A-compact sets
A-compact operators
Approximation properties
Approximation properties determined by
A-compact sets
Pablo Turco
Facultad de Ciencias Exactas y Naturales, UBA
IMAS-CONICET
(Joint work with Silvia Lassalle)
WidaBA14
Buenos Aires, Argentina
July 22, 2014
A-compact sets
A-compact operators
Approximation properties
Definition
A Banach space E has the approximation property if and only if
IdE can be approximated by finite rank operators uniformly on
compact sets.
A-compact sets
A-compact operators
Approximation properties
Definition
A Banach space E has the approximation property if and only if
IdE can be approximated by finite rank operators uniformly on
compact sets.
Our goal
To study when IdE can be approximated by by finite rank
operators on ”some” compact sets.
A-compact sets
A-compact operators
Approximation properties
Definition
A Banach space E has the approximation property if and only if
IdE can be approximated by finite rank operators uniformly on
compact sets.
Our goal
To study when IdE can be approximated by by finite rank
operators on ”some” compact sets.
Precedents:
Reinov (1984)
Bourgain and Reinov (1985)
Willis (1992)
Sinha and Karn (2002)
A-compact sets
A-compact operators
Approximation properties
Our ”some” compact sets are the A-compact sets of Carl
and Stephani (1984)
Definition (Carl and Stephani)
Fix Banach operator ideal A. A subset K ⊂ E is relatively
A-compact if there exist a Banach space F , an operator
S ∈ A(F ; E) and a compact set L ⊂ F such that
K ⊂ S(L).
A-compact sets
A-compact operators
Approximation properties
Our ”some” compact sets are the A-compact sets of Carl
and Stephani (1984)
Definition (Carl and Stephani)
Fix Banach operator ideal A. A subset K ⊂ E is relatively
A-compact if there exist a Banach space F , an operator
S ∈ A(F ; E) and a compact set L ⊂ F such that
K ⊂ S(L).
Relatively compact sets are relatively F-compact.
Relatively p-compact sets of Sinha and Karn are relatively
N p -compact sets, 1 ≤ p < ∞.
A-compact sets
A-compact operators
We want to measure the ”size” of an A-compact set.
Definition
Let K ⊂ E be a relatively A-compact set, then
mA (K; E) = inf{kSkA : K ⊂ S(L)},
where S ∈ A(F ; E) and L ⊂ BF is a compact set.
Approximation properties
A-compact sets
A-compact operators
Properties of the mA -size
mA (K; E) = mA (co{K}; E)
mA (K1 + K2 ; E) ≤ mA (K1 ; E) + mA (K2 ; E)
supx∈K kxk ≤ mA (K; E)
Approximation properties
A-compact sets
A-compact operators
Properties of the mA -size
mA (K; E) = mA (co{K}; E)
mA (K1 + K2 ; E) ≤ mA (K1 ; E) + mA (K2 ; E)
supx∈K kxk ≤ mA (K; E)
mF (K; E) = supx∈K kxk,
if K is compact.
Approximation properties
A-compact sets
A-compact operators
Approximation properties
Properties of the mA -size
mA (K; E) = mA (co{K}; E)
mA (K1 + K2 ; E) ≤ mA (K1 ; E) + mA (K2 ; E)
supx∈K kxk ≤ mA (K; E)
mF (K; E) = supx∈K kxk,
if K is compact.
If exists a compact set in E wich is not A-compact, then
there exist A-compact sets Lm ⊂ E such that
lim supx∈Lm kxk = 0
m→∞
and
lim mA (Lm ; E) = ∞
m→∞
A-compact sets
A-compact operators
Approximation properties
The Banach operator ideal KA
Definition (Carl and Stephani)
An operator T ∈ L(E; F ) is A-compact if T (BE ) is relatively
A-compact in F .
A-compact sets
A-compact operators
Approximation properties
The Banach operator ideal KA
Definition (Carl and Stephani)
An operator T ∈ L(E; F ) is A-compact if T (BE ) is relatively
A-compact in F . And
kT kKA = mA (T (BE ); F ).
