorily settled using
the Hahn-Banach theorem and its various consequences. Extending linear
operators, however, turns out to be a troublesome enterprise.
Analysing the proofs of the known extension theorems for linear map-
pings we come up with two principal arguments:
)I Continuous linear operators defined on dense subspaces possess a
continuous linear extension to the whole space.
2) If H is a linear subspace of an arbitrary real vector space E and F
is a Dedekind complete vector lattice, then a linear operator
T : H - F dominated by a sublinear mapping P : E ~ F can be extended
to a linear operator ~ defined on E under the domination of P. (The
vector-valued version of the Hahn-Banach theorem).
It is not intended to offer a survey of the numerous results based
on these two theorems.
On the other hand, it soon becomes evident that the majority of the
extension problems for linear operators resists a direct solution via
the methods formulated in (I) and (2). Moreover, several counterex-
amples (e.g. those concerning the non-existence of certain norm-pre-
serving extensions) are indicating that the available equipment is
seriously deficient.
V
Since the fundamental work of Lindenstrauss (see 45,46) mathematical
research on the extension of linear operators is indeed dominated by
efforts in attaining individual results (e.g. on the existence of cer-
tain projections).
The author hopes to interrupt this tradition presenting a complete
theory of positive and norm-preserving positive extensions for linear
operators in LP-spaces. In addition, several extension theorems are
proved that are also applicable in non-classical Banach lattices.
In Section 2, we shall deduce vector-valued generalizations of exten-
sion theorems that are due to H~rmander 32 and Anger/Lembcke 2 for
linear forms. The main aspect of these theorems exceeding the Hahn-
Banach theorem consists of the fact that the sublinear functionals or
mappings do no longer attain their values in ~ or a Dedekind complete
vector lattice but map into ~ U +~ or into an abstract cone. In ge-
neral, these cones cannot be imbedded into a vector lattice and pos-
sess "infinitely big" elements. A short exposition of some fundamental
properties of abstract cones, repeatedly used later, is presented in
Section .I
To get an idea in which way the concept opens a successful approach to
extension problems, consider a positive linear operator T from a lin-
ear subspace H of LP(~), I < p < ~, into itself, where ~ is a a-finite
positive measure (cid:12)9 For every positive linear extension T o :LP(u) -- LP(~)
of T we obtain
Tof ~ Pf := infTh : h 6 H, h ~ f f( 6 LP(~)),
the infimum being formed in the cone of all ~-measurable, numerical
functions possessing a lower bound in LP(~). (Here we identify ~-al-
most everywhere coinciding functions and set inf ~ = +~). In general,
we cannot expect Pf to be an element of LP(~) for each f 6 LP(u). In
fact, this will only be true when each function f E LP(u) is dominated
by some element h E H, an assumption on H that is obviously too strong(cid:12)9
IV
Thus, in cases of practical interest, we have to admit that the func-
tion Pf attains the value +~ on a set of positive measure.
So far, there is nothing new about this approach. Actually, sublinear
mappings of this type have already been discussed in literature (79,
54,58i). Extension theorems, however, have been obtained only when
the set of all elements f in the domain of P satisfying Pf < ~, has
non-empty interior with respect to the finest locally convex topology.
This assumption, however, is a severe obstacle to applications.
The extension theorems proved here will not suffer from this handicap.
On the other hand, things will not work without any regularity condi-
tions (such as lower semicontinuity) on the sublinear mapping P. It
must be approximated pointwise from below by locally defined concave
mappings dominated by P. Although we shall check this condition in
several examples (see 2.13.3), it may be difficult to be verified in
various other applications. Moreover, we shall show that the extension
technique developed in Section 2 cannot be used to construct norm-pre-
serving extensions of linear operators in non-AM-spaces.
To procede further, we therefore open a new approach in Section .3 If
E and F are normed spaces, then every continuous linear operator T
from E into the topological dual F' of F induces a continuous bilinear
form b T on E x F. This, in turn, corresponds to a linear form T | on
E | F continuous with respect to the projective norm q given by
T| | )f = bT(e,f) = Te(f) for all e E E, f E F.
