ON The SM -Operators | Darmawijaya | BIMIPA 13901 28573 1 PB
Soeparna dkk, On The SM-operators
ON THE SM -OPERATORS
Soeparna Darmawijaya1 , Muslim Ansori 2 , dan Supama3
1,3
2
Mathematics Departement, FMIPA UGM , Yogyakarta
email: [email protected]
Mathematics Departement, FMIPA Universitas Lampung
Jln. Soematri Brodjonegoro No 1 Bandar Lampung
email: [email protected]
ABSTRACT
This is a partial part of our results in studying generalization of Hilbert-Schmidt and
Carleman operators in Banach spaces. This problem can be done if we preserve some
intrinsic properties of Hilbert spaces involved; for examples, reflexivity and separability.
The result of the generalization of Hilbert-Schmidt operator will be called SM-operator.
Infact, almost all of properties of the SM-operator preserve almost all of propertie s of the
Hilbert-Schmidt operators. The application on some classical Banach spaces will appear in
the next publications.
Keywords: Orthonormal Schauder bases,separable and reflexive Banach spaces, HilbertSchmidt operator
Makalah diterima 17 Septemb er 2005
1. INTRODUCTION
One of the most important classes of
bounded operators is the class of
Hilbert-Schmidt operators. Let H 1 and H 2
be Hilbert spaces. A bounded operator
A : H 1 → H 2 is called a Hilbert-Schmidt
operator if there exists an orthonormal bases
{en } of H 1 such that
∞
∑
Aen
2
d k
∞
= ∑ < en , A∗dk∗ > dk .
k =1
It implies
∞
∞
∑∑
n =1 k =1
∞
n =1 k =1
x = ∑ α k ek .
∞
k =1
en = 1 for every n . We
define a sequence of vector {e∗n } ⊂ X ∗ ,
which is called biorthonormal system of {e } ,
as follows:
n
∞
k =1
x, en∗ = e∗n ( x ) = en∗ ∑ α k ek
∞
= ∑ α k e ∗n ek = α n ,
k =1
for every n ∈ N . It is true that e∗n ∈ X ∗ , for
e∗n
is linear and bounded, ek , en∗ = 0 for
every
k≠n
and
ek , en∗ = 1
for every
k = n . The sequence {e } forms a bases of
∗
n
the closed subspace [{e }] ⊂ X . Especially,
∗
n
∗
we have [{e∗n }] = X ∗ if and only if {en } is
shrinking, i.e.,
∞
∑ ek , x e = o ,
lim
n →∞ k = n
∗
∗
k
∗
for every x ∗ ∈ X ∗ (Lindenstrauss and
Tzafriri, 1996, Proposition 1.b.1).
Again, in what follows we shall always
assume that {en } and {d n} are orthonormal
Schauder bases, or in short, OSB of X
and Y , respectively. If A ∈ L c ( X , X ) ,
where Lc (X ,Y ) is the collection of
continuously linear operators from Banach
space X into Banach space Y , the operator
A∗ ∈ Lc ( Y∗ , X ∗ ) is called the adjoint operator
of A if for any x ∈ X and y∗ ∈ Y ∗ , we
have
Ax, y ∗ = x , A∗ y ∗ .
Then, we have
50
∞
= ∑ ∑ < en , A∗ d k∗ >.
Further, for simplicity and some reason
we assume that
∞
Aen , d ∗k = ∑∑ < en , A∗d k∗ >
k =1 n =1
The dual space X ∗ also has a dual. It is
usually denoted by X ∗∗ , is called the second
dual of ( X , ) and consists of all
continuously linear functionals on X ∗ . For
)
each fixed x ∈ X define x ( f ) to be
)
f ( x ) for all f in X ∗ . It is clear that x
is a linear functional on X ∗ , and since
)
x ( f ) = f ( x) ≤ f x ,
)
we see that x in X ∗∗ . Hence we can define
a map φ from X into X ∗∗ by letting
)
φ ( x ) = x for each x in X . Since, for
any nonzero element x0 in
(X, ) ,
then
there is an element f ∗ ∈ X ∗ such that
f ∗ = 1 and f ∗ ( x0 ) = x0 .
