Random unconditionality for bases in Banach
spaces
Pedro Tradacete (UC3M)
Joint work with J. López-Abad (CSIC)
Relations Between Banach Space Theory and
Geometric Measure Theory
08 - 12 June 2015, Mathematics Institute, University of Warwick
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
1 / 13
Outline
1
Introduction: RUC and RUD bases
2
Basic examples
3
Duality, reflexivity and uniqueness
4
Relation with unconditionality
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
2 / 13
A basis (xn )n∈N of a Banach
Pspace X is unconditional provided for
every x ∈ X its expansion n∈N an xn converges unconditionally.
TFAE:
(xn ) is an unconditional basis.
For every A ⊂ N,
X
X
an xn converges ⇒
an xn converges.
n∈N
n∈A
For every choice of signs (n )n∈N ,
X
X
an xn converges ⇒
n an xn converges.
n∈N
n∈N
There is C > 0 such that for any scalars and signs
k
m
X
n an xn k ≤ Ck
n=1
P. Tradacete (UC3M)
m
X
an xn k.
n=1
RUC/RUD bases
Warwick 2015
3 / 13
A basis (xn )n∈N of a Banach
Pspace X is unconditional provided for
every x ∈ X its expansion n∈N an xn converges unconditionally.
TFAE:
(xn ) is an unconditional basis.
For every A ⊂ N,
X
X
an xn converges ⇒
an xn converges.
n∈N
n∈A
For every choice of signs (n )n∈N ,
X
X
an xn converges ⇒
n an xn converges.
n∈N
n∈N
There is C > 0 such that for any scalars and signs
k
m
X
n an xn k ≤ Ck
n=1
P. Tradacete (UC3M)
m
X
an xn k.
n=1
RUC/RUD bases
Warwick 2015
3 / 13
Definition (Billard-Kwapien-Pelczynski-Samuel, 1985)
A basis (xn )n∈N is of Random unconditional convergence (RUC) if
X
X
an xn converges ⇒
n an xn converges a.s.
n∈N
n∈N
(xn )n∈N is an RUC-basis iff there is K ≥ 1 such that
m
m
m
X
X
X
X
1
n an xn = m
E n an xn ≤ K an xn .
2
m
n=1
n=1
(n )∈{−1,+1}
n=1
Definition
A basis (xn )n∈N is of Random unconditional divergence (RUD) when
X
X
an xn diverges ⇒
n an xn diverges a.s.
n∈N
n∈N
(xn )n∈N is an RUD-basis iff there is K ≥ 1 such that
m
m
X
X
an xn ≤ K E n an xn .
n=1
P. Tradacete (UC3M)
n=1
RUC/RUD bases
Warwick 2015
4 / 13
Definition (Billard-Kwapien-Pelczynski-Samuel, 1985)
A basis (xn )n∈N is of Random unconditional convergence (RUC) if
X
X
an xn converges ⇒
n an xn converges a.s.
n∈N
n∈N
(xn )n∈N is an RUC-basis iff there is K ≥ 1 such that
m
m
m
X
X
X
X
1
n an xn = m
E n an xn ≤ K an xn .
2
m
n=1
n=1
(n )∈{−1,+1}
n=1
Definition
A basis (xn )n∈N is of Random unconditional divergence (RUD) when
X
X
an xn diverges ⇒
n an xn diverges a.s.
n∈N
n∈N
(xn )n∈N is an RUD-basis iff there is K ≥ 1 such that
m
m
X
X
an xn ≤ K E n an xn .
n=1
P. Tradacete (UC3M)
n=1
RUC/RUD bases
Warwick 2015
4 / 13
Definition (Billard-Kwapien-Pelczynski-Samuel, 1985)
A basis (xn )n∈N is of Random unconditional convergence (RUC) if
X
X
an xn converges ⇒
n an xn converges a.s.
n∈N
n∈N
(xn )n∈N is an RUC-basis iff there is K ≥ 1 such that
m
m
m
X
X
X
X
1
n an xn = m
E n an xn ≤ K an xn .
