Constructing Banach ideals using upper
`p -estimates
Ben Wallis
Northern Illinois University
Dekalb, IL, USA
Tuesday, 2015 June 09
Relations between Banach space theory
and geometric measure theory
at the University of Warwick,
Coventry, UK
Ben Wallis
Upper `p -estimates
1/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ).
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal.
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal. A Banach ideal is any
operator ideal G equipped with an “ideal norm” ρ such that all
component spaces G(X , Y ) are complete with respect to the norm
ρ.
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal. A Banach ideal is any
operator ideal G equipped with an “ideal norm” ρ such that all
component spaces G(X , Y ) are complete with respect to the norm
ρ.
Let 1 ≤ ξ < ω1 be a countable ordinal and let Sξ denote the ξth
Schreier family.
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal. A Banach ideal is any
operator ideal G equipped with an “ideal norm” ρ such that all
component spaces G(X , Y ) are complete with respect to the norm
ρ.
Let 1 ≤ ξ < ω1 be a countable ordinal and let Sξ denote the ξth
Schreier family. The Schreier families contain only finite subsets
of N.
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal. A Banach ideal is any
operator ideal G equipped with an “ideal norm” ρ such that all
component spaces G(X , Y ) are complete with respect to the norm
ρ.
Let 1 ≤ ξ < ω1 be a countable ordinal and let Sξ denote the ξth
Schreier family. The Schreier families contain only finite subsets
of N. For example, S1 is the set of all A ⊆ N satisfying
#A ≤ min A. Note that this includes the empty set.
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal. A Banach ideal is any
operator ideal G equipped with an “ideal norm” ρ such that all
component spaces G(X , Y ) are complete with respect to the norm
ρ.
Let 1 ≤ ξ < ω1 be a countable ordinal and let Sξ denote the ξth
Schreier family. The Schreier families contain only finite subsets
of N. For example, S1 is the set of all A ⊆ N satisfying
#A ≤ min A. Note that this includes the empty set.
For convenience, denote by Sω1 the family of all finite subsets of N.
Ben Wallis
Upper `p -estimates
2/13
Banach ideals and Schreier families
A class G of continuous linear operators is said to have the ideal
property just in case BTA ∈ G(W , Z ) for all A ∈ L(W , X ),
B ∈ L(Y , Z ), and T ∈ G(X , Y ). If in addition G(X , Y ) is always a
linear space containing the finite-rank operators F(X , Y ), then we
say that class G is an operator ideal. A Banach ideal is any
operator ideal G equipped with an “ideal norm” ρ such that all
component spaces G(X , Y ) are complete with respect to the norm
ρ.
Let 1 ≤ ξ < ω1 be a countable ordinal and let Sξ denote the ξth
Schreier family. The Schreier families contain only finite subsets
of N. For example, S1 is the set of all A ⊆ N satisfying
#A ≤ min A. Note that this includes the empty set.
For convenience, denote by Sω1 the family of all finite subsets of N.
(This is nonstandard notation, but it greatly simplifies the writing.)
Ben Wallis
Upper `p -estimates
2/13
(∞,ξ)
The classes WD`p
Let us fix some 1 < p ≤ ∞ and 1 ≤ ξ ≤ ω1 .
Ben Wallis
Upper `p -estimates
3/13
(∞,ξ)
The classes WD`p
Let us fix some 1 < p ≤ ∞ and 1 ≤ ξ ≤ ω1 .
(∞,ξ)
We denote by WD`p (X , Y ) the set of all operators
T ∈ L(X , Y ) for which there exists a uniform constant C > 0 with
the following property: Each normalized weakly null sequence
(xn ) ⊂ X admits a subsequence (xnk ) such that for all (αk ) ∈ c00
with support in Sξ we get
k
X
αk Txnk k ≤ C k(αk )k`p .
Ben Wallis
Upper `p -estimates
3/13
Using Schreier families
An operator T : X → Y is called strictly singular (SS) just in
case for every normalized basic sequence
(xn ) ⊆ X P
and every > 0
P
there exists (αn ) ∈ c00 such that k αn Txn k < k αn xn k.
