Internat. J. Math. & Math. Sci.
18 NO. 2 (1995) 341-356
341
OL.
TOPOLOGICAL PROPERTIES AND MATRIX TRANSFORMATIONS OF
CERTAIN ORDERED GENERALIZED SEQUENCE SPACES
MANJUL GUPTA and KALIKA KAUSHAL
Department of Mathematics
Indian Institute of Technology
Kanpur 208016, India
(Received December 17, 1991)
ABSTRACT.
In this note, we carry out investigations related to the mixed
impact of ordering and topological structure of a locally convex solid
Riesz space (X, ) and a scalar valued sequence space
on the vector
,,
valued sequence space
of
x and X, and
x (X) which
is formed and
topologized with the help
vice versa.
Besides,we also characterize o-matrix trans(X) to themselves, cs(X) to c(X) and derive necessary conditions for a matrix of linear operators to transform .1 (X) into a
simple ordered vector valued sequence space ^(X).
formations from
c(X),
=
KEY WORDS AND PHRASES. Riesz spaces, order complete, ideal, diagonal
property; positive, sequentially order continuous and o-precompact linear
operators; locally convex solid Riesz spaces, Riesz seminorms, Lebesgue,
pre-Lebesgue, Fatau property; ordered vector valued sequence spaces,
matrix transformations.
1992 AMS SUBJECT CLASSIFICATION CODES.
6A05, g6A5.
1.
Primary g6AgO, 06F20; Secondary
INTRODUCTION.
In the direction of generalizing the K6the theory of perfect sequence
spaces [13], [16], the concept of a vector valued sequence space (VVSS)
or a generalized sequence space defined with the help of a scalar valued
sequence space (SVSS) was introduced by P ietsch [21], who also used
these spaces in the study of absolutely summing operators and nuclear
spaces, cf. [20], [22]. Moreover, the topological properties of VVSS which
are defined by using the seminorms generating the topology of a locally
convex space and a SVSS and have been found useful in the study of
,-nuclearity, X-bases, operators of ,-type, absolutely x-summing operators,
[], ,[5], (see also [i] and [15] for bases theory and its applications
related to nuclearity etc. in 1ocally convex spaces) and completeness of
the space via their mixed structure [9], are sufficiently explored now
[], [8], [18], [2]. However, if the underlying locally convex space and
M. GUPTA AND K. KAUSHAL
342
the SVSS have additional structure of ordering, it is natural to inquire
the mixed impact of the topology and the order structure on the VVSS.
This study when X has an order structure, was taken up by Walsh [27]
for certain type of VVSS, who applied these spaces to study K-absolutely
summing and majorizing operators introduced for Banach lattices in [25].
In the present paper, we carry out investigations related to this mixed
impact of ordering and topological structure of the locally convex solid
Riesz spaces and the SVSS on the VVSS and vice-versa.
Besides, we also
our
of
o-matrix transformations initiated in [10] for partistudy
co.ntinue
cular ordered VVSS introduced in
2.
[11].
PREREQUISITES.
In this section, we
mention the salient features of Riesz spaces,
locally convex solid Riesz spaces, scalar and vector valued sequence
spaces required for our present work. However for unexplained terms in
these theories, the reader is urged to look into [1], [17], [i9], [26].
Throughout the sequel, X denotes a Riesz space ordered by the cone
K. The notations x
(resp. x +) are used for an increasing (resp.
x) provided x ’/ and
in X and x
decreasing) net {x
x (resp. x
x).
x (resp. x
sup x
and inf x
(o)
A net {x a in X’ order converges to X, written as
x, if
there exists a net (ya} in X with y
0
< Ya for each a;
and
x
a
and it converges relatively uniformly to x, if there exists u
K such
> o, there exists
that for any
the
condition
satisfying
o
<
a
> a
u,
where u is known as the regulator of convergence.
A sequence {x
in X is said to be order-Cauchy (respectively, relatively
n
O in X such that
uniformly Cauchy) if there exists some sequence Yn
<
> o, there exists k
for all m,k _> n (resp. for any
N
< u for n,m _> k); accordingly, X is order-Cauchy
such that
complete [[3],[12]] (resp. uniformly co____mplete) if every order-Cauchy (resp.
relatively uniformly Cauchy) sequence is order convergent (resp. relatively
uniformly convergent) in X.
Ix
x
Ixa-xl
^
IXm-Xk[
Yn
xn-xml
Relating the above concepts of convergence and completeness, we have
the following well known result contained in [i], [19], [3].