2
m
n=1
n=1
(n )∈{−1,+1}
n=1
Definition
A basis (xn )n∈N is of Random unconditional divergence (RUD) when
X
X
an xn diverges ⇒
n an xn diverges a.s.
n∈N
n∈N
(xn )n∈N is an RUD-basis iff there is K ≥ 1 such that
m
m
X
X
an xn ≤ K E n an xn .
n=1
P. Tradacete (UC3M)
n=1
RUC/RUD bases
Warwick 2015
4 / 13
Definition (Billard-Kwapien-Pelczynski-Samuel, 1985)
A basis (xn )n∈N is of Random unconditional convergence (RUC) if
X
X
an xn converges ⇒
n an xn converges a.s.
n∈N
n∈N
(xn )n∈N is an RUC-basis iff there is K ≥ 1 such that
m
m
m
X
X
X
X
1
n an xn = m
E n an xn ≤ K an xn .
2
m
n=1
n=1
(n )∈{−1,+1}
n=1
Definition
A basis (xn )n∈N is of Random unconditional divergence (RUD) when
X
X
an xn diverges ⇒
n an xn diverges a.s.
n∈N
n∈N
(xn )n∈N is an RUD-basis iff there is K ≥ 1 such that
m
m
X
X
an xn ≤ K E n an xn .
n=1
P. Tradacete (UC3M)
n=1
RUC/RUD bases
Warwick 2015
4 / 13
Questions
What properties of an unconditional basis work for RUC/RUD
bases?
Does every RUC/RUD basis have an unconditional subsequence
(resp. blocks)?
Is every block of an RUC/RUD basis, also RUC/RUD?
Can reflexivity be characterized somehow? (in the spirit of James
theorem)
What are these bases good for?
...
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
5 / 13
Questions
What properties of an unconditional basis work for RUC/RUD
bases?
Does every RUC/RUD basis have an unconditional subsequence
(resp. blocks)?
Is every block of an RUC/RUD basis, also RUC/RUD?
Can reflexivity be characterized somehow? (in the spirit of James
theorem)
What are these bases good for?
...
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
5 / 13
Questions
What properties of an unconditional basis work for RUC/RUD
bases?
Does every RUC/RUD basis have an unconditional subsequence
(resp. blocks)?
Is every block of an RUC/RUD basis, also RUC/RUD?
Can reflexivity be characterized somehow? (in the spirit of James
theorem)
What are these bases good for?
...
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
5 / 13
Questions
What properties of an unconditional basis work for RUC/RUD
bases?
Does every RUC/RUD basis have an unconditional subsequence
(resp. blocks)?
Is every block of an RUC/RUD basis, also RUC/RUD?
Can reflexivity be characterized somehow? (in the spirit of James
theorem)
What are these bases good for?
...
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
5 / 13
Questions
What properties of an unconditional basis work for RUC/RUD
bases?
Does every RUC/RUD basis have an unconditional subsequence
(resp. blocks)?
Is every block of an RUC/RUD basis, also RUC/RUD?
Can reflexivity be characterized somehow? (in the spirit of James
theorem)
What are these bases good for?
...
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
5 / 13
Example
The summing basis (sn ) in c0 does not have any RUC nor RUD
subsequence
n
m
X
X
aj .
an sn = sup 1≤n≤m
n=1
j=1
In particular,
m
Z
X
E n an sn =
n=1
1
n
m
X
X
1
2
sup aj rj (t)dt ≈
aj2 ,
0 1≤n≤m
j=1
j=1
(by Levy’s and Khintchine’s inequalities).
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
6 / 13
Example
The summing basis (sn ) in c0 does not have any RUC nor RUD
subsequence
n
m
X
X
aj .
an sn = sup 1≤n≤m
n=1
j=1
In particular,
m
Z
X
E n an sn =
n=1
1
n
m
X
1
X
2
sup aj rj (t)dt ≈
aj2 ,
0 1≤n≤m
j=1
j=1
(by Levy’s and Khintchine’s inequalities).
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
6 / 13
Example
The summing basis (sn ) in c0 does not have any RUC nor RUD
subsequence
n
m
X
X
aj .
an sn = sup 1≤n≤m
n=1
j=1
In particular,
m
Z
X
E n an sn =
n=1
1
n
m
X
1
X
2
sup aj rj (t)dt ≈
aj2 ,
0 1≤n≤m
j=1
j=1
(by Levy’s and Khintchine’s inequalities).
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
6 / 13
Example
The unit basis (un ) of James space is RUD.
k
X
an un kJ = sup
X
m
n∈N
(apk − apk +1 )
2
1
2
: p1 < p2 < . . . < pm+1 .
k=1
It holds that
k
m
X
ai ui kJ ≤
√
m
X
2 E i ai ui .
i=1
i=1
J
Example
The Haar basis in L1 [0, 1] is an RUD basis.
