Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 487-492
487
THE UNIFORM BOUNDEDNESS PRINCIPLE FOR ORDER BOUNDED OPERATORS
CHARLES SWARTZ
Department of Mathematical Sciences
New Mexico State University
88003
Las Cruces, NM
(Received April 18, 1988)
ABSTRACT:
of
Under appropriate hypotheses on the spaces, it is shown that a sequence
order bounded llnear operators which is poJntwJse order bounded is uniformly
order bounded on order bounded
subsets.
This
result
to
used
is
establish
a
Banach-Stelnhaus Theorem for order bounded operators.
KEY WORDS AND PHRASES:
order
lattice,
bounded
operator,
uniform
boundedness
principle, Banach-Steinhaus Theorem.
1980 NATHEKATICS SUBJECT CLASSIFICATION:
1.
48A40.
INTRODUCTION
In this note we consider the problem of obtaining a version of the classical
Uniform Boundedness PrlncJple of functional analysis for linear operators between
vector lattices.
If
X
is a Banach space and
Y
is a normed linear space, the
Uniform Boundedness Principle then asserts that any sequence
linear operators from
X
into
the sequence of operator norms
the condition that the
sequence
(,Till)
is bounded {[1] t4).
X
is such that
It is easy to see that
are bounded is equivalent to the condition that the
X
{see the discussion
Thus, a possible version of the Uniform Boundedness Principle for order
bounded linear operators
(Tj)
which is poJntwise bounded on
Js uniformly bounded on bounded subsets of
(T i)
in [1] 14).
(llTi.)
Y
of continuous
(Tj)
(Ti)
between vector lattices
is pointwise order bounded on
order bounded subsets of
X.
X, then
We will show
(Ti}
below
X
and
Y
might be that if
is uniformly order bounded on
in
Example
that
such
d
straightforward analogue of the Uniform Boundedness Principle does not hold even Jf
both
X
and
Y
are Banach lattices.
However, by Imposing additional conditions on
488
C. SWARTZ
the spaces and by employing the matrix methods of
[1], we will obtain an order
Using the Uniform
version of the Uniform Boundedness Principle in Theorem 3 below.
Boundedness Principle, we also establish a version of the Banach-Stetnhaus Theorem
for order bounded operators which generalizes a result of Nakano.
RESULTS
2.
the
conslder
First,
example
following
a straightforward
that
shows
which
analogue of the Uniform Boundedness Principle does not ho]d for order bounded linear
(A linear operator
operators between Banach lattices.
X
or Riesz spaces
X
T
is order bounded If
and
,
into order bounded subsets of
between vector lattices
T
carries order bounded subsets of
A
where a subset
-u
such
u)
X
[-u, u]
(v
(in genera], we conform to the notation and terminology of [5]).)
v
order bounded if there is an order Interval
A c [-u, u]
that
X
of a vector lattice
EXAMPLE 1.
X
Let
LI[O,1]
and
For
f e
the usual polntwise ordering.
Y
c
O
L1[0,1],
and assume that these spaces have
fk /0 f(t)sin ktdt.
set
Define
J
k
T
X
k
Y
Tkf
by
where
fjej,
co
is the element of
ej
with a
in the
J=l
jth
coordinate and
if
Ifl
<
(Tk)
g
[Tkf
in the other coordinates.
pointwise
[fjJej
e c
0
og
If j[
X, then
in
is
0
order
a
bounded
([fj[)
(note
and
T
Each
<
ITk f
X
on
e c
o
by
is order bounded since
k
k
AI so
aej
the
f e X,
if
since
the sequence
then
Lemma).
Rlemann-Lebesgue
j=l
However,
(Tk} is not uniformly order bounded on
if
Vk(t) sin kt, then (Fk) is order
(Tl(i)} (el/2) is not order bounded in c O
Note that both
X
and
Y
order bounded subsets of
bounded
(IFkl
X
in
X
I)
since
but
in this example are Banach lattices and both are
Dedekind complete.
In order to obtain our version of the Uniform Boundedness Principle for ordered
spaces, we first obtain a matrix theorem which is the analogue for ordered spaces of
the matrix result given in [I] 2.1.
X and Y
Is u-convergent to
Throughout the remainder of this note we let
(vector lattices),
there
u
lim x
sequence
to
x
a
exists
k
x.
