On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
On Decomposition Method And PN-Spaces
Received: 16/12/2003
Accepted: 15/3/2004
A. Tallafha* & W. Shatanawi**
‫ملخص‬
‫طريقة تجزئة فوجت هي إحدى األدوات الرئيسية لبرهان أن فضاء فريشييت النووي‬
‫ طالفحة عرف‬.‫ ترزي أوجلو عمم طريقة تجزئة فوجت‬.‫مكافئ لفضاء مستقر من نوع الالنهائي‬
‫ مؤخ ار شطناوي أعطى تعديل جديد‬.‫أن‬.‫ أن وأثبت نتيجة ترزي أوجلو على فضاء بي‬.‫فضاء بي‬
.‫على نتيجة ترزي أوجلو‬
Abstract
|
Vogt decomposition method is one of the main tools to show whether a
certain nuclear Frèchet space is isomorphic to a stable power series space of
infinite type. Terzioğlu generalized the Vogt decomposition method. Tallafha
defined what he called a PN-space and he proved the Terzioğlu’s result on PNspace. Later on Shatanawi gave a new modification for Terzioğlu result. In this
paper we shall generalize Tallafha and Shatanawi’s results.
1. Introductions.
.
x 
Let H be a Frèchet space with an absolute basis n , and n be the
fundamental system of semi-norms for H. Then H is isomorphic to a Köthe
a k  xn k : n, k  
Space λ(A) where the Köthe set A={ n
}. From DyninMitiagin basis theorem, we conclude that every basis in a nuclear Fréchet
space is absolute, thus we can identify nuclear Frèchet spaces with nuclear
Köthe spaces. Therefore the attraction of sequence spaces to a large
number of analysists is understandable.
*
Department of Mathematics, University of Jordan.
**
Department of Mathematics, The Hashemite University.
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In set theory it is known that if a set A is equipotent to a subset of a set
B and B in turn is equipotent to a subset of A, then A and B are equipotent.
By imitating this fact, we ask the following question: If E and F are two
locally convex spaces, E is isomorphic to a complemented subspace F of E
and F is isomorphic to a complemented subspace of E, then under what
conditions on E and F to be E isomorphic to F. Among the pioneer workers
in this are M-Valdivia and D-Vogt. They proved what are called later
Valdivia Decomposition method [14] and Vogt decomposition method
[15].
Vogt decomposition method is one of the main tools to show whether a
certain nuclear Frèchet space is isomorphic to a stable power series space
of infinite type. In [1], Aytune, Krone and Terzioğlu defined what they
called a local imbedding and they modified Vogt decomoposition method.
Later on, in [12], Terzioğlu gave a modification of Aytune, Krone and
Terzioğlu ‘s result. In [9], Tallafha generalized the result of Terzioğlu.
Also in [10], Tallafha and Jarrah gave another generalization. Thereafter in
[8], Shatanawi modified Terzioğlu’s result.
In this paper, we shall generalize the result obtained by Tallafha in [9],
and the result obtained by Shatanawi in [8].
2. Basic Concepts.
A set A of sequences of non-negative real numbers is called a Köthe
set if it satisfies the following conditions:
1. For each pair of elements a, b  A there is c  A with an = O(cn) and bn
=O(c ) where a = O(c ) means that there is a   0 such that
n
an   cn
n
n
for all n   .
2. For every integer r   there exists a  A with ar  0 .
Given a Köthe set A, the space of all sequences x=(xn) such that
p a ( x) :  xn a n   
n
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for all a  A is called the Köthe space generated by A, denoted by
λ(A). The semi-norms {pa : a A} define a locally convex Hausdorf
topology on λ(A), called the normal topology of λ(A).
Definition 1.1 [11] A Köthe set P will be called a power set of infinite
type if it satisfies the following conditions:
1. For each a  P, 0 < an ≤ an+1 for all n.
2. For each a  P, there exists b  P such that
a n2  O(bn )
.
A Köthe space of the form λ(P) where P is a power set of infinite type
is called a G - space or a smooth sequence space of infinite type.
Definition 1.2 [9] A Köthe space λ(P) will be called a PN-space if each
a  a n   P
is positive and non decreasing.
Remark. Every G -space is a PN-space.
A Köthe space λ(A) is called stable if λ(A)  λ(A)  λ(A). The stability
of a nuclear G -space λ(P) is equivalent to the condition that, for each a
 P there is b  P such that a2n = O(bn)[13].
Definition 1.2 [1] A continuous linear map i from a stable nuclear G space λ(P) into a locally convex space E is called a local imbedding if
.
there is a sequence   ( n )  P and a continuous semi-norm
on E with
x  : n xn  n  ix
For two sequences x = (xn) and y = (yn) we will denote the sequence
(x1, y1, x2, y2, …) by x* y. If K and L are Köthe sets, then
K  L : {a  b : a  K , b  L}
is also a Köthe set [13].
Definition 1.3 Let x1, x2, …, xn be elements of a linear space E.
The subspace sp{x1,x2, …, xn} of E is defined by
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 n