PROPOSITION 2.1. (i) In a o-order complete Riesz space X with the
diagonal property, the order, and uniform convergence coincide for sequences; (ii) every order complete Riesz space is uniformly complete; and
(iii) every -order complete Riesz space is order-Cauchy complete.
We now equip X with a locally convex topology
generated by the
<
in X,
family=) of Riesz seminorms p’s, i.e., p(x) < p(y) for
We refer
so that (X, z) is a locally convex solid (l.c.s.) Riesz space.
Ixl
[I]
IYl
for various terms and results on the theory of locally convex solid
However, we say a Riesz seminorm p on X is,
0
(i) o-Lebesgue if p(x
in X; (ii) Lebesgue
o for any sequence x
n
n
in X;
in X which decreases to
o for any net {x
if p(x
Riesm spaces.
ORDERED GENERALIZED SEQUENCE SPACES
(iii)
pre-Lebesgue
p(x n
343
if for any disjoint order bounded sequence
o; and (iv)o-Fatou
(resp.
x (resp.
Fatou) if p(x
n
0 < x
x).
(x
in X,
n
p(x) (resp.
p(x)
p(x)) whenever 0 < x
We denote the family of all linear operators from a Riesz space X to
another Riesz space Y, by (X,Y) and the subspace of .(X,Y) consisting
of all order bounded operators by
The subspaces of
containing order continuous and sequentially order continuous operators
.b(X,Y).
.b(x,Y)
=-
respectively denoted by ’C(x,Y) and .Z*S(X,Y).
For Y
set of real numbers, we write X b
), X c
,C(x,[) and
are
-= .b(x,
xSO _-.L’SO(x,
the
)..
Concerning these spaces, we have
PROPOSITION 2.2. If Y is an order complete Riesz space, then
is an order complete Riesz space ordered by the cone K
{T
T(x) >
x a K] where for T in
X,
and x
lyl _<lxi ). Further,
ITI(II) sup {IT(y)
,C(x,Y)
are bands in
,b(x,Y)
b(x,v):
,
,b(x,Y)
2S(x,Y)and
.b(x,Y).
For monotone nets of operators in .b(x,Y), we have [26]
^)
e
THEOREM 2.3. {Let T
be an increasing or decreasing net
(o), Tx,
Y) such that
of operators in
X, for some
x
(o))
,b(x,Y)
T
and T
(X,Y). Then T
T in .b(x,Y).
,
b(x
Tax
In case Y is also equipped with a
subclass of
b(x,Y)
l.c.s,
topology, we have another
[7].
as given in
DEFINITION 2.4. An operator T in .b(x,Y), where X is a Riesz
space and (y,T) is an order complete l.c.s. Riesz space, is said to be
o__p______mpact if it maps order bounded subsets of X into precompact subsets of Y.
The class of all o-precompact operators from X to Y is
denoted by
(X,Y).
op
PROPOSITION 2.5. The space
(X,Y) forms a band in b(x,Y),
op
where X is a Riesz space and (Y,T) is an order complete l.c.s. Riesz
space with Lebesgue property.
NOTE. The above result is due to P.G. Dodds and D.H. Fremlin [6]
.
.
for the special case of Banach lattices; however, in its present form it
is to be found in [2] (with a simplified proof due to A.R. Schep).
We follow [18], [23] for the fundamentals of vector valued sequence
spaces (VVSS). For the sake of convenience, let us recall [23] the
vector spaces a(X) and o(X) consisting of all sequences and finitely nonA VVSS ^(X) is a subspace of
zero sequences from a vector space X.
a(X) containing (X). Corresponding to a dual pair < X,y> of vector
spaces defined over the some field, the generalized KBthe dual
^(X) is the vector space given by
^(X) x
-= ^ x (Y)
{{Yi
i>l
Y
y’
l<xi,Y >{
>
<
(R),
^X(y)
and
{x i}
A(X)).
of
344
M. GUPTA AND K. KAUSHAL
,,
The generalized K;the dual of ^X(y) is denoted.by
Members of n(X) are denoted by
etc., i.e.,
X,
xi,y
>1.
so on.
[Yt }’
N, the set of natural numbers,
X and
For x
^XX(x) and
[xi}’ Y
we write
x
x, 0, 0
i-th co--ordinate
0, 0
ai
Observe that (X) is spanned by[a
x.
X,
N ].
^(X)
Let us also recall [18] that a VVSS ^(X) is normal if {a. x.
<
>
.]
[a
^(X)
{x.