Thus, the map is linear and φ ( x ) = x
for
each x in X . As a consequence, we have
φ ( x ) = sup x∗ , φ ( x )
x ∗ =1
= sup x, x∗
x ∗ =1
(1.1)
= x.
Thus φ is also an isometry and sets up a
congruence between X and X ∗∗ . The
normed space is imbedded X into X ∗∗ by
the canonical imbedding
φ
in a
isometrically
isomorfic
way
and
φ ( X ) = X ∗∗ . Thus X can be considered as
the normed space X ∗∗ .
Soeparna dkk, On The SM-operators
Proof: (i) For every x ∈ X , we have
3. MAIN RESULTS
∞
Based on the results of the last
discussion we start with the following
definition.
Ax = ∑ x , en∗ Aen
k =1
and
∞
Ax ≤ ∑ x Aen
Definition 1. An operator A ∈ L c ( X ,Y ) is
called an SM-operator from X into Y , if
∞
∞
∑∑
n =1 m =1
∞
n =1
∞
∑
= x
Aen
n =1
∞
Aen , d m∗ = ∑ ∑ en , A∗ d m∗ < ∞
= x
m =1 n =1
for every OSB {en } of X and {d m } of
Y.
∞
∞
∑ ∑ < Ae , d
n
n =1
∞
> dk
k =1
∞
∑∑
≤ x
∗
k
Aen , d k∗
dk
n =1 k =1
It is clear that if A is an SM-operator,
then the number
A :
∞
n =1 m =1
is nonnegative and it does not depend on the
choice of an OSB {en } of X and an
OSB {d n} of Y . Let SM (X ,Y ) be the
collection of SM-operators from a Banach
space X into a Banach space Y .
By Definitio n 1 and (1.1), we have the
following theorem.
∞
∑∑
Ae n , d k∗
n =1 k =1
= x
∞
A = ∑∑ Aen , d m∗
∞
= x
A ,
A ≤ A .
(ii). The space SM (X ,Y ) is a normed space
which implies
with respect to the norm
∞
(ii.a).
. , for:
∞
A = ∑∑ Aen , dm∗ ≥ 0 , for every
n =1 m =1
A ∈ SM (X , Y ).
∞
Theorem 2. An operator A ∈ L c ( X ,Y ) is an
SM-operator if only if A∗ is an SM-operator,
that is, A ∈ SM (X ,Y ) if and only if
A∗ ∈ SM (Y ∗ , X ∗ ) , and
∞
A = 0 ⇔ ∑ Aen , d m∗ = 0 ⇔ A = O
m =1
(null operator),
(ii.b).For every scalar α and A∈ SM(X,Y),
we have
∞
A = ∑ ∑ Aen , d m∗
n =1 m =1
∞
∞
= ∑ ∑ en , A∗ d m∗
∞
α A = ∑ α Aen , d m∗
m =1
=α
m =1 n =1
= A ,
(i )
( ii )
A ≤ A
, for every A ∈ SM (X ,Y ) ,
SM (X ,Y ) is a Banach space with
respect to
( iii ) If
. ,
A ∈ SM (X ,Y ) , then A is compact.
Ae n , d m∗ = α
A ,
m =1
∗
for every OSB {en } of X and {d k } of
Y.
Theorem 3. Let {en } and {d n} be an
OSB of Banach space X and Y ,
respectively. Then,
∞
∑
and
(ii.c). For every A, B ∈ SM (X ,Y ) , we have
∞
∞
A + B = ∑∑
n =1 m =1
∞
( A + B) e , d
n
∗
m
∞
= ∑∑ Aen + Ben , d m∗
n =1 m =1
∞
∞
= ∑∑ Aen , d m∗ + Be n , d m∗
n =1 m =1
∞
∞
= ∑∑ Aen , d m∗ + Ben , d m∗
n =1 m =1
51
Berkala MIPA, 16(1), Januari 2006
∞
∞
∞
∞
∞
= ∑ ∑ Aen , d m∗ + ∑∑ Ben , d m∗
n =1 m =1
∞
∑ ∑ ( A − A) e , d
n =1 m =1
j=1
n
k =1
= A + B
∞
j=1
A+ B ≤ A + B .