A-compact sets
A-compact operators
Approximation properties
The Banach operator ideal KA
Definition (Carl and Stephani)
An operator T ∈ L(E; F ) is A-compact if T (BE ) is relatively
A-compact in F . And
kT kKA = mA (T (BE ); F ).
Proposition
KA = (A ◦ F)sur = Asur ◦ K
KKA = KA
A-compact sets
A-compact operators
Approximation properties
The Banach operator ideal KA
Definition (Carl and Stephani)
An operator T ∈ L(E; F ) is A-compact if T (BE ) is relatively
A-compact in F . And
kT kKA = mA (T (BE ); F ).
Proposition
KA = (A ◦ F)sur = Asur ◦ K isometrically.
KKA = KA isometrically.
A-compact sets
A-compact operators
Approximation properties
A very useful lemma
Fix ε > 0 and let K ⊂ E be an absoluelty convex and compact
set, then there exist a Banach space F , a compact set L ⊂ BF
and a compact operator T ∈ K(F ; E) such that K ⊂ T (L)
and
supx∈K kxk ≤ kT k ≤ supx∈K kxk + ε.
A-compact sets
A-compact operators
Approximation properties
A very useful lemma
Fix ε > 0 and let K ⊂ E be an absoluelty convex and A-compact
set, then there exist a Banach space F , a compact set L ⊂ BF
and an A-compact operator T ∈ KA (F ; E) such that K ⊂ T (L)
and
mA (K; E) ≤ kT kKA ≤ mA (K; E) + ε.
A-compact sets
A-compact operators
Approximation properties
A very useful lemma
Fix ε > 0 and let K ⊂ E be an absoluelty convex and A-compact
set, then there exist a Banach space F , a compact set L ⊂ BF
and an A-compact operator T ∈ KA (F ; E) such that K ⊂ T (L)
and
mA (K; E) ≤ kT kKA ≤ mA (K; E) + ε.
Proposition
KA = (A ◦ F)sur = Asur ◦ K
KKA = KA
A-compact sets
A-compact operators
Approximation properties
A very useful lemma
Fix ε > 0 and let K ⊂ E be an absoluelty convex and A-compact
set, then there exist a Banach space F , a compact set L ⊂ BF
and an A-compact operator T ∈ KA (F ; E) such that K ⊂ T (L)
and
mA (K; E) ≤ kT kKA ≤ mA (K; E) + ε.
Proposition
KA = (A ◦ F)sur = Asur ◦ K
KKA = KA
Consequence
A-compact sets = Asur -compact sets= KA -compact sets
A-compact sets
A-compact operators
Approximation properties
A very useful lemma
Fix ε > 0 and let K ⊂ E be an absoluelty convex and A-compact
set, then there exist a Banach space F , a compact set L ⊂ BF
and an A-compact operator T ∈ KA (F ; E) such that K ⊂ T (L)
and
mA (K; E) ≤ kT kKA ≤ mA (K; E) + ε.
Proposition
KA = (A ◦ F)sur = Asur ◦ K isometrically.
KKA = KA isometrically.
Consequence
A-compact sets = Asur -compact sets= KA -compact sets and
mA = mAsur = mKA .
A-compact sets
A-compact operators
Approximation properties
Shared properties between A-compact sets and KA
Given Banach spaces E ⊂ F and K ⊂ E. If K is A-compact in E,
then is A compact in F and mA (K; F ) ≤ mA (K; E).
A-compact sets
A-compact operators
Approximation properties
Shared properties between A-compact sets and KA
Given Banach spaces E ⊂ F and K ⊂ E. If K is A-compact in E,
then is A compact in F and mA (K; F ) ≤ mA (K; E).
Does the converse holds?
A-compact sets
A-compact operators
Approximation properties
Shared properties between A-compact sets and KA
Given Banach spaces E ⊂ F and K ⊂ E. If K is A-compact in E,
then is A compact in F and mA (K; F ) ≤ mA (K; E).
Does the converse holds?
Proposition
The following are equivalent
The Banach operator ideal KA is injective.