Ben Wallis
Upper `p -estimates
4/13
Using Schreier families
An operator T : X → Y is called strictly singular (SS) just in
case for every normalized basic sequence
(xn ) ⊆ X P
and every > 0
P
there exists (αn ) ∈ c00 such that k αn Txn k < k αn xn k. An
operator T : X → Y is called Sξ -strictly singular (SS ξ ) whenever
(αn ) can always be chosen to have support in Sξ .
Ben Wallis
Upper `p -estimates
4/13
Using Schreier families
An operator T : X → Y is called strictly singular (SS) just in
case for every normalized basic sequence
(xn ) ⊆ X P
and every > 0
P
there exists (αn ) ∈ c00 such that k αn Txn k < k αn xn k. An
operator T : X → Y is called Sξ -strictly singular (SS ξ ) whenever
(αn ) can always be chosen to have support in Sξ .
An operator T : X → Y is weakly compact (W) just in case for
every normalized basic P
sequence (xn ) ⊆PX and every > 0 there is
(αn ) ∈ c00 such that k αn Txn k < k αn sn k, where (sn ) is the
summing basis for c0 .
Ben Wallis
Upper `p -estimates
4/13
Using Schreier families
An operator T : X → Y is called strictly singular (SS) just in
case for every normalized basic sequence
(xn ) ⊆ X P
and every > 0
P
there exists (αn ) ∈ c00 such that k αn Txn k < k αn xn k. An
operator T : X → Y is called Sξ -strictly singular (SS ξ ) whenever
(αn ) can always be chosen to have support in Sξ .
An operator T : X → Y is weakly compact (W) just in case for
every normalized basic P
sequence (xn ) ⊆PX and every > 0 there is
(αn ) ∈ c00 such that k αn Txn k < k αn sn k, where (sn ) is the
summing basis for c0 . We say that T is Sξ -weakly compact
(Wξ ) whenever we can always choose (αn ) to have support in Sξ .
Ben Wallis
Upper `p -estimates
4/13
Using Schreier families
An operator T : X → Y is called strictly singular (SS) just in
case for every normalized basic sequence
(xn ) ⊆ X P
and every > 0
P
there exists (αn ) ∈ c00 such that k αn Txn k < k αn xn k. An
operator T : X → Y is called Sξ -strictly singular (SS ξ ) whenever
(αn ) can always be chosen to have support in Sξ .
An operator T : X → Y is weakly compact (W) just in case for
every normalized basic P
sequence (xn ) ⊆PX and every > 0 there is
(αn ) ∈ c00 such that k αn Txn k < k αn sn k, where (sn ) is the
summing basis for c0 . We say that T is Sξ -weakly compact
(Wξ ) whenever we can always choose (αn ) to have support in Sξ .
The Rosenthal (R) and Sξ -Rosenthal (Rξ ) operators can be
similarly defined using the canonical basis for `1 instead of the
summing basis for c0 .
Ben Wallis
Upper `p -estimates
4/13
Using Schreier families
Theorem (Odell-Teixeira, 2010)
There exists X such that SS 1 (X ) is not closed under addition. In
particular, SS 1 is not an operator ideal.
Ben Wallis
Upper `p -estimates
5/13
Using Schreier families
Theorem (Odell-Teixeira, 2010)
There exists X such that SS 1 (X ) is not closed under addition. In
particular, SS 1 is not an operator ideal.
Question.
Is Wξ an operator ideal?
Ben Wallis
Upper `p -estimates
5/13
Using Schreier families
Theorem (Odell-Teixeira, 2010)
There exists X such that SS 1 (X ) is not closed under addition. In
particular, SS 1 is not an operator ideal.
Question.
Is Wξ an operator ideal?
Proposition.
(∞,ξ)
WD`p
is an operator ideal.