^(X) and
V
where
with
and
is si.__mple if for each
equipped ith a locally convex topology J
in ^(X) such that for each
bounded set A, there exists an element
_< 1,
for some sequence {ai of scalars with ]a
[Yi
Yi ai
> 1, i.e., A is contained in the normal hull of the set }.
x
fail
x.
^’
il
In addition, if X is also a Riesz space, the VVSS ^(X)
vector space relative to the co-ordinate wise ordering [10].
is an ordered
Indeed we
10].
have
If X is an order complete Riesz space, then n(X)
PROPOSITION 2.6.
and (X) are also order complete Riesz spaces and a Riesz subspace^ (X)
of (X) is order complete if and only if it is an ideal in fl(X). Let us
A(X) is normal if and only
note that in the particular case when X
if it is a solid subspace of a(X), or equivalently, an ideal in a(X).
PROPOSITION 2.7. Let X be an order complete Riesz space such that
^(X) is an ideal in a(X). Then a linear functional T on ^(X) is in
z
in AX(x s)
^(X)] s if and only if there is a unique
,
X s for n >
z
T(,)
such that
<,i >
[
<
X
n=l
n’ Zn>,
,
V
x
n
^(X).
As examples of ordered vector valued sequence spaces (OVVS), we
recall the following spaces from [11], which are needed in the sequel and
are defined over an order complete Riesz space X such that < X,X s> forms
a dual pair.
l(x)
’(X)
c(X)
n
{{x }: x
n
{x }: x
n
{x }: x
n
(X)
n
{x
n
n
n
and
i=1
X, n >
X, n >
n
x
X, n >
X, n >
and sup
n
Ixil)
{Ix n
order converges in X },
)exists in
X),
(2.8)
and {x n
order converges in X),
and x
o in X }.
n
One can easily verify that .I(x) (R)(X) and c o (X) are ideals of
and c(X) is a Riesz subspace of a(X), which is not an ideal.
ORDERED GENERALIZED SEQUENCE SPACES
345
We also define
n
cs(X)
x
{{Xn}
X, n >
n
x
and
(2.9)
order converges in X }.
i=l
Note that the space cs(X) is an ordered vector subspace of n(X),
which is not a Riesz space.
In particular, when X
IR, the VVSS ^(X) written as X in the sequel,
space (SVSS).
_=
is known as a scalar valued sequence
Using a normal SVS5 x and a locally convex space {X,T), another
type of a VVSS introduced in [6] is given as
{
t(X)
{x
}:
x
c
_>
X,
and
{p(x i)
A,
p .)
(2.10)
If X is a locally convex space and t is equipped with a normal
(-topology
compatible with the dual pair <A ,Ax> and generated by
{a
(
of seminorms, where for ff
S
the family {Ps
i} A
T
ps(a)
S
i>
is a family of normal hulls of balanced convex
and
x,
x,
-
then the topology
covering x
subsets of x
{P
S
ps o p
the family of seminorms
s
()
o p)()
pS({P(Xi) })
(Ps
PS
for
3.
(2.11)
lail IBi[
sup
{B.}
Tt(X)
o(xx,.)-bounded
on x(X) generated by
p )
defined as
(2.12)
X(X), is a Hausdorff locally convex topology; cf. [4], p.130.
TOPOLOGICAL PROPERTIES OF ,(X).
c
defined as in (2.10)
Recalling the space x(X) and the topology
and (2.12), corresponding to a normal SVSS X and a locally convex space
X, we prove in this section the impact of topologies of X and X on
Tx(X)
Tx
To begin with, we have
PROPOSITION 3.1. If (X,T) is a 1.c.s. Riesz space generated by the
of Riesz seminorms, then x(X) is an ideal in n(X) and T x(X)
family
is a locally convex solid topology on t(X).
and vice-versa.
PROOF. Straightforward.
A partial converse of the above result is contained in
PROPOSITION 3.2. Let X be a Riesz space equipped with a locaUy
convex topology z generated by the family o) of seminorms, and
Then z is a
be a locally convex solid Riesz space.
(x (X), T
locally convex solid topology on X.