The proof of the completeness of the
space is as follows. Let { An} ⊂ SM (X , Y ) be
an arbitrary Cauchy sequence. Then, for any
number ε > 0 , there is a positive integer n0
such that for every two positive integers
m , n ≥ n0 , we have
ε
Am − An < .
2
We want to prove that there is
A ∈ SM (X ,Y ) such that
lim
An − A = 0 .
n→
there is A ∈ L c ( X ,Y ) such that
ε
A − An < ,
2
for every n ≥ n0 or
k =1
∗
n
s
t
j=1
k= 1
≤∑∑
m
(A
n
j
k
dk
∈ SM (X ,Y ) and hence A = An +
in SM (X ,Y ) . Moreover,
− Am )e j , d k
for every
j=1
k =1
n ≥ n0 . Hence,
If A ∈ SM (X ,Y ) and x ∈ X , we have
(iii)
∞
Ax = ∑ Ax, d ∗k d k
k =1
and for every positive integer n , we define
an operator Bn : X → Y :
Bn x = ∑ Ax, d k∗ d k .
k =1
It is clear that Bn ∈ Lc ( X ,Y ), Bn is a finite
rank operator, and
lim A − Bn = 0 .
BA ≤ B
then BA ∈ SM ( X , Z ) and
For
∞
n =1 j =1
OSB
every
X , {d m } ⊂ Y and
∑∑
{en }
{ f } ⊂ Z , we have
A .
of
j
∞
BAen , f j∗ ≤ ∑ B Aen
f j∗
n =1
≤ B
∞
∞
∑∑
Aen , d ∗m
n =1 m =1
∑ ∑ ( A − A) e ,d
n
j
≤∑∑
s
j=1
t
k=1
(A
n
∗
k
dk
− A ) ej , dk∗
ε
2
for any integers s,t and m , n ≥ n0 . Letting
s → ∞ and t → ∞ , we have
≤
n
lim
An − A = 0 .
n→
∞
for any integers s,t and m , n ≥ n0 . By
letting m → ∞ , we have
t
(A− A )
An − A < ε ,
Proof:
∗
ε
2
s
Therefore
Theorem 4. Let X ,Y and Z be Banach
spaces. If A ∈ SM (X ,Y ) and B ∈ Lc (Y , Z ) ,
≤ An − Am
<
ε
2
An − A
− A) e j , d k∗ <
Therefore, A is a compact operator.
Thus, we have
j=1
n
n→∞
An − A = 0 .
lim
n →∞
∑ ∑ ( A − A )e , d
dk
k
n
Am − An ≤ Am − An ,
t
k =1
∗
(A
every n ≥ n0 .
for
Since Lc (X ,Y ) is complete and
s
∞
≤ ∑∑
or
52
j
= B A ,
that is BA ∈ SM ( X , Z ) and
BA ≤ B
A .
Let SM ( X ) and Lc ( X ) stand for
SM ( X , X ) and Lc (X , X ) , respectively.
Combining the results of Theorem 3 and
Theorem 4, we have proved that SM(X) is
∗ − algebra, where ∗ is an involution from
SM(X) into SM(X) satisfying :
Soeparna dkk, On The SM-operators
(A )
∗
∗
= A ; ( AB ) = B∗ A∗
∗
ACKNOWLEDGEMENTS
and
(α A + B )
∗
=α A + B
∗
∗
for every A, B ∈ SM ( X ) and a real scalar
α , as stated in the following theorem.