Ben Wallis
Upper `p -estimates
5/13
Using Schreier families
Theorem (Odell-Teixeira, 2010)
There exists X such that SS 1 (X ) is not closed under addition. In
particular, SS 1 is not an operator ideal.
Question.
Is Wξ an operator ideal?
Proposition.
(∞,ξ)
WD`p
is an operator ideal.
Proposition (Causey-Freeman-anonymous)
\
(∞,ξ)
(∞,ω )
WD`p (X , Y ) = WD`p 1 (X , Y ).
1≤ξ<ω1
Ben Wallis
Upper `p -estimates
5/13
Operator ideal families with `p parameters
Several other families of operator ideals have been constructed
which have parameters related to the `p spaces. For example:
Ben Wallis
Upper `p -estimates
6/13
Operator ideal families with `p parameters
Several other families of operator ideals have been constructed
which have parameters related to the `p spaces. For example:
Np = the ideal of p-nuclear operators.
Ben Wallis
Upper `p -estimates
6/13
Operator ideal families with `p parameters
Several other families of operator ideals have been constructed
which have parameters related to the `p spaces. For example:
Np = the ideal of p-nuclear operators.
Ip = the ideal of p-integral operators.
Ben Wallis
Upper `p -estimates
6/13
Operator ideal families with `p parameters
Several other families of operator ideals have been constructed
which have parameters related to the `p spaces. For example:
Np = the ideal of p-nuclear operators.
Ip = the ideal of p-integral operators.
Πp = the ideal of p-summing operators.
Ben Wallis
Upper `p -estimates
6/13
Operator ideal families with `p parameters
Several other families of operator ideals have been constructed
which have parameters related to the `p spaces. For example:
Np = the ideal of p-nuclear operators.
Ip = the ideal of p-integral operators.
Πp = the ideal of p-summing operators.
V = the norm-closed ideal of completely continuous operators.
Ben Wallis
Upper `p -estimates
6/13
Operator ideal families with `p parameters
Several other families of operator ideals have been constructed
which have parameters related to the `p spaces. For example:
Np = the ideal of p-nuclear operators.
Ip = the ideal of p-integral operators.
Πp = the ideal of p-summing operators.
V = the norm-closed ideal of completely continuous operators.
The new family of ideals sits here:
(∞,ξ)
Np ( Ip ( Πp ( V ( WD`p
Ben Wallis
Upper `p -estimates
.
6/13
(∞,ξ)
A seminorm for the spaces WD`p
(∞,ξ)
For each T in class WD`p
(X , Y )
, we denote
C(p,ξ) (T ) := inf C ,
where the “inf” ranges over all possible (uniform) domination
(∞,ξ)
constants in the definition of WD`p .
Ben Wallis
Upper `p -estimates
7/13
(∞,ξ)
A seminorm for the spaces WD`p
(∞,ξ)
For each T in class WD`p
(X , Y )
, we denote
C(p,ξ) (T ) := inf C ,
where the “inf” ranges over all possible (uniform) domination
(∞,ξ)
constants in the definition of WD`p .
Proposition.
(∞,ξ)
C(p,ξ) is a seminorm on the space WD`p
Ben Wallis
(X , Y ).
Upper `p -estimates
7/13
(C ,ξ)
Classes WD`p
For each 0 ≤ C < ∞, define
n
o
(C ,ξ)
(∞,ξ)
WD`p (X , Y ) = T ∈ WD`p (X , Y ) : C(p,ξ) (T ) ≤ C .
Ben Wallis
Upper `p -estimates
8/13
(C ,ξ)
Classes WD`p
For each 0 ≤ C < ∞, define
n
o
(C ,ξ)
(∞,ξ)
WD`p (X , Y ) = T ∈ WD`p (X , Y ) : C(p,ξ) (T ) ≤ C .