Recall, L1 [0, 1] has no unconditional basis. Actually, L1 [0, 1] does not
embed in a space with unconditional basis [Pelczynski (1961)].
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
7 / 13
Example
The unit basis (un ) of James space is RUD.
k
X
an un kJ = sup
X
m
n∈N
(apk − apk +1 )
2
1
2
: p1 < p2 < . . . < pm+1 .
k=1
It holds that
k
m
X
ai ui kJ ≤
√
m
X
2 E i ai ui .
i=1
i=1
J
Example
The Haar basis in L1 [0, 1] is an RUD basis.
Recall, L1 [0, 1] has no unconditional basis. Actually, L1 [0, 1] does not
embed in a space with unconditional basis [Pelczynski (1961)].
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
7 / 13
Example
The unit basis (un ) of James space is RUD.
k
X
an un kJ = sup
X
m
n∈N
(apk − apk +1 )
2
1
2
: p1 < p2 < . . . < pm+1 .
k=1
It holds that
k
m
X
ai ui kJ ≤
√
m
X
2 E i ai ui .
i=1
i=1
J
Example
The Haar basis in L1 [0, 1] is an RUD basis.
Recall, L1 [0, 1] has no unconditional basis. Actually, L1 [0, 1] does not
embed in a space with unconditional basis [Pelczynski (1961)].
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
7 / 13
Duality
Proposition
Let (xn ) be a basis, and (xn∗ ) bi-orthogonal functionals.
(xn ) RUC ⇒ (xn∗ ) RUD.
(xn∗ ) RUC ⇒ (xn ) RUD.
However, if we take (sn ), the summing basis of c0
(n)
z }| {
sn = (1, . . . , 1, 0 . . .),
this is not RUC, although
(n−1)
sn∗
z }| {
= (0, . . . , 0, 1, −1, 0, . . .)
form an RUD basis in `1 .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
8 / 13
Duality
Proposition
Let (xn ) be a basis, and (xn∗ ) bi-orthogonal functionals.
(xn ) RUC ⇒ (xn∗ ) RUD.
(xn∗ ) RUC ⇒ (xn ) RUD.
However, if we take (sn ), the summing basis of c0
(n)
z }| {
sn = (1, . . . , 1, 0 . . .),
this is not RUC, although
(n−1)
sn∗
z }| {
= (0, . . . , 0, 1, −1, 0, . . .)
form an RUD basis in `1 .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
8 / 13
Reflexivity
Theorem (James)
A Banach space X with unconditional basis which does not contain `1
nor c0 subspaces, is reflexive.
Theorem
1
Let (xn ) be a basis of a Banach space X such that every block is
RUD. (xn ) is shrinking ⇔ `1 6⊂ X
2
Let (xn ) be a basis of a Banach space X such that every block is
RUC. (xn ) is boundedly complete ⇔ c0 6⊂ X .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
9 / 13
Reflexivity
Theorem (James)
A Banach space X with unconditional basis which does not contain `1
nor c0 subspaces, is reflexive.
Theorem
1
Let (xn ) be a basis of a Banach space X such that every block is
RUD. (xn ) is shrinking ⇔ `1 6⊂ X
2
Let (xn ) be a basis of a Banach space X such that every block is
RUC. (xn ) is boundedly complete ⇔ c0 6⊂ X .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
9 / 13
Uniqueness
Theorem (Lindenstrauss-Zippin)
X has a unique unconditional basis iff X ≈ `1 , `2 or c0 .
Theorem (BKPS)
X has a unique RUC basis iff X ≈ `1 .
Theorem
If X has an RUD basis, then there are non-equivalent RUD basis in X .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
10 / 13
Uniqueness
Theorem (Lindenstrauss-Zippin)
X has a unique unconditional basis iff X ≈ `1 , `2 or c0 .
Theorem (BKPS)
X has a unique RUC basis iff X ≈ `1 .
Theorem
If X has an RUD basis, then there are non-equivalent RUD basis in X .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
10 / 13
Uniqueness
Theorem (Lindenstrauss-Zippin)
X has a unique unconditional basis iff X ≈ `1 , `2 or c0 .
Theorem (BKPS)
X has a unique RUC basis iff X ≈ `1 .