Let K ⊂ E. If for every Banach space F such that E ⊂ F , K
is A-compact in F , then K is A-compact in E and
mA (K; E) = mA (K; F ).
A-compact sets
A-compact operators
Proposition (Galicer, Lassalle, T.)
The ideal KN p (1 ≤ p < ∞) is not injective.
Approximation properties
A-compact sets
A-compact operators
Approximation properties
Proposition (Galicer, Lassalle, T.)
The ideal KN p (1 ≤ p < ∞) is not injective.
There exist Banach spaces E ⊂ F and a set K ⊂ E such that
K is N p -compact in F but is not in E.
A-compact sets
A-compact operators
Approximation properties
Proposition (Galicer, Lassalle, T.)
The ideal KN p (1 ≤ p < ∞) is not injective.
There exist Banach spaces E ⊂ F and a set K ⊂ E such that
K is N p -compact in F but is not in E.
Consequence
There exist Banach spaces E ⊂ F and N p -compact sets Lm ⊂ E
such that
lim mN p (Lm ; F ) = 0
m→∞
and
lim mN p (Lm ; E) = ∞
m→∞
A-compact sets
A-compact operators
Approximation properties
Proposition (Galicer, Lassalle, T.)
The ideal KN p (1 ≤ p < ∞) is not injective.
There exist Banach spaces E ⊂ F and a set K ⊂ E such that
K is N p -compact in F but is not in E.
Consequence
There exist Banach spaces E ⊂ F and N p -compact sets Lm ⊂ E
such that
lim mN p (Lm ; F ) = 0
m→∞
and
lim mN p (Lm ; E) = ∞
m→∞
What happens if F = E 00 ?
A-compact sets
A-compact operators
Approximation properties
Shared properties between A-compact sets and KA
Proposition
The following are equivalent
The Banach operator ideal KA is regular.
K ⊂ E is A-compact in E 00 , then K ⊂ E is A-compact and
mA (K; E) = mA (K; E 00 ).
A-compact sets
A-compact operators
Approximation properties
Shared properties between A-compact sets and KA
Proposition
The following are equivalent
The Banach operator ideal KA is regular.
K ⊂ E is A-compact in E 00 , then K ⊂ E is A-compact and
mA (K; E) = mA (K; E 00 ).
Proposition
The Banach operator ideal KA is regular for any Banach
operator ideal A.
A-compact sets
A-compact operators
Approximation properties
Shared properties between A-compact sets and KA
Proposition
The following are equivalent
The Banach operator ideal KA is regular.
K ⊂ E is A-compact in E 00 , then K ⊂ E is A-compact and
mA (K; E) = mA (K; E 00 ).
Proposition
The Banach operator ideal KA is regular for any Banach
operator ideal A.
K ⊂ E is A-compact in E if and only if is A-compact in E 00
and mA (K; E) = mA (K; E 00 ).
A-compact sets
A-compact operators
Approximation properties
On L(E; F ) we consider two topologies.
Approximation properties
A-compact sets
A-compact operators
Approximation properties
Approximation properties
On L(E; F ) we consider two topologies.
τA : the topology of uniform convergence on A-compact sets
A-compact sets
A-compact operators
Approximation properties
Approximation properties
On L(E; F ) we consider two topologies.
τA : the topology of uniform convergence on A-compact sets,
determined by the seminorms
qK (T ) = supx∈K kT xk
where K ranges over all the absolutely convex and A-compact sets.
A-compact sets
A-compact operators
Approximation properties
Approximation properties
On L(E; F ) we consider two topologies.
τA : the topology of uniform convergence on A-compact sets,
determined by the seminorms
qK (T ) = supx∈K kT xk
where K ranges over all the absolutely convex and A-compact sets.
τsA : determined by the seminorms
qK (T ) = mA (T (K); F )
where K ranges over all the absolutely convex and A-compact sets.
A-compact sets
A-compact operators
Approximation properties
Approximation properties
On L(E; F ) we consider two topologies.
