A new approach to the Akcoglu–Sucheston
dilation theorem for positive contractions
on Lp -spaces
Emergent trends of Complex Analysis and Functional
Analysis, Będlewo (joint work with J. Glück)
Who? Stephan Fackler
From?
Institute of Applied Analysis, Ulm University (Germany)
When? April 27, 2017
What is a dilation?
X class of Banach spaces
Definition T ∈ B(Y ) has a dilation in X if there exist X ∈ X ,
U ∈ B(X ) invertible isometry, J : Y → X and Q : X → Y
bounded with
T n = QU n J
for all n ∈ N0 .
Remark T ∈ B(Y ) has a dilation ⇒ kT n k ≤ kQk kJk for all
n ∈ N0 , i.e. T is power bounded.
The choice of X is crucial
Example If X is the class of all Banach spaces, then every power
bounded operator T ∈ B(Y ) has a dilation in X .
The choice of X is crucial
Example If X is the class of all Banach spaces, then every power
bounded operator T ∈ B(Y ) has a dilation in X .
Proof. We explicitly construct a dilation:
X = `∞ (Z; Y ),
Jy = (T n y1N0 (n))n∈Z with kJk ≤ supn∈N0 kT n k,
U(xm )m∈Z = (xm+1 )m∈Z ,
Q(xm )m∈Z = x0 .
The choice of X is crucial
Example If X is the class of all Hilbert spaces, then there exists
some power bounded T ∈ B(H) without a dilation in X .
The choice of X is crucial
Example If X is the class of all Hilbert spaces, then there exists
some power bounded T ∈ B(H) without a dilation in X .
Let U be unitary. For all complex polynomials p
kp(U)k ≤ sup |p(z)| ,
|z|=1
i.e. U is polynomially bounded.
The choice of X is crucial
Example If X is the class of all Hilbert spaces, then there exists
some power bounded T ∈ B(H) without a dilation in X .
Let U be unitary. For all complex polynomials p
kp(U)k ≤ sup |p(z)| ,
|z|=1
i.e. U is polynomially bounded.
T has a dilation ⇒ T is polynomially bounded.
The choice of X is crucial
Example If X is the class of all Hilbert spaces, then there exists
some power bounded T ∈ B(H) without a dilation in X .
Let U be unitary. For all complex polynomials p
kp(U)k ≤ sup |p(z)| ,
|z|=1
i.e. U is polynomially bounded.
T has a dilation ⇒ T is polynomially bounded.
Construct power bounded T that is not polynomially
bounded.
The choice of X is crucial
In practice, one wants X ∈ X to have the same nice
geometric properties as Y (for T ∈ B(Y )).
Example Y = H Hilbert space: a good choice for X is the class of
all Hilbert spaces.
Example Y = Lp -space for some p ∈ (1, ∞): a good choice for X is
the class of all Lp -spaces.
Two fundamental dilation results
Theorem X all Hilbert spaces: T ∈ B(H) on Hilbert space H with
(Sz.-Nagy) kT k ≤ 1 has a dilation to a unitary operator in X .
Two fundamental dilation results
Theorem X all Hilbert spaces: T ∈ B(H) on Hilbert space H with
(Sz.-Nagy) kT k ≤ 1 has a dilation to a unitary operator in X .
Corollary (von T ∈ B(H) with kT k ≤ 1 satisfies for all complex
Neumann polynomials p
inequality)
kp(T )k ≤ sup kp(z)k .
|z|=1
Theorem X all Lp -spaces for some p ∈ (1, ∞): T ∈ B(Lp (Ω, Σ, µ))
(Akcoglu– with kT k ≤ 1 and T ≥ 0 (f ≥ 0 implies Tf ≥ 0) has a
Sucheston) dilation to a positive invertible isometry in X .
Theorem X all Lp -spaces for some p ∈ (1, ∞): T ∈ B(Lp (Ω, Σ, µ))
(Akcoglu– with kT k ≤ 1 and T ≥ 0 (f ≥ 0 implies Tf ≥ 0) has a
Sucheston) dilation to a positive invertible isometry in X .
Corollary Let T ∈ B(Lp (Ω, Σ, µ)) for p ∈ (1, ∞) with T ≥ 0 and
(Akcoglu) kT k ≤ 1. Then
N
1 X k
(T f )(x)
lim
N→∞ N
k=1
exists for a.e. x ∈ Ω.
In the main steps the usual
proofs construct the dilations
rather explicitly.
In the main steps the usual
proofs construct the dilations
rather explicitly.
We now give a more generic
approach.
Our standard assumptions on X
1
X is ultra-stable: (Xi )i∈I ⊂ X and U free ultrafilter on
directed set I. Then the ultraproduct (Xi )U lies in X .
Our standard assumptions on X
1
2
X is ultra-stable: (Xi )i∈I ⊂ X and U free ultrafilter on
directed set I. Then the ultraproduct (Xi )U lies in X .