  i xi :  i C 
.
sp{x1, x2, …, xn} =  i 1
Definition 1.4 A sequence (xn) in a locally convex space E is said to be a
basic sequence if it is basis for sp{xn : n  } .
The following well known result is essential for our subsequent
arguments.
Theorem 1.1 (Grothendieck-Pietsch criterion for nuclearity) A Köthe
space λ(A) is nuclear if and only if for every a  A, there is b  A such
that (an /bn)  ℓ1.
Our goal in this paper is to generalize the following result
Theorem 2.2 [9] Let  (A) be a PN-space where the Köthe set A satisfies
the following conditions: For each a  A there is a b  B such that
a 2 n  Obn 
. If there is a local imbedding map i from λ(A) into a locally
convex space E and if E is isomorphic to a closed subspace of λ(A), then E
has a complemented subspace isomorphic to λ(A)
2
Main Results.
It is worth starting this section by discussing the nuclearity of the
sequence space of the form   A* A .
Proposition 3.1 The Köthe space   A is nuclear if and only if   A* A is
nuclear.
Proof. Suppose that   A is nuclear. Given c  a1 , a1 , a2 , a2 ,... in A*A.
Then a  a1 , a2 ,...  A .Since   A is nuclear, there is b  b1 , b2 ,...  A
d  b1 , b1, b2 , b2 ,...
Let
. Then d  A * A and
. Therefore   A* A is nuclear. Conversely, let a  A . Then
a  a1 , a1 , a2 , a2 ,...  A * A . Since   A* A is nuclear, there is
an / bn    1 .
such that
cn / d n    1
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
b  b1 , b1 , b2 , b2 ,...  A * A such that
Therefore   A is nuclear. ■
an / bn    1 .
So
an / bn    1 .
One can easily see the following proposition.
Proposition 3.2 The sequence space   A is a G -space if and only if
  A* A is a G -space.
By defining a map h from   A* A into   A    A via
hxn   x2 n1 , x2 n  , one can easily see the following proposition.
Proposition 3.3 The sequence space
  A    A .
  A* A is isomorphic to
By the definition of stability and the aid of Proposition 3.3, we have the
following result.
Corollary 3.1 If   A is stable, then   A* A is isomorphic to   A .
To proceed in our work and to achieve our goals, we introduce the
following definition.
Definition 3.1 A Köthe set A is called a Super Köthe set if for each
a, b  A there is c  A such that
a

sup  i : i  1,2,..., n  Ocn 
 bi

.
Theorem 2.1 Suppose that λ(A) is a nuclear Köthe space and A is a super
Köthe set. If there is a local imbedding map j from λ(A*A) into a locally
convex space E and if E is isomorphic to a closed subspace of λ(A), then E
has a complemented subspace isomorphic to λ(A).
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
Proof. Since j is a local imbedding, there are a sequence   ( n )  A * A
which is of the form a1 , a1 , a2 , a2 ,... and a continuous semi-norm p on E
such that
x   n xn  n  p( jx).
Since every subspace of nuclear is nuclear and nuclearity is topological
property, E is nuclear (see Pietsch [6]). Also, since E is nuclear, p is given
by a semi-inner product  ,  which is an inner product on the subspace
j(λ(A  A)); (see Pietsch [6]).
We will construct a basic sequence (gn) in j(λ(A  A)), satisfying the
following conditions:
1.
g n  sp{ je1 , je2 ,..., je2 n }
2.
 g n , e    g n , g i   0 for  1,..., n and i  1,..., n  1.
3. p(gn) = 1
where (en) is the canonical basis of λ(A  A), which is assumed to be
orthonormal with respect to  ,  . This construction is possible since sp{je ,
1
…je2n} is a 2n-dimensional space and p is a norm on j(λ(A  A)). If gn =
2n
j i 1  ni ei
, then we have



2n
i 1
Let
n  i 
i

2n
i 1
 n ei
i

 p( g n )  1.
.
be any continuous semi-norm on E. By the continuity of j we can
find c  c1 , c2 ,...  A * A which is of the form c  a1 , a1 , a2 , a2 ,... and
  0 such that

g n  j i 1  ni ei
2n

  i 1  nn ei
2n
  i 1  ni ci .
2n
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
So we have
g n   i 1  ni ci
2n
i
i
c

  sup  i : i  1,2,...,2n
 i

a

  sup  i : i  1,2,..., n.
 ai

 
Since A is a super Köthe set, there is b  bn  A such that
a

sup  i : i  1,2,..., n  Obn .
 ai

So we have
g n  O(bn )
.
Since E is isomorphic to a subspace of λ(A) and {en: n  N} is a base of
x  v   e
λ(A), then for x  E, write
. We have
g n , x  v  v g n , ev  v n g n , ev  v
.
So for each a  A ,
g n , x a n  v n g n , e  v av