τA : the topology of uniform convergence on A-compact sets,
determined by the seminorms
qK (T ) = mF (T (K); F )
where K ranges over all the absolutely convex and A-compact sets.
τsA : determined by the seminorms
qK (T ) = mA (T (K); F )
where K ranges over all the absolutely convex and A-compact sets.
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
τA
IdE ∈ F(E; E) .
F(F ; E) are τA -dense in L(F ; E) for any Banach space F .
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
τA
IdE ∈ F(E; E) .
F(F ; E) are τA -dense in L(F ; E) for any Banach space F .
F(F ; E) is k · k-dense in KA (F ; E) for any Banach space F .
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
τA
IdE ∈ F(E; E) .
F(F ; E) are τA -dense in L(F ; E) for any Banach space F .
F(F ; E) is k · k-dense in KA (F ; E) for any Banach space F .
Definition
A Banach space E with any of the above properties is said to have
the KA -uniform approximation property (KA -uAP).
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
IdE ∈ F(E; E)
τsA
.
F(F ; E) are τsA -dense in L(F ; E) for any Banach space F .
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
IdE ∈ F(E; E)
τsA
.
F(F ; E) are τsA -dense in L(F ; E) for any Banach space F .
F(F ; E) is k · kKA -dense in KA (F ; E) for any Banach
space F .
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
IdE ∈ F(E; E)
τsA
.
F(F ; E) are τsA -dense in L(F ; E) for any Banach space F .
F(F ; E) is k · kKA -dense in KA (F ; E) for any Banach
space F .
Definition
A Banach space E with any of the above properties is said to have
the KA -approximation property (KA -AP).
A-compact sets
A-compact operators
Approximation properties
Proposition
Let E be a Banach space. The following are equivalent.
IdE ∈ F(E; E)
τsA
.
F(F ; E) are τsA -dense in L(F ; E) for any Banach space F .
F(F ; E) is k · kKA -dense in KA (F ; E) for any Banach
space F .
Definition
A Banach space E with any of the above properties is said to have
the KA -approximation property (KA -AP).
The above definition fits the definition of Oja (2012).
A-compact sets
A-compact operators
Approximation properties
Relations between the AP’s
Examples
F-AP=F-uAP = Classical approximation property.
KN p -uAP = p-approximation property of Sinha and Karn.
KN p -AP = κp -approximation property of Delgado, Piñeiro
and Serrano.
A-compact sets
A-compact operators
Approximation properties
Relations between the AP’s
Examples
F-AP=F-uAP = Classical approximation property.
KN p -uAP = p-approximation property of Sinha and Karn.
KN p -AP = κp -approximation property of Delgado, Piñeiro
and Serrano.
KA -AP ⇒ KA -uAP.
A-compact sets
A-compact operators
Approximation properties
Relations between the AP’s
Examples
F-AP=F-uAP = Classical approximation property.
KN p -uAP = p-approximation property of Sinha and Karn.
KN p -AP = κp -approximation property of Delgado, Piñeiro
and Serrano.
KA -AP ⇒ KA -uAP.
If A ⊂ B, then KB -uAP ⇒ KA -uAP.
A-compact sets
A-compact operators
Approximation properties
Relations between the AP’s
Examples
F-AP=F-uAP = Classical approximation property.
KN p -uAP = p-approximation property of Sinha and Karn.
KN p -AP = κp -approximation property of Delgado, Piñeiro
and Serrano.
:
KA -AP ⇒ KA -uAP.
:
If A ⊂ B, then KB -uAP ⇒KA -uAP.
Every space has the KN p -uPA for 1 ≤ p ≤ 2 (Sinha and Karn).
Every space has the KN 2 -PA (Delgado, Piñeiro and Serrano).
For every p 6= 2 there exists a Banach space without KN p -PA
(Delgado, Piñeiro and Serrano).
A-compact sets
A-compact operators
Approximation properties
Relations between the AP’s
Examples
F-AP=F-uAP = Classical approximation property.
KN p -uAP = p-approximation property of Sinha and Karn.