X is `p -stable for some p ∈ (1, ∞): For X ∈ X and n ∈ N
one has `p ({1, ..., n}; X ) ∈ X .
Our standard assumptions on X
1
2
3
X is ultra-stable: (Xi )i∈I ⊂ X and U free ultrafilter on
directed set I. Then the ultraproduct (Xi )U lies in X .
X is `p -stable for some p ∈ (1, ∞): For X ∈ X and n ∈ N
one has `p ({1, ..., n}; X ) ∈ X .
The spaces in X are reflexive.
Our standard assumptions on X
1
2
3
X is ultra-stable: (Xi )i∈I ⊂ X and U free ultrafilter on
directed set I. Then the ultraproduct (Xi )U lies in X .
X is `p -stable for some p ∈ (1, ∞): For X ∈ X and n ∈ N
one has `p ({1, ..., n}; X ) ∈ X .
The spaces in X are reflexive.
Example The class of all Hilbert spaces is ultra-stable and `2 -stable.
Our standard assumptions on X
1
2
3
X is ultra-stable: (Xi )i∈I ⊂ X and U free ultrafilter on
directed set I. Then the ultraproduct (Xi )U lies in X .
X is `p -stable for some p ∈ (1, ∞): For X ∈ X and n ∈ N
one has `p ({1, ..., n}; X ) ∈ X .
The spaces in X are reflexive.
Example The class of all Hilbert spaces is ultra-stable and `2 -stable.
Example For p ∈ [1, ∞] the class of all Lp -spaces is ultra-stable
and `p -stable.
The key definition
Definition T ⊂ B(Y ) admits a simultaneous dilation in X if for all
m ∈ N andT1 , ..., Tm ∈ T there are
X ∈ X,
invertible isometries U1 , ..., Um ∈ B(X ),
J : Y → X and Q : X → Y bounded
such that for all non-commutative polynomials p in m
variables
p(T1 , ..., Tm ) = Qp(U1 , ..., Um )J.
Example (Easy) For X ∈ X all invertible isometries on X have a
simultaneous dilation in X .
Stability result for dilations – our toolkit
Theorem Let Y ∈ X and suppose that T ⊂ B(Y ) dilates
wot
simultaneosuly in X . Then the same holds for conv T ,
the closed convex hull of T in the weak operator topology.
wot
Further, conv T
is multiplicatively closed.
Stability result for dilations – our toolkit
Theorem Let Y ∈ X and suppose that T ⊂ B(Y ) dilates
wot
simultaneosuly in X . Then the same holds for conv T ,
the closed convex hull of T in the weak operator topology.
wot
Further, conv T
is multiplicatively closed.
Corollary Let Y ∈ X . Then the closed convex hull of all invertible
isometries on Y with respect to the weak operator topology
simultaneously dilates in X .
Dilation theorem of Sz.-Nagy
Use our toolkit together with the following facts:
1
H finite dimensional Hilbert space. Then every T ∈ B(H)
is a finite convex combination of unitaries.
2
H infinite dimensional Hilbert space. For every K ⊂ H
finite dimensional subspace {PK TPK : kT kB(H) ≤ 1}
dilates simultaneously.
3
ultraproduct argument: {T ∈ B(H) : kT k ≤ 1} dilates
simultaneously.
Our simultaneous dilation theorem recovers the
noncommutative von Neumann inequality by Bożejko:
kp(T1 , T2 , . . . , Tn )k ≤ sup{kp(U1 , . . . , Un )k},
where p is a noncommutative polynomial and the
supremum runs over all finite dimensional unitaries.
Dilation theorem of Akcoglu–Sucheston
Use our toolkit together with the following fact:
1
The positive contractions on Lp ([0, 1]) are the closed
convex hull of all positive invertible isometries with
respect to the weak operator topology.
Dilation theorem of Akcoglu–Sucheston
Use our toolkit together with the following fact:
1
The positive contractions on Lp ([0, 1]) are the closed
convex hull of all positive invertible isometries with
respect to the weak operator topology.
Alternatively, one can use the toolkit to pass via
p
separation preserving operators `n → Lp ([0, 1]).
Dilation theorem of Akcoglu–Sucheston
Use our toolkit together with the following fact:
1
The positive contractions on Lp ([0, 1]) are the closed
convex hull of all positive invertible isometries with
respect to the weak operator topology.
Alternatively, one can use the toolkit to pass via
p
separation preserving operators `n → Lp ([0, 1]).
Theorem The set of all positive contractions on some Lp -space for
p ∈ (1, ∞) dilates simultaneously to positive invertible
isometries in the class of all Lp -spaces.
As a consequence we obtain a noncommutative
Matsaev / von Neumann inequality for positive
contractions on Lp -spaces:
kp(T1 , T2 , . . . , Tn )k ≤ sup{kp(U1 , . . . , Un )k},
where p is a noncommutative polynomial and the
supremum runs over all finite dimensional positive
invertible isometries.
Thank you for your attention!