Note that IITII ~ I if and only if bT(e,f) ~ lleli llfll. or, equivalently,
T ~ ~ q. We thus seem to have reduced the problem of norm-preserving
linear extensions to the classical Hahn-Banach theorem for linear
functionals. Unfortunately, this conclusion is false. To expose the
fallacies note first that the operator T to be extended is defined on
some subspace H c E. Let us replace the functional (e,f) ~ llfll.llelI
IIV
by an arbitrary bisublinear functional p : E x F ~ ~. Passing to the
tensor product E | F we obtain
p~(t) = inf~ P(ei,f i) : eiE E, fiEF, ~ ei| f i = t~, t 6 E|
i i
which will define a sublinear form p| or the constant -~.
At any rate a bilinear form b : E | F ~ ~ is dominated by p if and only
if b | is dominated by p| Thus we first must make sure that there ex-
ists at least one linear operator S : E ~ F' such that b S is dominated
by p. If p(e,f) = ~IlelI , then this condition is clearly satisfied
by the zero-operator.
The second obstacle is not so evident. Once we know that
Th(f) < p(h,f) for all h E H, f 6 F,
is it true that T| | f) < p| | )f for all h 6 H, f E F?
This is trivial when H = E, but there are counterexamples for H ~ E,
This observation led us to the introduction of so-called subbilinear
forms, which, by definition, satisfy the equation
p| | f) = p(e,f) for all e E E, f E F. (see Definiton 3.6).
For p(e,f) = ,lIfiloiIeII p is such a subbilinear form.
There is, however, a third fallacy in the arguments sketched above.
Suppose that p actually is a subbilinear form. Then the inequality
Th(f) < p(h,f) obviously implies T| | ~) < p|174 f) for all h E H,
f 6 F. But what we need is the domination of T | by p| on all of H | F
not only on the elementary tensors h | f. When forming p~(t) for
t 6 H | F, however, we have to take into account all the representa-
tions e i | fi = t, e i 6 E, fi 6 F, just choosing e i s H will not do!
i
(This is the reason why counterexamples for the existence of norm-pre ~
serving extensions can be found even in finite-dimensional spaces).
There is only one possibility to solve extension problems by tensor
product methods. We have to be able to compute p| for arbitrary
IIIV
t E E | F, not only for elementary tensors! This can be successfully
done for the bisublinear functional belonging to norm-preserving posi-
tive extensions of positive linear operators whenever F = G', where
(E,G) is one of the following pairs of Banach lattices:
)I E an arbitrary Banach lattice, G an L1-space (see 4.3),
2) E an LP-space, G an Lq-space, where q < p; p,q 6 I,~, (see 5.5),
3) E an AM-space, there exists a positive contractive projection from
G" onto G and G' possesses a topological orthogonal system,
(see 5.10).
For such pairs of Banach lattices, called adapted pairs, we obtain a
complete solution of the positive and of the positive norm-preserving
extension problem. In addition, starting with a positive continuous
operator T : H ~ G, instead of T : H ~ F' = G", we end up with an ex-
tension T O : E ~ G, instead of T O : E ~ G". Given M ~ O, IITII ~ M, T has
a positive linear extension T O with the norm loTII I ~ M if and only if
II ~/ (Thi)+II ~ IIM Vh~II for every finite family (h i ) in H,
i6I i~I -
where ~/ denotes the supremum operator (see 4.4). From this condition
several extension and projection theorems in classical Banach lattices
can be easily deduced. One of the most remarkable consequences is the
following strikingly simple solution of the positive extension prob-
lem for adapted pairs (E,G) of Banach lattices (see 4.7):
A positive linear operator T : H ~ G possesses a positive linear ex-
tension T o : E ~ G if and only if T(A) is bounded from above in G for
each subset A c H bounded from above in E.
By a counterexample we show that the stated extension theorems are
specific for classical Banach lattices (see 4.8).