KN p -AP = κp -approximation property of Delgado, Piñeiro
and Serrano.
:
KA -AP ⇒ KA -uAP.
:
If A ⊂ B, then KB -uAP ⇒KA -uAP.
If A ⊂ B, then KB -AP ; KA -AP.
Every space has the KN p -uPA for 1 ≤ p ≤ 2 (Sinha and Karn).
Every space has the KN 2 -PA (Delgado, Piñeiro and Serrano).
For every p 6= 2 there exists a Banach space without KN p -PA
(Delgado, Piñeiro and Serrano).
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -uniform approximation
property
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -uniform approximation
property
Proposition
If E has the approximation property then E has the KA -uniform
approximation property for every Banach operator ideal A.
Proof:
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -uniform approximation
property
Proposition
If E has the approximation property then E has the KA -uniform
approximation property for every Banach operator ideal A.
Proof:
IdE can be approximated by finite rank operators uniformly on all
compact sets then
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -uniform approximation
property
Proposition
If E has the approximation property then E has the KA -uniform
approximation property for every Banach operator ideal A.
Proof:
IdE can be approximated by finite rank operators uniformly on all
compact sets then IdE can be approximated by finite rank
operators uniformly on some compact sets.
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -approximation property
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -approximation property
Definition
A Banach space E has the λ-bounded approximation property if
and only if IdE can be approximated by finite rank operators with
norm ≤ λ uniformly on compact sets.
A-compact sets
A-compact operators
Approximation properties
Approximation property Vs. KA -approximation property
Definition
A Banach space E has the λ-bounded approximation property if
and only if IdE can be approximated by finite rank operators with
norm ≤ λ uniformly on compact sets.
Proposition
If E has the λ-bounded approximation property, then E has the
KA -approximation property for every Banach operator ideal A.
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
Let K ⊂ E be A-compact and ε > 0. Then, there exist a finite
dimensional subspace W ⊂ E, a bounded set L1 ⊂ W and L2 ⊂ E
an A-compact set with mA (L2 ; E) ≤ ε such that K ⊂ L1 + L2 .
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
Let K ⊂ E be A-compact and ε > 0. Then, there exist a finite
dimensional subspace W ⊂ E, a bounded set L1 ⊂ W and L2 ⊂ E
an A-compact set with mA (L2 ; E) ≤ ε such that K ⊂ L1 + L2 .
Since E has the λ-BAP and dim W < ∞ then there exits
R ∈ F(E; E) with kRk < 2λ and Rx = x ∀x ∈ W .
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
Let K ⊂ E be A-compact and ε > 0. Then, there exist a finite
dimensional subspace W ⊂ E, a bounded set L1 ⊂ W and L2 ⊂ E
an A-compact set with mA (L2 ; E) ≤ ε such that K ⊂ L1 + L2 .
Since E has the λ-BAP and dim W < ∞ then there exits
R ∈ F(E; E) with kRk < 2λ and Rx = x ∀x ∈ W .
qK (R − IdE ) = mA ((R − IdE )(K); E) ≤
mA ((R − IdE )(L1 ); E) + mA ((R − IdE )(L2 ); E)
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
Let K ⊂ E be A-compact and ε > 0. Then, there exist a finite
dimensional subspace W ⊂ E, a bounded set L1 ⊂ W and L2 ⊂ E
an A-compact set with mA (L2 ; E) ≤ ε such that K ⊂ L1 + L2 .
Since E has the λ-BAP and dim W < ∞ then there exits
R ∈ F(E; E) with kRk < 2λ and Rx = x ∀x ∈ W .
qK (R − IdE ) = mA ((R − IdE )(K); E) ≤
m ((R − IdE )(L1 ); E) +mA ((R − IdE )(L2 ); E)
|A
{z
}
=0
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
Let K ⊂ E be A-compact and ε > 0. Then, there exist a finite
dimensional subspace W ⊂ E, a bounded set L1 ⊂ W and L2 ⊂ E
an A-compact set with mA (L2 ; E) ≤ ε such that K ⊂ L1 + L2 .