Theorem 5. Let X be a Banach space
having a shrinking OSB . Then, SM ( X ) is
a Banach ∗ − algebra and an ideal of
Lc ( X ).
CONCLUSION
Generalization
of
Hilbert-Schmidt
operators into Banach spaces can be done by
preserving the instrinsic properties of Hilbert
spaces, i.e., separable and reflexivity. The
results, denoted by SM(X,Y) has in general the
same properties of those of Hilbert-Schmidt
operators.
The biorthonormal system
∗
n
and
{({d }, {d } ) : {d
m
∗
m
m
Conway, J.B., 1990. A Course in Functional
Analysis, Springer Verlag, New York.
Dapa P.S., 2000. On strong M-bases in
Banach spaces with PRI, Collect. Math.
51, 3, 277-284.
Johnson W.B.,Roshental,H.P., and Zippin,M.,
1971. On bases, finite-dimensional
decompositions and weaker structures
in Banach spaces, Israel J. Math. 9,
488-506.
∗
} ⊂ Y , {d } ⊂ Y
Morrison, T.J.,2001. Functional Analysis: An
Introduction to Banach Space Theory,
John Wiley & Sons. Inc. New York.
∗
n
n
REFERENCES
Lindenstrauss J and Tzafriri L, 1996.
Classical Banach Spaces I and II,
Springer Verlag, New York.
{({e }, {e }) : {e } ⊂ X ,{e } ⊂ X }
n
The second author was partially
supported by BPPS DIKTI. We also thank all
the anonymous referees for reading the paper
carefully.
∗
m
∗
}
is the key to solve the condition of
orthonormality in Hilbert-Schmidt operators,
used later in SM (X ,Y ) . For further works,
we have been using the operator in classical
Banach spaces Lp and l p ,1 < p < ∞ .
Weidmann, J., 1980. Linear Operators in
Hilbert Spaces, Springer Verlag. New
York.
Zippin, M., 1968. A remark on bases and
reflexivity in Banach spaces, Israel J.
Math. 6, 74-79
53
Berkala MIPA, 16(1), Januari 2006
54
ON THE SM -OPERATORS
Soeparna Darmawijaya1 , Muslim Ansori 2 , dan Supama3
1,3
2
Mathematics Departement, FMIPA UGM , Yogyakarta
email: [email protected]
Mathematics Departement, FMIPA Universitas Lampung
Jln. Soematri Brodjonegoro No 1 Bandar Lampung
email: [email protected]
ABSTRACT
This is a partial part of our results in studying generalization of Hilbert-Schmidt and
Carleman operators in Banach spaces. This problem can be done if we preserve some
intrinsic properties of Hilbert spaces involved; for examples, reflexivity and separability.
The result of the generalization of Hilbert-Schmidt operator will be called SM-operator.
Infact, almost all of properties of the SM-operator preserve almost all of propertie s of the
Hilbert-Schmidt operators. The application on some classical Banach spaces will appear in
the next publications.
Keywords: Orthonormal Schauder bases,separable and reflexive Banach spaces, HilbertSchmidt operator
Makalah diterima 17 Septemb er 2005
1. INTRODUCTION
One of the most important classes of
bounded operators is the class of
Hilbert-Schmidt operators. Let H 1 and H 2
be Hilbert spaces. A bounded operator
A : H 1 → H 2 is called a Hilbert-Schmidt
operator if there exists an orthonormal bases
{en } of H 1 such that
∞
∑
Aen
2
d k
∞
= ∑ < en , A∗dk∗ > dk .
k =1
It implies
∞
∞
∑∑
n =1 k =1
∞
n =1 k =1
x = ∑ α k ek .