vn
v

vn
g n , e av

 v  n  v g n a .
.a
Since
is a continuous semi-norm on E, there are d   A and   0
such that
gn
a
  d n
for all n  N.
Therefore we have
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
g n , x a n   d n v n  v
(1)
  v n  v d v'   x d ' .
(2)
Since E is a linear space, g n  E for each n  N, and


n 1
g n , x g n spg n : n  ,
Tx  n g n , x g n
we define a map T : E  E by putting
. For any
.
continuous semi norm
on E,
Tx  n g n , x g n .
Since
that
.
is a continuous semi norm on E, there are a  A and   0 such
g n   an
for all n  N.
So we have
Tx   n g n , x a n .
By nuclearity of λ(A) and inequality (2), there are b  A , d  A and
  0 such that a n / bn    1 and g n , x bn   x d . So we have
Tx   n g n , x
 x
d

n
an
bn
bn
an
.
bn
Therefore T is continuous. Moreover, since Tgn=gn, and
gn : n  
{
} is a basic sequence, T is a projection of E onto
G  g n : n  . One can easily show that G is complemented in E. To
finish our proof it remains to show that G is isomorphic to λ(A). Define a
map J :  ( A)  G by butting Jen  g n . It is clear that J is well-defined,
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
one-to-one and continuous. So the only remaining part is to show that J is
x  n g n , x g n
onto and open. To show J is onto, let x  G be given. Then
.
Let c   g n , x  , which is in λ(A) by nuclearity of λ(A) and inequality (1).
Moreover



i 1
i 1
i 1
Jc  J ( g n , x en )  g n , x J (en )  g n , x g n  x
Hence J is onto. For each a  A
x
a
 n g n , x a n
defines a continuous semi norm on G. To show that J is closed, let ua
be any zero neighborhood of  ( A) . We claim
J (u a )  { y  G : y
a
 1}.
z  J (u a ).
y  ua
To show our claim, let
Then there is
such that
z  Jy. Also there is    g n , z    ( A) such that z  J . Since J is oneto-one we have   y, hence
p a ( )  p a ( y )  n g n , z a n  z
1
.
y a 1
Conversely, let y  G with
be
   g n , y    ( A)
given. Then there is
such that J  y. Hence
Therefore
z { y  G : y
a
a
 1}.
p a ( )  n g n , y a n  y
and so
 ( A)  G.
a
 1,
  u a . Thus y  J (u a ) and hence J is closed. Therefore
Definition 2.2 [8] A continuous linear map j from λ(B  A) into a locally
convex space E is called strong stable local imbedding if there is a
.
sequence   ( n ) and a continuous semi-norm on E with
x  : n xn  n  jx
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
where  satisfies the following condition:
(L2) :
b  B.
 n  0, 1 /  2n  Oan , and 1 /  2n1  O(bn ) for some a A and
It is clear that if λ(A) is a G  space and if we assume that   A* A ,
then condition L2 is automatically satisfied.
Applying Theorem 3.1 we have the following result:
Corollary 3.2 [8] Suppose that   A is a nuclear G -space. If there is a
strong stable local imbedding j from   A* A into a locally convex space
E and if E is isomorphic to a closed subspace of   A , then E has a
complemented subspace isomorphic   A .
It is an easy task to show that if   A is a PN-space and if A satisfies
the following condition: For each a  A there is b  A such that
a 2 n  Obn 
, then   A    A, A , and A is a super Köthe set. So we have
the following corollary.
Corollary 3.3 [9] Let  ( A) be a PN-space where the Köthe set A satisfies
the following condition: For each a  A there is b  A such that
a 2 n  Obn 
. If there is a strong stable local imbedding map j from  ( A)
into a locally convex space E and if E is isomorphic to a closed subspace of
 ( A) , then E has a complemented subspace isomorphic to  ( A) .
We close our paper by giving an example of a sequence space   A
which is not a G -space and the Köthe set A doesn’t satisfy the condition
of Corollary 3.3. But   A satisfies the conditions of Theorem 3.1.
k
A  a nk n : k  
Example 3.1 Let
where an  kln k n  1 . Then we
have the following assertions:
n
1. A is a super Köthe set .
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On Decomposition Method and PN- Spaces ………....… A. Tallafha & W. Shatanawi
2.   A is not a G -space.
3.
  A is nuclear.
4.
There is no s   such that
 .
a 12 n  O a ns
Proof. It is clear that A is a super Köthe set . To prove that   A is not a
G -space, it is enough to show that there is no s   such that
a 
1 2
n
 .
 O a ns
Assume the contrary, i.e; there is s   such that
ln n  12 n  
2
a1n   Oans . So there is a ρ>0 such that sln sn  1n
n   ,
2
which contradicts the fact that ln n  1 / sln sn  1 is divergent for
any s   . Hence   A is not a G -space. Being   A is nuclear follows
from the following
a nk  kln k n  1 
1
      1
 
2k
an
 2
 2kln 2k n  1 
.
n
n
To this end, one can prove 4 by similar arguments in proving
  A is not a G -space.
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