X with [x <_ lY[ and a member S
x,y
PROOF. Consider p c)
N. Then
s > 0 ro om
such that M
o
su (I
(i
x(X))
sil:
_<
o
) is a Riesz seminorm.
implies that p(x) < p(y). Thus each p
From now onwards in this sectio_n_, we assume that the pair (X,)
H. GUPTA AND K. KAUSHAL
346
stands for a Hausdorff locally convex solid Riesz space, where the topology
is equipped
of Riesz seminorms and
which is the same as the absolute weak
by the family
is generated
with the normal topology n (,
xx)
1o[ (,x),
cf. [I]; and which is an
-topology where
corresponds to the normal balanced convex hulls of singleton sets in
In this case, we write the seminorms defined in (2.12) as
topology
-
V ()
.
siJ
i--I
,
p(x i’
e,
where P corresponds to p
(P_
xx}
e
xx
x(x)
8.} e
and
X
(3.3)
for the family
of Riesz seminorms.
p
The interrelationship of various properties of Riesz seminorms, for
the families
and
are exhibited in
e
(x)
satisfies Lebesgue property
PROPOSITION 3.. A seminorm p
e xx,-’) satisfy
in X if and only if the corresponding seminorms
the same in (X).
and.a
PROOF. For proving the necessity, consider
8l}. e
B in X(X).
Then p(x n)
o for
sequence (n] in X(X) such that
[P
xx
n
I
each
SiJ p(x),
and
i=l
>o, thee
fixed
exists
<-,
n
$
I.
Hence for an arbitrary
N such that
e
O
Consequently, for each n
o
iI
x
n
For provinB the converse, consider a sequence Ix n In X such that
e
N corresponds to
such that
of
in X and an element
x
+ e
the non-zero co-ordinate of 8.
X
for
n
+
-
in z(X)
PROPOSITION 3.5.
In
Thus p(x
n)
o
satisfies the Lebesgue property
A member p of
x satisfies
Pfor "g"
order to prove that
Pif p does, consider a net lx
if and only if each
PROOF
Then
X
for
Lebesgue property
-a
e in (X). Then p( xai) /o,
x
> o as well as for fixed a
A
O
>
the same in (X)
x
satisfies the
a
^t in :(X) with
and for an arbitrary fixed
we can find
O
N such that
ORDERED GENERALIZED SEQUENCE SPACES
i-i
o
(<)
P
:>
..IBil p(x?)
=I
B
o
p(x)
le
[
As
347
i=I
..IBi
<
p(x i) +
P_ (e
o,
V
>
o.
B
Replacing sequence by net in the proof of the converse part of
preceding result, we may infer the Lebesgue property of p from that of
PROPOSITION 3.6. A seminorm p in
possesses pre-Lebesgue
P_.
B
x
property if and only if each P for
possesses the same
B
PROOF. For proving the necessity,
consider an order bounded disjoint
for every n
and for some
sequence {n} in (X). Then ]n] !
{x.
e (X)
and so {x n} is an order bounded disjoint sequence in X
e
for each
N. Hence p(x n).
e
N. Now for
-for each
o as n
e
and e > o, choose
N such that
any B e
o
,
i=i
o
e
nov proceed as in he poof of necessary
orde o ge he result.
For sufficiency, observe ha for each
bounded dis]oin sequence in (X) fo any
par
>
,
disoin
of Proposition 3.4 in
x
{.n}
order bounded sequence
Nov use he pre-Lebesgue propery of P
{x
in X.
n
result.
PROPOSITION 3.7.
p in
is a
is an order
fo deriving he
a-Faou seminorm if and only if P
is so.
Le p satisfy he a-Faou propex.
PROOF.
popert of P
e <
x
x
i’
x
hee
e
V
and so
>
consider
p(x)_
ently, V m
m
,Bi, P(in)
i=I
Fo poving he a-Faou
x in (X). Then
N
e
Consequfo each
--n
x
f
n
p(xi)
m
8
p(x
i=l
n
If a is any upper bound of the sequence
m
p_ (n)
then
i=l
Hence P
()
< a, i.e. P
(n)
p
()
The proof of the converse is analogous to the earlier proofs of con-
verse parts and so omitted.
PROPOSITION 3.8.
P
p in
is a
Fatou seminorm if and only if each
is so for 8
8
PROOF.
Analogous to the proof of the preceding result and so omitted.
M. GUPTA AND K. KAUSHAL
348
Concerning the spaces X and (X), the preceding propositions immediately lead to
THEOREM 3.9.
(X,) satisfies o-Lebesgue (resp. Lebesgue, preLebesgue, o-Fatou or Fatou) property if and only if ((X), T(X)) does
so.
PROOF.
4.
Straightforward.
o-MATRIX TRANSFORMATIONS.
This section which is divided into two subsections incorporates
results on o-matrix transformations from one OVVSS to another OVVSS defined
corresponding to Riesz spaces. Whereas the first subsection deals with the
o-precompactness of the o-matrix transformation, the second subsection
includes characterizations of such transformations on particular OVVSS in
terms of component linear operators.