∞
k =1
en = 1 for every n . We
define a sequence of vector {e∗n } ⊂ X ∗ ,
which is called biorthonormal system of {e } ,
as follows:
n
∞
k =1
x, en∗ = e∗n ( x ) = en∗ ∑ α k ek
∞
= ∑ α k e ∗n ek = α n ,
k =1
for every n ∈ N . It is true that e∗n ∈ X ∗ , for
e∗n
is linear and bounded, ek , en∗ = 0 for
every
k≠n
and
ek , en∗ = 1
for every
k = n . The sequence {e } forms a bases of
∗
n
the closed subspace [{e }] ⊂ X . Especially,
∗
n
∗
we have [{e∗n }] = X ∗ if and only if {en } is
shrinking, i.e.,
∞
∑ ek , x e = o ,
lim
n →∞ k = n
∗
∗
k
∗
for every x ∗ ∈ X ∗ (Lindenstrauss and
Tzafriri, 1996, Proposition 1.b.1).
Again, in what follows we shall always
assume that {en } and {d n} are orthonormal
Schauder bases, or in short, OSB of X
and Y , respectively. If A ∈ L c ( X , X ) ,
where Lc (X ,Y ) is the collection of
continuously linear operators from Banach
space X into Banach space Y , the operator
A∗ ∈ Lc ( Y∗ , X ∗ ) is called the adjoint operator
of A if for any x ∈ X and y∗ ∈ Y ∗ , we
have
Ax, y ∗ = x , A∗ y ∗ .
Then, we have
50
∞
= ∑ ∑ < en , A∗ d k∗ >.
Further, for simplicity and some reason
we assume that
∞
Aen , d ∗k = ∑∑ < en , A∗d k∗ >
k =1 n =1
The dual space X ∗ also has a dual. It is
usually denoted by X ∗∗ , is called the second
dual of ( X , ) and consists of all
continuously linear functionals on X ∗ . For
)
each fixed x ∈ X define x ( f ) to be
)
f ( x ) for all f in X ∗ . It is clear that x
is a linear functional on X ∗ , and since
)
x ( f ) = f ( x) ≤ f x ,
)
we see that x in X ∗∗ . Hence we can define
a map φ from X into X ∗∗ by letting
)
φ ( x ) = x for each x in X . Since, for
any nonzero element x0 in
(X, ) ,
then
there is an element f ∗ ∈ X ∗ such that
f ∗ = 1 and f ∗ ( x0 ) = x0 .
Thus, the map is linear and φ ( x ) = x
for
each x in X . As a consequence, we have
φ ( x ) = sup x∗ , φ ( x )
x ∗ =1
= sup x, x∗
x ∗ =1
(1.1)
= x.
Thus φ is also an isometry and sets up a
congruence between X and X ∗∗ . The
normed space is imbedded X into X ∗∗ by
the canonical imbedding
φ
in a
isometrically
isomorfic
way
and
φ ( X ) = X ∗∗ . Thus X can be considered as
the normed space X ∗∗ .
Soeparna dkk, On The SM-operators
Proof: (i) For every x ∈ X , we have
3. MAIN RESULTS
∞
Based on the results of the last
discussion we start with the following
definition.
Ax = ∑ x , en∗ Aen
k =1
and
∞
Ax ≤ ∑ x Aen
Definition 1. An operator A ∈ L c ( X ,Y ) is
called an SM-operator from X into Y , if
∞
∞
∑∑
n =1 m =1
∞
n =1
∞
∑
= x
Aen
n =1
∞
Aen , d m∗ = ∑ ∑ en , A∗ d m∗ < ∞
= x
m =1 n =1
for every OSB {en } of X and {d m } of
Y.
∞
∞
∑ ∑ < Ae , d
n
n =1
∞
> dk
k =1
∞
∑∑
≤ x
∗
k
Aen , d k∗
dk
n =1 k =1
It is clear that if A is an SM-operator,
then the number
A :
∞
n =1 m =1
is nonnegative and it does not depend on the
choice of an OSB {en } of X and an
OSB {d n} of Y . Let SM (X ,Y ) be the
collection of SM-operators from a Banach
space X into a Banach space Y .
By Definitio n 1 and (1.1), we have the
following theorem.