For the treatment of the applications in Sections 6 to 8 we require
detailed information about the set E T of all positive linear exten-
sions ~ of the positive operator T : H ~ G. In particular, we have to
determine the elements e E E for which the set ~e : ~ E E T is a single-
XI
ton, i.e. for which all positive linear extensions ~ of T coincide at
e. This problem is the connecting link of all sections. It is solved
for adapted pairs of Banach, lattices in Sections 4 and .5 The result-
ing description of ~e :~ 6 E T is extensively applied in the following
sections.
While until the end of Section 5 examples and applications have been
interspersed occasionally, Section 6 to 8 are concerned with conver-
gence theorems for nets and of positive linear operators. Our starting
point is the theorem of Korovkin 39. It states that for a given se-
quence (Tn)n6 ~ of positive linear operatos on C(a,b) into itself,
a,b ER, a < b, (Tnf)n6~ converges uniformly to f for each f 6 C(a,b)
provided that
lim T ( idJ) = id 3 for j = O,1,2 (uniformly),
n
n~
where id denotes the identity mapping on a,b. In addition, Korovkin
proved that a minimal "test set", i.e. a minimal set of functions that
replaces the set idO,id,id 2 is a Chebyshev triple. His results have
been generalized to arbitrary compact spaces instead of a,b by var-
ious mathematicians, most notably by ~a~kin 63,643 .
Korovkin theorems are most naturally treated within the setting of
topological vector lattices or, in the normed case, within the frame-
work of Banach lattices. In fact, such a general investigation is not
only illuminating from the theoretical point of view but also covers
new applications in spaces of integrable functions and spaces of con-
tinuous functions vanishing at infinity. Consider two real Banach lat-
tices E and F, a linear subspace H of E usually called "test space"
and a class T of nets of positive linear operators of E into F. Given
a vector lattice homomorphism S : E -- F, the Korovkin closure or shadow
KorT,s(H) of H with respect to T and S is the set of all elements eE E
that satisfy the following condition:
For each net T( )i E T, (Tie) converges to Se, provided that (Tih) con-
verges to Sh for all h 6 H.
A test space H such that KorT,s(H) = E is called a Korovkin space
(with respect to T and S). If E = F := C(a,b) and if ~2 denotes the
linear subspace of all polynomials of degree at most 2, then Korov-
kin's classical theorem states that H2 is a Korovkin space in C(a,b)
with respect to the set Pe of all sequences of positive linear opera-
tors on E. In fact, H2 is also a Korovkin space with respect to the
class P of all nets of positive linear operators on E.
To point out the connection with the extension problem of linear ope-
rators let E and F be arbitrary Banach lattices. In order to show that
a given element e E E does not belong to Korp,s(H) it (cid:127) necessary to
construct a net (Ti)i61 in P such that lim Tih = Sh for all h 6 H but
iEI
(Tie)iE I does not converge to Se. If the index set I contains only a
single element this amounts to the construction of a positive linear
extension T i of the restriction SIH such that Tie ~ Se. For an arbi-
trary index set there are still strong connections between the deter-
mination of Korovkin closures and the extension of positive, linear
operators. Indeed, in many cases the former problem can be reduced to
the latter.
A characterization of KOrp,s(H) for arbitrary Banach lattices E and F
has been given in 20,213. It carries over to Korp~,s(H), whenever H
possesses a countable algebraic basis. Moreover, the description re-
mains valid for locally convex vector lattices E and F. Examining
these results in more detail we find out that we persued the wrong
trace. To see this, note first that, in Korovkin's classical theorem,
sequences (T n) 6 P~ satisfying lim Tnh = h for all h 6 92 are automati-
n~
cally equicontinuous. If, for arbitrary Banach lattices E and F, P
e
and P~ denote the classes of all equicontinuous nets and sequences in
e
P and P~, respectively, the Banach Steinhaus theorem shows that
,eY~_roK S(H) = K~ provided that KOrp~,s(H) is of second cate-
gory in E. Since we are mainly interested in cases where Korovkin