Since E has the λ-BAP and dim W < ∞ then there exits
R ∈ F(E; E) with kRk < 2λ and Rx = x ∀x ∈ W .
qK (R − IdE ) = mA ((R − IdE )(K); E) ≤
m ((R − IdE )(L1 ); E) + mA ((R − IdE )(L2 ); E)
|A
{z
} |
{z
}
=0
≤(2λ+1)ε
A-compact sets
A-compact operators
Approximation properties
τsA
Sketch of the proof: Let see that IdE ∈ F(E; E)
.
Let K ⊂ E be A-compact and ε > 0. Then, there exist a finite
dimensional subspace W ⊂ E, a bounded set L1 ⊂ W and L2 ⊂ E
an A-compact set with mA (L2 ; E) ≤ ε such that K ⊂ L1 + L2 .
Since E has the λ-BAP and dim W < ∞ then there exits
R ∈ F(E; E) with kRk < 2λ and Rx = x ∀x ∈ W .
qK (R − IdE ) = mA ((R − IdE )(K); E) ≤
m ((R − IdE )(L1 ); E) + mA ((R − IdE )(L2 ); E)≤ (2λ + 1)ε.
|A
{z
} |
{z
}
=0
≤(2λ+1)ε
A-compact sets
A-compact operators
What can we say about the classic-AP and the KA -AP?
Approximation properties
A-compact sets
A-compact operators
What can we say about the classic-AP and the KA -AP?
Amin = F ◦ A ◦ F.
Approximation properties
A-compact sets
A-compact operators
Approximation properties
What can we say about the classic-AP and the KA -AP?
Definition
A Banach operator ideal A is right accessible if Amin = A ◦ F
isometrically.
A-compact sets
A-compact operators
Approximation properties
What can we say about the classic-AP and the KA -AP?
Definition
A Banach operator ideal A is right accessible if Amin = A ◦ F
isometrically.
Proposition
Let A be a right accessible Banach operator ideal
KA = (Amin )sur .
A-compact sets
A-compact operators
Approximation properties
What can we say about the classic-AP and the KA -AP?
Definition
A Banach operator ideal A is right accessible if Amin = A ◦ F
isometrically.
Proposition
Let A be a right accessible Banach operator ideal
KA = (Amin )sur .
KA is right accessible.
A-compact sets
A-compact operators
Approximation properties
What can we say about the classic-AP and the KA -AP?
Definition
A Banach operator ideal A is right accessible if Amin = A ◦ F
isometrically.
Proposition
Let A be a right accessible Banach operator ideal
KA = (Amin )sur .
KA is right accessible.
k · kKA = k · kKmin on finite rank operators.
A
A-compact sets
A-compact operators
Approximation properties
Proposition
Let A be a right accessible Banach operator ideal. If E has the
approximation property, then E has the KA -approximation
property.
A-compact sets
A-compact operators
Approximation properties
Proposition
Let A be a right accessible Banach operator ideal. If E has the
approximation property, then E has the KA -approximation
property.
Proof: Let see that F(F ; E) is k.kKA -dense in KA (F ; E)
KA (F ; E)
A-compact sets
A-compact operators
Approximation properties
Proposition
Let A be a right accessible Banach operator ideal. If E has the
approximation property, then E has the KA -approximation
property.
Proof: Let see that F(F ; E) is k.kKA -dense in KA (F ; E)
min (F ; E)
KA (F ; E) = KA
A-compact sets
A-compact operators
Approximation properties
Proposition
Let A be a right accessible Banach operator ideal. If E has the
approximation property, then E has the KA -approximation
property.
Proof: Let see that F(F ; E) is k.kKA -dense in KA (F ; E)
k·kKmin
min (F ; E) = F(F ; E)
KA (F ; E) = KA
A
A-compact sets
A-compact operators
Approximation properties
Proposition
Let A be a right accessible Banach operator ideal. If E has the
approximation property, then E has the KA -approximation
property.