Theorem
If X has an RUD basis, then there are non-equivalent RUD basis in X .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
10 / 13
Uniqueness
Theorem (Lindenstrauss-Zippin)
X has a unique unconditional basis iff X ≈ `1 , `2 or c0 .
Theorem (BKPS)
X has a unique RUC basis iff X ≈ `1 .
Theorem
If X has an RUD basis, then there are non-equivalent RUD basis in X .
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
10 / 13
Things we know
Does every weakly null sequence have an RUD subsequence?
NO [e.g. Maurey-Rosenthal space (Studia 1977).]
Is every block sequence of an RUD basis also RUD?
NO [A modification of M-R.]
Given an RUD sequence, does it have an unconditional subsequence?
NO [e.g. a weakly null sequence in L1 [0, 1] without unconditional
subsequences (Johnson-Maurey-Schechtman, JAMS 2007)]
Theorem
Every block sequence of the Haar basis in L1 is RUD.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
11 / 13
Things we know
Does every weakly null sequence have an RUD subsequence?
NO [e.g. Maurey-Rosenthal space (Studia 1977).]
Is every block sequence of an RUD basis also RUD?
NO [A modification of M-R.]
Given an RUD sequence, does it have an unconditional subsequence?
NO [e.g. a weakly null sequence in L1 [0, 1] without unconditional
subsequences (Johnson-Maurey-Schechtman, JAMS 2007)]
Theorem
Every block sequence of the Haar basis in L1 is RUD.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
11 / 13
Things we know
Does every weakly null sequence have an RUD subsequence?
NO [e.g. Maurey-Rosenthal space (Studia 1977).]
Is every block sequence of an RUD basis also RUD?
NO [A modification of M-R.]
Given an RUD sequence, does it have an unconditional subsequence?
NO [e.g. a weakly null sequence in L1 [0, 1] without unconditional
subsequences (Johnson-Maurey-Schechtman, JAMS 2007)]
Theorem
Every block sequence of the Haar basis in L1 is RUD.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
11 / 13
Things we know
Does every weakly null sequence have an RUD subsequence?
NO [e.g. Maurey-Rosenthal space (Studia 1977).]
Is every block sequence of an RUD basis also RUD?
NO [A modification of M-R.]
Given an RUD sequence, does it have an unconditional subsequence?
NO [e.g. a weakly null sequence in L1 [0, 1] without unconditional
subsequences (Johnson-Maurey-Schechtman, JAMS 2007)]
Theorem
Every block sequence of the Haar basis in L1 is RUD.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
11 / 13
Things we know
Does every weakly null sequence have an RUD subsequence?
NO [e.g. Maurey-Rosenthal space (Studia 1977).]
Is every block sequence of an RUD basis also RUD?
NO [A modification of M-R.]
Given an RUD sequence, does it have an unconditional subsequence?
NO [e.g. a weakly null sequence in L1 [0, 1] without unconditional
subsequences (Johnson-Maurey-Schechtman, JAMS 2007)]
Theorem
Every block sequence of the Haar basis in L1 is RUD.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
11 / 13
Things we know
Does every weakly null sequence have an RUD subsequence?
NO [e.g. Maurey-Rosenthal space (Studia 1977).]
Is every block sequence of an RUD basis also RUD?
NO [A modification of M-R.]
Given an RUD sequence, does it have an unconditional subsequence?
NO [e.g. a weakly null sequence in L1 [0, 1] without unconditional
subsequences (Johnson-Maurey-Schechtman, JAMS 2007)]
Theorem
Every block sequence of the Haar basis in L1 is RUD.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
11 / 13
Things we don’t know
Does every Banach space contain an RUD/RUC basic sequence?
Is every basis of `1 an RUD basis?
Suppose every block basis of (xn ) is RUD. Can we find
unconditional blocks?
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
12 / 13
Things we don’t know
Does every Banach space contain an RUD/RUC basic sequence?
Is every basis of `1 an RUD basis?
Suppose every block basis of (xn ) is RUD. Can we find
unconditional blocks?
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
12 / 13
Things we don’t know
Does every Banach space contain an RUD/RUC basic sequence?
Is every basis of `1 an RUD basis?
Suppose every block basis of (xn ) is RUD. Can we find
unconditional blocks?
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
12 / 13
Thank you for your attention.
P. Tradacete (UC3M)
RUC/RUD bases
Warwick 2015
13 / 13