Before we pass on to these results,
let us recall [i0].
DEFINITION 4.1. Let X and Y be two order complete Riesz spaces
such that ^(X) and (Y) are ideals in n(X) and n(Y) respectively. A
linear map Z from ^(X) to u(Y) is said to be an o-matrix transformation
if there exists a matrix
every
[Zij
of linear maps from X to Y such that for
^(X) and each
[x.) in
e
order converges to some
Yi Y and Z()
case we write
Zij ]. The transpose Z
Z -=
X
Y, i,j >
is defined
Zij
of adjoint maps, i.e., Z
]"
PROPOSITION 4.2. Let Z -=
be an
^(X) to (Y). The the following statements
(i) For x
N
X and
linear maps
[Zji
[Zij]
of z(
(ii)
i,j
Zt](x)
x.);
k
N, the sequence
,
r.
j=l
where
Zij(xj))
{Yi ];
of a matrix Z
in such a
[Zij]
of
as the transpose of the matrix
o-matrix transformations from
hold:
(Z(6 x
j))i’
the
th
co-ordinate
Z is positive if and only if the Z..’s are positive for each
N.
PROPOSITION 4.3.
Let X and Y be order complete Riesz spaces such
that <X,XS> and
form dual pairs. Assume that the ideals ^(X)
and (Y) in a(X) and l(Y) are in duality with ^X(xS) and ,X(ySO)
<y,ySO>
respectively.
If Z is an order bounded, sequentially order continuous
linear map from ^(X) to (Y), given by the matrix
[Zij]
of linear maps
from X to Y for i,j >_ i, then the adjoint Z of Z is an order bounded,
sequentially order continuous o-matrix transformation from uX(ySO) to
^X(xS) such that Z
o-PRECOMPACTNESS OF o-MATRIX TRANSFORMATIONS. In order to
consider the o-precompactness of the o-matrix transformation, we consider
the range space (Y) as X(Y) defined as in (2.10) corresponding to a
normal SVSS x and a 1.c.s. Riesz space Y where Y is equipped with the
Lebesgue topology so that the topology T(y) on (Y) is generated by
}, where Q_ is defined as in (3.3).
seminorms
q
[Zji].
x
ORDERED GENERALIZED SEQUENCE SPACES
349
With these underlying assumptions, we prove
PROPOSITION &.. A positive o-matrix transformation Z-from
^(X) to (Y) is o-precompact if and only if each Z.. is o-precompact.
(z(6X.)) i’ for each
PROOF. For necessity, use the fact that Zij(x)
N,
In order to prove the converse part, for i,j
i,j >1 and x e X.
^(X)
,(Y) by
define linear operators Z
[Zij]
Zij(x
zjix)’-"
^(X).
xiI
6.,
’s are o-precompact operators from X to Y, Z i’ s are also
Since Z
x
’
with
o-precompact; indeed, for B
Bi o, (the case when 8
follows)
and any -neighbourhood U
{ a (Y): Q_
trivially
8
8
of
in (Y),
.
"
< [
zji[-,]
/
f
{6
()
< )
u
f e F t, F being the finite subset of
q(y) <
y
ponding to the neighbourhood U
(n).
x() by
^(X)
i,n e N, define Z
where
o,
obtained
e/IBilt.
cortes-
Now for
n
z(n)
Z
j=l
^(X),
These linear operators are clearly o-precompact and for
^(X)
(V) is defined by
Z () as n /- where Z
Zi(n)() (o)______.>
Zi()
6
(Z()).
_>
V
.I. As (,(Y),T(y))
Definition
converges to Z () in
3.5
cf.
zn)()
is a Lebesgue space by Proposition
T
(y)"
In order-to show the o-precompactness of each Z i, fix
of
> [ in ^(X) and a neighbourhood
interval [-,] for
such that
of
Then we can find another neighbourhood
N such that
Choose n
O
N, an
in (Y).
"
z(n)()-
Z ()
z.
o
"
V
n > n
o
in (Y) with
and a finite set
(n
z
[-,1 c
+
g.
Hence
zi[-,]
arid st
Zn
n
r.
Z.
Zi,
is
c
+
o-precompact.
Consequently,
z(n):
^(X)
W n > I, are also o-precompact operators.