Proof: Let see that F(F ; E) is k.kKA -dense in KA (F ; E)
k·kKmin
min (F ; E) = F(F ; E)
KA (F ; E) = KA
A
k·kKA
= F(F ; E)
.
A-compact sets
A-compact operators
Note that
k·kKmin
min
KA
(F ; E) = F(F ; E)
A
Approximation properties
A-compact sets
A-compact operators
Approximation properties
Note that if A is right accessible
k·kKmin
min
KA
(F ; E) = F(F ; E)
A
k·kKA
= F(F ; E)
.
A-compact sets
A-compact operators
Approximation properties
Note that if A is right accessible
k·kKmin
min
KA
(F ; E) = F(F ; E)
A
k·kKA
= F(F ; E)
.
Proposition
Let A be a right accessible Banach operator ideal. E has the
KA -approximation property if and only if
min
KA (F ; E) = KA
(F ; E)
for all Banach space F .
isometrically,
A-compact sets
A-compact operators
For a Banach operator ideal A.
BAP
/ AP
/ KA − uAP
KA − AP
Approximation properties
A-compact sets
A-compact operators
For a Banach operator ideal A.
Approximation properties
When A is right accessible.
/ AP
BAP
/ AP
BAP
/ KA − uAP
KA − AP
KA − AP
w
/ KA − uAP
E has the KA -AP, iff
min (F ; E) isometrically
KA (F ; E) = KA
for all Banach space F .
A-compact sets
A-compact operators
For a Banach operator ideal A.
Approximation properties
When A is right accessible.
/ AP
BAP
/ AP
BAP
/ KA − uAP
KA − AP
KA − AP
w
/ KA − uAP
E has the KA -AP, iff
min (F ; E) isometrically
KA (F ; E) = KA
for all Banach space F .
A-compact sets
A-compact operators
For a Banach operator ideal A.
BAP
Approximation properties
When A is right accessible.
/ AP
/ AP
BAP
/ KA − uAP
KA − AP
???
w
KA − AP
w
/ KA − uAP
E has the KA -AP, iff
min (F ; E) isometrically
KA (F ; E) = KA
for all Banach space F .
Question
Does the approximation property imply the KA -approximation
property for any Banach operator ideal A?
A-compact sets
A-compact operators
For a Banach operator ideal A.
Approximation properties
When A is right accessible.
/ AP
/ AP
BAP
/ KA − uAP
KA − AP
BAP
???
w
KA − AP
???
w
/ KA − uAP
E has the KA -AP, iff
min (F ; E) isometrically
KA (F ; E) = KA
for all Banach space F .
Question
Does the approximation property imply the KA -approximation
property for any Banach operator ideal A?
A-compact sets
A-compact operators
For a Banach operator ideal A.
Approximation properties
When A is right accessible.
/ AP
/ AP
BAP
/ KA − uAP
KA − AP
BAP
???
w
KA − AP
???
w
/ KA − uAP
E has the KA -AP, iff
min (F ; E) isometrically
KA (F ; E) = KA
for all Banach space F .
Question
Does the approximation property imply the KA -approximation
property for any Banach operator ideal A?
We don’t know!!!
A-compact sets
A-compact operators
Approximation properties
References
J. M. Delgado, E. Oja, C. Piñeiro, E. Serrano. The
p-approximation property in terms of density of finite rank
operators, J. Math. Anal. Appl. 354 (2009), 159–164.
J. M. Delgado, C. Piñeiro, E. Serrano. Density of finite rank
operators in the Banach space of p-compact operators, J.
Math. Anal. Appl. 370 (2010), 498–505.
S. Lassalle, P.T. The Banach ideal of A-compact operators
and related approximation properties, J. Funct. Anal. 265
(2013), 2452–2464
E. Oja, A remark on the approximation of p -compact
operators by finite-rank operators J. Math. Anal. Appl. 387
(2) (2012), 949–952
D. P. Sinha, A. K. Karn. Compact operators whose adjoints
factor through subspaces of `p , Studia Math. 150 (2002),
17–33.
A-compact sets
A-compact operators
THANK YOU!
¡GRACIAS!
Approximation properties