∞
∑∑
Ae n , d k∗
n =1 k =1
= x
∞
A = ∑∑ Aen , d m∗
∞
= x
A ,
A ≤ A .
(ii). The space SM (X ,Y ) is a normed space
which implies
with respect to the norm
∞
(ii.a).
. , for:
∞
A = ∑∑ Aen , dm∗ ≥ 0 , for every
n =1 m =1
A ∈ SM (X , Y ).
∞
Theorem 2. An operator A ∈ L c ( X ,Y ) is an
SM-operator if only if A∗ is an SM-operator,
that is, A ∈ SM (X ,Y ) if and only if
A∗ ∈ SM (Y ∗ , X ∗ ) , and
∞
A = 0 ⇔ ∑ Aen , d m∗ = 0 ⇔ A = O
m =1
(null operator),
(ii.b).For every scalar α and A∈ SM(X,Y),
we have
∞
A = ∑ ∑ Aen , d m∗
n =1 m =1
∞
∞
= ∑ ∑ en , A∗ d m∗
∞
α A = ∑ α Aen , d m∗
m =1
=α
m =1 n =1
= A ,
(i )
( ii )
A ≤ A
, for every A ∈ SM (X ,Y ) ,
SM (X ,Y ) is a Banach space with
respect to
( iii ) If
. ,
A ∈ SM (X ,Y ) , then A is compact.
Ae n , d m∗ = α
A ,
m =1
∗
for every OSB {en } of X and {d k } of
Y.
Theorem 3. Let {en } and {d n} be an
OSB of Banach space X and Y ,
respectively. Then,
∞
∑
and
(ii.c). For every A, B ∈ SM (X ,Y ) , we have
∞
∞
A + B = ∑∑
n =1 m =1
∞
( A + B) e , d
n
∗
m
∞
= ∑∑ Aen + Ben , d m∗
n =1 m =1
∞
∞
= ∑∑ Aen , d m∗ + Be n , d m∗
n =1 m =1
∞
∞
= ∑∑ Aen , d m∗ + Ben , d m∗
n =1 m =1
51
Berkala MIPA, 16(1), Januari 2006
∞
∞
∞
∞
∞
= ∑ ∑ Aen , d m∗ + ∑∑ Ben , d m∗
n =1 m =1
∞
∑ ∑ ( A − A) e , d
n =1 m =1
j=1
n
k =1
= A + B
∞
j=1
A+ B ≤ A + B .
The proof of the completeness of the
space is as follows. Let { An} ⊂ SM (X , Y ) be
an arbitrary Cauchy sequence. Then, for any
number ε > 0 , there is a positive integer n0
such that for every two positive integers
m , n ≥ n0 , we have
ε
Am − An < .
2
We want to prove that there is
A ∈ SM (X ,Y ) such that
lim
An − A = 0 .
n→
there is A ∈ L c ( X ,Y ) such that
ε
A − An < ,
2
for every n ≥ n0 or
k =1
∗
n
s
t
j=1
k= 1
≤∑∑
m
(A
n
j
k
dk
∈ SM (X ,Y ) and hence A = An +
in SM (X ,Y ) . Moreover,
− Am )e j , d k
for every
j=1
k =1
n ≥ n0 . Hence,
If A ∈ SM (X ,Y ) and x ∈ X , we have
(iii)
∞
Ax = ∑ Ax, d ∗k d k
k =1
and for every positive integer n , we define
an operator Bn : X → Y :
Bn x = ∑ Ax, d k∗ d k .
k =1
It is clear that Bn ∈ Lc ( X ,Y ), Bn is a finite
rank operator, and
lim A − Bn = 0 .