, (Y),
As
where
z(n)() (o..._)_>
A(X) cf. [I0], Proposition 3.5(i) and the topology
Z() in (Y), V
Z() in T (y). Now proceeding as in
t(Y)
is
Lebesgue,
of
the preceding paragraph, we infer the o-precompactness of Z. This
completes the proof.
z(n)()
M. GUPTA AND K. KAUSHAL
350
As an order bounded, sequentially order continuous o-matrix transformZ
can be written as Z-- [Z +
], cf. Propositions
(vi)
.2.(v),
and .3 of [I0], the above result immediately leads to
THEOREM .5. Let Z
be an order bounded, sequentially order
[Zij
Ition
ij]
Zj
[Zij]
continuous o-matrix transformation from ^(X) to x(Y).
Then Z is o-precompact if and only if Z.. is o-precompact for each i,j _> I.
CERTAIN o-MATRIX TRANSFORMATIONS. In this subsection, we characterize o-matrix transformations from c(X), ’(X) to themselves, cs(X) to c(X)
and derive necessary conditions for a matrix of linear operators to transform l(x) into a simple OVVSS ^(X).
We begin with the following general
result:
PROPOSITION .6. Let X be a Riesz space, Y an order complete Riesz
space and T n a sequence of positive linear operators from X to Y. Then
n
a necessary and sufficient condition for the sequence
T (x)) to order
,
i=l
order converges in X is that
converge in Y whenever {x n
T.1order
i--l
converges in b(x,y).
PROOF. It suffices to prove the sufficient condition as necessity is
immediate from Theorem 2.3.
For an order converging sequence {Xn in X with
_< x, ’, n _> I,
..IXnl
write
n
n
i=l
Ti(x
n
N
T
(x)I
Then for n > m,
n
It n
[
m
i=m
n
<
[
T (x)
i=m
n
[
Since
Ti(x)},
i=m
being order convergent, is order-Cauchy in Y,
tn
is
order-Cauchy and so order converges in Y by Proposition 2.1.(iii).
Making use of the above result, we prove
PROPOSITION .7. Let X be an order complete Riesz space with diagonal property and Zij s, positive linear operators from X into itself Then
[Z ij is an o-matrix transformation from c(X) to c(X) so that the
Z
transformed sequence order converges to the same limit if and only if
n
(i)
j=l
(o)
Zij
2
Z. as n
(R),
for some positive linear operator
_> I;
Z i,
(ii)
,Zi(x)
(o)
x as
x
x; and
N.
-,
0 as
X and for each
x
Z ij (x)
(ii)
and (tii).
PROOF. We first derive the necessary conditions (i),
(iii)
n
(i)
for each
order converges in X, for each
Z ij
j=l
{x:} c(X), (i) follows from Proposition .6.
Since
(xj)
N and
ORDERED GENERALIZED SEQUENCE SPACES
(ii)
For x
351
X, consider the constant sequence {x
x
n
n
(o)
c(X).
in
Then Z.(x)
x by the hypothesis
(iii) This follows from the hypothesis, equalities
Zii(x
(o)
V i,j
N and
as
in X, for each
_> i.
For proving the sufficiency, let us consider a sequence
x, V n
_>
(z(6X)) i’
(6xj)i
n
xj (o))x
with
in X.
say, in X for each
we can find
> I.
Hence by Proposition 2.1(i), for any
e >
o,
N such that
o
Ixj
c(X)
Zij(xj) (o).. ti,
j=l
Then by Proposition 4.6,
{x.
<
x
Jo
>
V
u,
where u is the regulator of convergence of the sequence (x
n
be such that Z.(u) < v, V
> I.
Then
n
_<
lo-lim
n+(R)
Z
--i
j=l
ij
Let v
X
n
Zij(xj)
x)
(xj
-o-lira
n+(R)
j=l
Zij(x)
x)[
Zij(x
+
+
+
IZi(x
IZi(x)
x
x[
o
o
<
j=l
where
x)
Zij(x
J=Jo
Zij([xj
o-lira
n
x[)
+ v +
[
Zij(x
J=Jo
(iii) and Proposition 2.1(i), t.
IZi(x)
x).
x in X.
x[
Consequently, by
(i,i),
This completes the proof.
NOTE. Observe that the restriction of Z on the space c (X) has the
o
range contained in c o (X) under the conditions (i), (ii) and (iii).
For the space (R)(X), we have the following characterization of omatrix transformation.
Let X be an order complete Riesz space. Then a matrix
Z -:
of positive linear operators Zij from X to X, transforms ,(R) ()
into itself if and only if
THEOREM 4.8.--
[Zij]
n
(i)
for each
in
N,
(o)
Z. as n
(R),
for some positive
j=l
linear operator
(ii)
Z. on X; and
X,
for each x
Z.(x):
1} is an order bounded subset of X.