{x k)
scalar
u
in
tk40
sequence
The element
is
(x k)
h sequence
X
such
IXk
that
denote Riesz spaces
x, where
x
tku;
is called a convergence regulator for
reltivel niformly convergent to x
for some convergence regulator
u
O; we write
if
r
(xk)
llm x
k
u
O, if
we
write
(Xk).
The
is u-convergent
x.
UNIFORM BOUNDEDNESS PRINCIPLE FOR ORDER BOUNDED OPERATORS
rows and columns are u-convergent to
Let
O.
an increasing sequence of positive integers
PROOF.
exists
P2
>
iS
>
P2
P3
IXp3pl
Put
Pl
Pl
such that
1.
X, be such that its
(pi)
IXpiPj
Then there exists
such that
for
iju
(xlj) and {Xil) are u-convergent to O,
IXlp21 _< 12u and IXp21 < 21u" Similarly,
there
Since
IXplP31
that
such
[Xlj], xij
lJ > O, 6iJ "
Let the infinite matrix
([4] 2.1)
LEMMA 2.
489
IXp2P31
el3U,
IXp3P21
23 u,
there
32 u
and
Now continue.
e31u.
We now prove our Uniform Boundedness Principle for order bounded operators.
g
Let
THEOREM 3.
be Dedekind o-complete and let
u.
have an order unit
T
If
(Tt([-w,
assume that
is uniformly order bounded on
(T i)
X, then
e )
w])
I-w, w]
and
m
m
Now consider the matrix
jth
bounded so the
(xj/J 2)
l
14j-u,
u].
in
X
such
there exist
For notational convenience,
Then
i.
I
TmtX
such that
i
Thus, for each
is not order bounded.
Tlx 1
sequence
be Dedeklnd complete and
If the conclusion fails, there is an interval
e [-w, w]
xi
Y
g.
order bounded subsets of
PROOF.
is order
is a sequence of order bounded linear maps
Y
X
1
which is pointwise order bounded on
that
Y
is regular ([5] VIII.2.2).
T
bounded if and only if
X
T
is Dedekind complete, then a linear map
Y
Recall that if
M
14j-u,
[(1/t)Tl(xj/J2)].
column of
M
(2.1)
u].
For each
J,
(Tl(xj/j2)) 1
is order
is relatively uniformly convergent to
O.
0, and since each
T
is relatively uniformly convergent to
The
t
is
sequentially continuous with respect to relative uniform convergence ([5] VIII .2),
th
row of M is also relatively uniformly convergent to 0. Thus, the rows
the I
and columns of
M
are u-convergent to
sequence of positive integers
#
J.
since
By Lemma 2 there is an increasing
I(1/pi)Tpi(Xpj/pj2)
Again for notational convenience, we assume that
-complete and
element
(pi)
O.
such that
x
Ix
Ixjl
w, the series
j/j2
Pi
i.
2-l-Jr
for
X
is
Since
is absolutely order convergent to an
X ([5] IV. 9); moreover, this series is actually w-convergent in X
n
T
From the continuity of
Ixjl/J2 ( 1/J2)w.
xj/j21
J=l
j=n+l
Jfn+l
with respect to relative uniform convergence, we have
490
C. SWARTZ
+1
<
.1=1
j=l
.
.j=l
<
l(]/,)T,(x)l
2
j=l
_<
(]/J)lTi(x)l
+
2-iv
(2.2)
Y
where we have used the order completeness of
right hand side of (2) are convergent.
Hence, there exists
order bounded.
t.
all
1
Putting
to insure that the series on the
Both terms on the right hand side of (2) are
k
(1/t)lTl(xi/t2)
such that
k[-u, u]
for
contradicts (1) and the result Is established.
k
Note that the range space
c
is Dedeklnd complete, but does not
in Example
O
have an order unit.
It is perhaps worthwhile noting that if
order bounded subsets of
property.
ITllx
For
sup(lTizl
is uniformly order bounded on
(ITII)
ITt[0,
X, then the sequence of modull
0
if
z $ x)
0
(T1)
w
x
u
and
([5] VIII.2) so
ITil[0,
also has this
u]
w,
then
u]
From Theorem 3 we can also obtain an order analogue of the equicontlnuity
conclusion of the classical Uniform Boundedness Principle.
Recall that if
a sequence of continuous linear operators from a normed space
space
Y, then
(T i)
on bounded subsets of
llm
J
Tix
0
is equlcontlnuous if and only if
X
if and only if
uniformly in
1
whenever
(TI)
X
(T 1)
is
into a normed
is uniformly bounded
Tix
0
xj
([1] 4.5).