It is evident that
(∞,ξ)
WD`p (X , Y )
=
(C ,ξ)
WD`p (X , Y )
[
C ≥0
Ben Wallis
=
∞
[
(n,ξ)
WD`p
(X , Y ).
n=1
Upper `p -estimates
8/13
(C ,ξ)
Classes WD`p
For each 0 ≤ C < ∞, define
n
o
(C ,ξ)
(∞,ξ)
WD`p (X , Y ) = T ∈ WD`p (X , Y ) : C(p,ξ) (T ) ≤ C .
It is evident that
(∞,ξ)
WD`p (X , Y )
=
(C ,ξ)
WD`p (X , Y )
[
C ≥0
=
∞
[
(n,ξ)
WD`p
(X , Y ).
n=1
We also have
(0,ξ)
WD`p
Ben Wallis
= V.
Upper `p -estimates
8/13
(C ,ξ)
Classes WD`p
For each 0 ≤ C < ∞, define
n
o
(C ,ξ)
(∞,ξ)
WD`p (X , Y ) = T ∈ WD`p (X , Y ) : C(p,ξ) (T ) ≤ C .
It is evident that
(∞,ξ)
WD`p (X , Y )
=
(C ,ξ)
WD`p (X , Y )
[
C ≥0
=
∞
[
(n,ξ)
WD`p
(X , Y ).
n=1
We also have
(0,ξ)
WD`p
= V.
(C ,ξ)
However, when C ∈ (0, ∞), the components WD`p
not form linear spaces.
Ben Wallis
Upper `p -estimates
(X , Y ) do
8/13
(∞,ξ)
WD`p
(∞,1)
fails to be norm-closed but WD`p
(C ,1)
In the special case ξ = 1, the sets WD`p
are always norm-closed in L(X , Y ).
Ben Wallis
is Fσ
(X , Y ), 0 ≤ C < ∞,
Upper `p -estimates
9/13
(∞,ξ)
WD`p
(∞,1)
fails to be norm-closed but WD`p
is Fσ
(C ,1)
In the special case ξ = 1, the sets WD`p (X , Y ), 0 ≤ C < ∞,
are always norm-closed in L(X , Y ). Consequently,
(∞,1)
WD`p (X , Y )
=
∞
[
(n,1)
WD`p
(X , Y )
n=1
is an Fσ -subset of L(X , Y ).
Ben Wallis
Upper `p -estimates
9/13
(∞,ξ)
WD`p
(∞,1)
fails to be norm-closed but WD`p
is Fσ
(C ,1)
In the special case ξ = 1, the sets WD`p (X , Y ), 0 ≤ C < ∞,
are always norm-closed in L(X , Y ). Consequently,
(∞,1)
WD`p (X , Y )
=
∞
[
(n,1)
WD`p
(X , Y )
n=1
is an Fσ -subset of L(X , Y ).
Proposition.
For any 1 < p ≤ ∞ and any 1 ≤ ξ ≤ ω1 , there exists a space X
(∞,ξ)
such that WD`p (X ) is not norm-closed.
Ben Wallis
Upper `p -estimates
9/13
(∞,1)
WD`p
is a Banach ideal
Define
kT k(p,ξ) := C(p,ξ) (T ) + kT kop .
Ben Wallis
Upper `p -estimates
10/13
(∞,1)
WD`p
is a Banach ideal
Define
kT k(p,ξ) := C(p,ξ) (T ) + kT kop .
(∞,ξ)
This is a norm on the space WD`p
a norm and a seminorm.
Ben Wallis
(X , Y ), since it is the sum of
Upper `p -estimates
10/13
(∞,1)
WD`p
is a Banach ideal
Define
kT k(p,ξ) := C(p,ξ) (T ) + kT kop .
(∞,ξ)
This is a norm on the space WD`p (X , Y ), since it is the sum of
a norm and a seminorm. In fact, it is easily seen to be an ideal
norm.
Ben Wallis
Upper `p -estimates
10/13
(∞,1)
WD`p
is a Banach ideal
Define
kT k(p,ξ) := C(p,ξ) (T ) + kT kop .
(∞,ξ)
This is a norm on the space WD`p (X , Y ), since it is the sum of
a norm and a seminorm. In fact, it is easily seen to be an ideal
norm.