>
It suffices to prove the sufficiency of the conditions (i) and
PROOF.
(ii) as necessity is immediate from the definition of o-matrix transformation
(R)(X), then "the sequence
Z from (X) into itself. Indeed, if
xj]
n
j=1
every
Zij,(xj):
>
is order-Cauchy by (i) and so order
n >_i
by Proposition 2.1(iii).
Let
n
t.
o-lira
n/(R)
j=l
Z ij (xj)
>
I.
converges for
M. GUPTA AND K. KAUSHAL
352
Ixjl
>
< x, for each
and for some x
In case of the matrices of linear operators transforming the space
cs(X) to c(X), we have the following two results dealing separately the
sufficient and necessary conditions:
Since
THEOREM 4.9. Let X be an order complete Riesz space wth dagona
a matrix of positive sequentially order continuous
property and Z
lnear operators satisfying the conditions
[Zj],
([)
for each
>
, o-m
n
IZj
j=l
n-
Z ij+l
ially order continuous linear operator Z. on X;
CZ.) is an order bounded sequence in
(ii)
(iii)
1,
for each
(o)
-.
Zij
Z i’ tot some sequent-
S(x)
S(x,x); and
s(x) where is
in
as
the identity map on X.
Then Z is an o-matrix transformation from cs(X) to c(X) so that the
transformed sequence order converges to the sum of the original sequence.
PROOF. e first prove the result for those elements
xj) cs(X),
n
which have the
zero sum;
cs(X), we show that
t.
(o)
indeed, if
[
o-lira
n-
ti
o-limn
i=
Zij(xj),
j=l
N, exists and
for each
in X.
m
in X and so
N. Then s m (o),
[ xj, m
j=l
>
<
N such that
for given t > o, we can find jo
that
such
(X) be
Let A,B
where u is the regulator of convergence.
I. Then using the equalities
! B,
! A and
Let us write s m
Zil
[Zil
m-I
m
(Zij-Zij+l)(Sj)
Zij(xj
Zim(Sm):
+
-
Jo’
(*)
I;
I, m
and
m+p-I
m+p
j=m
j=m (Zij-Zij+l)(sj)
Zij(xj)
+
Zim+p (sin+p)
Zim (sm-1)’
i,m,p e
N,
we infer that
m+p
Zij(xj)l
=m
<
W
(3A+2B)u"
m
>_
Jo
and
n
t.
(Zij
o-lira
(+)
>
Zij+l)(Sj)
j=l
the condition (iii), we deduce that
(+)
and
Also, from
n/(R)
Z()
(o)
e
Thus
c(X).
If
e
cs(X) is such that
[
x.
j=l
defined by
Yl
xl
x,
yj
xj,
for
x
e,
then the sequence
{yj)
> 2, is. in cs(X) with zero sum. As
ORDERED GENERALIZED SEQUENCE SPACES
m
353
m
Z ij (xj)
j=l
Zij(Yj)
j=l
+ Z
il
(x)
V
m >
(o)
x in X, the result follows from the preceding paragraph.
Let X be an order complete Riesz space and Z
a matrix of positive and sequentially order continuous linear operators
Zij(x)
and
-= [Zij],
THEOREM 4.10.
from X into itself such that
>
Zii+l for each i,j in N. Assume that
Z transforms cs(X) to c(X) so that the transformed sequence order converges
sum
to the
(i)
of the original sequence.
> i, Z.
for each
,S(x);
Z
>I and x
for each
(ii)
Zii
(iii)
Then
X, Z ij(x)
(o)
x as
Z. as n
".
In
-(0)3
where
(R),
and
X,
{Zi(x) is an order bounded sequence of X.
(i) This is immediate from the equalities
for each x
PROOF.
(z(6Xj))i
Zij(x)
V
N
i,
and
_
X.
x
(ii) As e < Z. (x)
N and X is
in X, for each x in K and
In
n
N. Hence,
order complete, Zin(X)_
for some
X and for each
if we define linear operators Z.’s on X as
v.
v.,
Zi(x)
Zin(X)
o-lim
n/=
eX, then in view of Proposition 2.2 and Theorem 2.3,
for x
S(x).
(iii)
e <
since
Zi(x)
Zin(x)
<
<
11
K, observe that
.for each x
e <
{Zi:
> 1
Zil(X)
Zil(X),
N and
for e,ach i,n
(iii)
Hence
follows from (i).
Zin(X) (o); Zi(x
as
in X by (ii).