It is easy to establish
order analogues of these equlcontlnuity conditions for operators which satisfy the
conclusion of Theorem 3.
We give order analogues of these conditions below and
consider the relationship between them.
X, we say that
a sequence
ia
tl0
uniformly for
is u-convergent to
such that
a e A.
If for each
IXla
ttu
0
a e A,
uniformly for
for all a
is a sequence in
{xia
a
A
A; we write
if there exists
u
ltm
Xia
0
UNIFORM BOUNDEDNESS PRINCIPLE FOR ORDER BOUNDED OPERATORS
(T1)
(i)
(11)
>
v e Y, v
some
(111)
r
(I)
Always
X
1
Y
be order bounded.
Consider
X.
is uniformly order bounded on order bounded subsets of
r
if
T
Let
PROPOSITION 4.
491
llm
O,
xj
v
then
llm
Tlx
e
uniformly for
0
for
J
O,
11m
0
Tlx
(II)
implies
Jim x
r
whenever
(ill).
implies
0.
Y
If
Archlmedean
is
the
has
and
boundedness property ([3] 1.5.12), then (iii) implies {I).
is relatively uniformly convergent to
{tjxj}
<_
ITixjl
v e Y
by (i) there is
order bounded,
Then
To establish {ii), there exists
Assume (i) holds.
PROOF.
Assume {Ill) and that
Clearly (li) implies (ill).
show
r
lim
that
Let
u >_ O, u e X.
(Tkl)
if
a
is
But, since
O.
tlTklX I
ITi{tjxj)l
such that
(tjxj)
Since
_( v
is
I,
for all
J.
Implles (I11.
(1/tj)v
boundedness property.
to
([4] VI.4).
0
such that
tj,
u
Ixi
subsequence,
lim
is Archimedean with the
Y
By the boundedness property, it suffices
<_
u
tl0,
and
then
O, this follows from (iii).
tix i
Theorem 3 gives sufficient conditions for the equlcontlnulty condition (I) and,
therefore, (ll) and {lli), to hold.
As in the classical case we can apply the Uniform Boundedness Principle given
in Theorem 3 to obtain a Banach-Stelnhaus type result for order bounded operators
([1] 5.1).
There is an order version of the Banach-Steinhaus Theorem for linear
to
functlonals due
Nakano
glven
in
[5]
IX.
1.1,
and
general
a
of
form
Banach-Steinhaus Theorem for operators between vector lattices given in [4].
the
The
results in [4] treat a different class of operators than that considered below In
Corollary 5 and the assumptions on the spaces are not as restrictive.
COROLLARY 5.
If
bounded.
0
Let
lim
X, Y
Tx
Tix
T
be as in Theorem 3 and let
T
x e X, then
exists for each
X-, Y
I
X
Y
be order
Is an order
bounded linear operator.
For each
PROOF.
convergent.
order interval
c
(TlX)
is order bounded
Therefore, from Theorem 3, (T i)
X.
If
I-w, w]
in
bounded subsets of
T{[u, v])
x,
I-w, w]
and
T
[u, v]
Y
T
X
Y
the sequence
is
order
is uniformly order bounded on order
is an order interval in
such that
Ti([u,
v])c_ [-w, w]
X, then there is an
for all
i.
Hence,
is order bounded.
The sequence of operators
the operator
since
(T i)
given by
in Example
Tf
fjej.
is pointwlse order convergent to
IIowever, T
is not order bounded
j=l
since
(TFj)
(ej/2),
where
Vj(t)
sln//Jt.
This shows that in general
the
492
C. SWARTZ
Banach-Stelnhaus
result
will
not
hold
the
if
range
space
is
merely
Dedektnd
complete.
The author would like to thank Joe KJst for his kind assistance.
REFERENCES
I.
P. Antoslk and C. Swartz, Matrix Methods
Mathematics 1113, Springer-Verlag, 1985.
2.
W.A.J. Luxemburg and A.C. Zaanen, Rlesz Spaes I, North Bolland, 1971.
A.L. Peressini, Ordered Topological Vector Soace, Harper and Row, 1967.
3.
4.
5.
in .nalysls,
Lecture
Notes
in
C. Swartz, The Banach-Stelnhaus Theorem for ordered spaces, proceedings of the
Conference 6FAC, Dubrovnlk, Yugoslavla, 1987, to appear.
B.Z. Vulikh, Introduction to the
Wolters-Noordhoff, 1967.
Theory
of partlallv Ordered
Spaces,