BA ≤ B
then BA ∈ SM ( X , Z ) and
For
∞
n =1 j =1
OSB
every
X , {d m } ⊂ Y and
∑∑
{en }
{ f } ⊂ Z , we have
A .
of
j
∞
BAen , f j∗ ≤ ∑ B Aen
f j∗
n =1
≤ B
∞
∞
∑∑
Aen , d ∗m
n =1 m =1
∑ ∑ ( A − A) e ,d
n
j
≤∑∑
s
j=1
t
k=1
(A
n
∗
k
dk
− A ) ej , dk∗
ε
2
for any integers s,t and m , n ≥ n0 . Letting
s → ∞ and t → ∞ , we have
≤
n
lim
An − A = 0 .
n→
∞
for any integers s,t and m , n ≥ n0 . By
letting m → ∞ , we have
t
(A− A )
An − A < ε ,
Proof:
∗
ε
2
s
Therefore
Theorem 4. Let X ,Y and Z be Banach
spaces. If A ∈ SM (X ,Y ) and B ∈ Lc (Y , Z ) ,
≤ An − Am
<
ε
2
An − A
− A) e j , d k∗ <
Therefore, A is a compact operator.
Thus, we have
j=1
n
n→∞
An − A = 0 .
lim
n →∞
∑ ∑ ( A − A )e , d
dk
k
n
Am − An ≤ Am − An ,
t
k =1
∗
(A
every n ≥ n0 .
for
Since Lc (X ,Y ) is complete and
s
∞
≤ ∑∑
or
52
j
= B A ,
that is BA ∈ SM ( X , Z ) and
BA ≤ B
A .
Let SM ( X ) and Lc ( X ) stand for
SM ( X , X ) and Lc (X , X ) , respectively.
Combining the results of Theorem 3 and
Theorem 4, we have proved that SM(X) is
∗ − algebra, where ∗ is an involution from
SM(X) into SM(X) satisfying :
Soeparna dkk, On The SM-operators
(A )
∗
∗
= A ; ( AB ) = B∗ A∗
∗
ACKNOWLEDGEMENTS
and
(α A + B )
∗
=α A + B
∗
∗
for every A, B ∈ SM ( X ) and a real scalar
α , as stated in the following theorem.
Theorem 5. Let X be a Banach space
having a shrinking OSB . Then, SM ( X ) is
a Banach ∗ − algebra and an ideal of
Lc ( X ).
CONCLUSION
Generalization
of
Hilbert-Schmidt
operators into Banach spaces can be done by
preserving the instrinsic properties of Hilbert
spaces, i.e., separable and reflexivity. The
results, denoted by SM(X,Y) has in general the
same properties of those of Hilbert-Schmidt
operators.
The biorthonormal system
∗
n
and
{({d }, {d } ) : {d
m
∗
m
m
Conway, J.B., 1990. A Course in Functional
Analysis, Springer Verlag, New York.
Dapa P.S., 2000. On strong M-bases in
Banach spaces with PRI, Collect. Math.
51, 3, 277-284.
Johnson W.B.,Roshental,H.P., and Zippin,M.,
1971. On bases, finite-dimensional
decompositions and weaker structures
in Banach spaces, Israel J. Math. 9,
488-506.
∗
} ⊂ Y , {d } ⊂ Y
Morrison, T.J.,2001. Functional Analysis: An
Introduction to Banach Space Theory,
John Wiley & Sons. Inc. New York.
∗
n
n
REFERENCES
Lindenstrauss J and Tzafriri L, 1996.
Classical Banach Spaces I and II,
Springer Verlag, New York.
{({e }, {e }) : {e } ⊂ X ,{e } ⊂ X }
n
The second author was partially
supported by BPPS DIKTI. We also thank all
the anonymous referees for reading the paper
carefully.
∗
m
∗
}
is the key to solve the condition of
orthonormality in Hilbert-Schmidt operators,
used later in SM (X ,Y ) . For further works,
we have been using the operator in classical
Banach spaces Lp and l p ,1 < p < ∞ .
Weidmann, J., 1980. Linear Operators in
Hilbert Spaces, Springer Verlag. New
York.
Zippin, M., 1968. A remark on bases and
reflexivity in Banach spaces, Israel J.
Math. 6, 74-79
53
Berkala MIPA, 16(1), Januari 2006
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