Proposition.
(∞,1)
In the special case ξ = 1, the space WD`p
(X , Y ) is complete
(∞,1)
under the norm k·k(p,1) . In particular, class WD`p
Banach ideal.
Ben Wallis
Upper `p -estimates
forms a
10/13
(∞,1)
WD`p
is a Banach ideal
Define
kT k(p,ξ) := C(p,ξ) (T ) + kT kop .
(∞,ξ)
This is a norm on the space WD`p (X , Y ), since it is the sum of
a norm and a seminorm. In fact, it is easily seen to be an ideal
norm.
Proposition.
(∞,1)
In the special case ξ = 1, the space WD`p
(X , Y ) is complete
(∞,1)
under the norm k·k(p,1) . In particular, class WD`p
Banach ideal.
(∞,ξ)
We do not yet know if the classes WD`p
when ξ 6= 1.
Ben Wallis
forms a
form Banach ideals
Upper `p -estimates
10/13
Significance of p parameter
Proposition.
(∞,ξ)
When ξ is fixed, the norm-closed operator ideals WD`p
distinct as p ranges over 1 ≤ p ≤ ∞.
Ben Wallis
Upper `p -estimates
are all
11/13
Significance of p parameter
Proposition.
(∞,ξ)
When ξ is fixed, the norm-closed operator ideals WD`p
distinct as p ranges over 1 ≤ p ≤ ∞.
are all
In particular, for 1 < q < p < ∞,
(∞,ξ)
WD`p
(∞,ξ)
(`q ) = K(`q ) 6= L(`q ) = WD`q
Ben Wallis
Upper `p -estimates
(`q ).
11/13
Significance of ξ parameter
Let Tp0 = Tp0 [Sξ , 12 ] denote the p 0 -convexified Tsirelson space of
order ξ (where p1 + p10 = 1).
Ben Wallis
Upper `p -estimates
12/13
Significance of ξ parameter
Let Tp0 = Tp0 [Sξ , 12 ] denote the p 0 -convexified Tsirelson space of
order ξ (where p1 + p10 = 1).
Example
(∞,ξ)
WD`p
(∞,ω1 )
(Tp∗0 ) = L(Tp∗0 ) 6= WD`p
Ben Wallis
Upper `p -estimates
(Tp∗0 ).
12/13
Significance of ξ parameter
Let Tp0 = Tp0 [Sξ , 12 ] denote the p 0 -convexified Tsirelson space of
order ξ (where p1 + p10 = 1).
Example
(∞,ξ)
WD`p
Recall again that
T
(∞,ω1 )
(Tp∗0 ) = L(Tp∗0 ) 6= WD`p
(∞,ξ)
1≤ξ<ω1
WD`p
Ben Wallis
(Tp∗0 ).
(∞,ω1 )
(X , Y ) = WD`p
Upper `p -estimates
(X , Y ).
12/13
Significance of ξ parameter
Let Tp0 = Tp0 [Sξ , 12 ] denote the p 0 -convexified Tsirelson space of
order ξ (where p1 + p10 = 1).
Example
(∞,ξ)
WD`p
Recall again that
T
(∞,ω1 )
(Tp∗0 ) = L(Tp∗0 ) 6= WD`p
(∞,ξ)
1≤ξ<ω1
WD`p
(Tp∗0 ).
(∞,ω1 )
(X , Y ) = WD`p
(X , Y ).
Proposition.
There exists a strictly increasing sequence (ξn ) of countable
ordinals 1 ≤ ξn < ξn+1 < ω1 , n ∈ N, and a sequence (Xn ) of
Banach spaces, such that for all m, n ∈ N with m < n we have
(∞,ξn )
WD`p
(∞,ξm )
(Xm ) ( WD`p
Ben Wallis
(Xm )
Upper `p -estimates
12/13
That’s all folks!
Thank you for listening!
Ben Wallis
Upper `p -estimates
13/13