Restricting further the linear operators Z.. in the hypothesis of the
above theorem, we have
THEOREM .11. Let Z
and X be as in Theorem 4.10. Assume
where
N. Then A
V i,j
further that
j,
i,j e N, is an o-matrix transformation from c(X) to
n
(R),
[Zij]
-= [Aij],
Zij >Zi+j,
Zij
Zij+l,
Aij
Co(X).
,
PROOF. Observe that we have one-one onto correspondence R between
where
c(X) and cs(X), defined by R()
{xj} c(X) and
>
2.
u
and
x
is
by
given
uj xj xj_ I,
{uj
I
(o)
{x n} in c(X) with x
x
Let us now consider a positive x
n
c(X) and
as defined above, Z()
X. Then with
in X, for some x
Also, from the proof of the part (li)
(Z())
x by the hypothesis.
.
of
the’
preceding theorem,
N’
Further, Z in(X n
Zin(X)
x)
n
(o)
v i, for some
>
8
as n
X and for each
v.t
(R),
V
_> I.
Hence
n
(Z())i
o-lim
n.(R)
j=l
(Z -Z
ij ij+l
)(xj)
+
vi
> I.
(*)
354
M. GUPTAAND K. KAUSHAL
Note that v.
Zi+l, j, i,j >1.
x under the additional restriction, namely
Zi..j
>
Hence if
n
s.
o-lira
n
(Z())
(Zij-Zij+l)(X )J
j=l
>
v
(o))
e in X. Thus A()
(s.)
then s.
c(X).
c (X) for
O
Lastly, we prove
THEOREM 4.12. Let X be an order complete Riesz space such that
<x,xS> forms a dual pair and o(X s,X)-bounded sets are order bounded
in X
Also, assume that A(X) is an ideal in a(X) and is a simple space
for the topology o( ^(X) ^x(xs) ).
For a matrix Z
of positive
linear operators from X into itself, consider the following
(i) Z is an order bounded sequentially order continuous, o-matrix
s.
=- [Zij]
.I(x)
transformation from
(ii)
The adjoint
Z*
to
A(X).
of Z is an order bounded, sequentially order
^x(Xs) to
(X SO ).
,=
continuous o-matrix transformation from
X
For x
(iii)
statement:
the sequence
[a x
> I}
where a
x.
{Z
ij
(x)
> 1}, is an order bounded set in ^(X).
Further, if (iii) holds, then the partThen (i)=> (ii) and (i)=> (iii).
ial statement of (ii), namely,
^X(xSO)
to
PROOF.
3.5,
a
x
=
"Z*
(X SO )" holds.
(i) => (ii). Since
is an o-matrix transformation from
(l(x))x
=(xs),
[ll].
cf.
Proposition
the implication follows from Proposition
(i) => (iii) For a given
K, we first prove that
in N and x
Z(6 x and x z
^(X). Indeed it is immediate as a
6j
x
In order to show that the set [ax
.bounded, consider a positive element
X so
the preceding implication, Z * (-)
_>
[fj
-=
=
such that
I<(Z * (f--))i ’x>l
Ij1 fj(Zj (x))
> t) is o(^(X)
Hence [aX
o(^(X),
1) is
AX(xS).
in
and so order bounded in
,
n
For the last statement, we first show that
j=l
...
n
j=l
T
(fj
in
Z* (fj)
AX(x s)
j=
Indeed
n
f.a.
j=l
n
n
Z
order
N,
z ji fj(x)
=> the set
(+)
> I
< g(x)
AX(xS))-bounded
for a positive element
Then from
X so
and so there exists g
^(X) by the hypothesis.
converges in X s
for x > e and n
,I(X).
^X(xS))
J fj(x):
n >
I}
is bounded in
R and so
Z
j=
. (fj)
s
order converges in X
by Theorem 2.2 and Proposition 2.3.
xS), for x > e, choose
For proving that Z * (T)
> 1.
Hence
<
that
for every
,=
aJl
A(X) such
355
ORDERED GENERALIZED SEQUENCE SPACES
I,:lz*7)),
>1
j=I
fj(Zj
j=t
_<
[
j-1
Thus
(Z
(f))t
ACKNOWLEDGEMENTS.
,,x )l
(x)
fj(yj),
> 1,
> 1} is order bounded in X s and the result follows.
The authors are thankful to the referee for his sugg-
estions, especially for pointing out the concepts of
order-Cauchy complete
Riesz space and Proposition 2.1 (iii) which have improved Proposition 4.6
and Theorem 4.8 in the present form, proved earlier for order complete
space Y with diagonal property.
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