Subordination on K -convex spaces
Christian Le Merdy – Bedlewo, April 27th, 2017
Joint work with F. Lancien
Operators constructed by subordination I
Let X be a Banach space
Let B(X ) be the algebra of bounded operators on X .
An operator T ∈ B(X ) is called power bounded if there exists C > 1
such that
∀ n > 0,
kT n k 6 C .
Operators constructed by subordination I
Let X be a Banach space
Let B(X ) be the algebra of bounded operators on X .
An operator T ∈ B(X ) is called power bounded if there exists C > 1
such that
∀ n > 0,
kT n k 6 C .
Construction I.
P
Let c = (ck )k>0 be a sequence of R+ with ∞
k=0 ck = 1 .
Operators constructed by subordination I
Let X be a Banach space
Let B(X ) be the algebra of bounded operators on X .
An operator T ∈ B(X ) is called power bounded if there exists C > 1
such that
∀ n > 0,
kT n k 6 C .
Construction I.
P
Let c = (ck )k>0 be a sequence of R+ with ∞
k=0 ck = 1 .
Let T ∈ B(X ) be a power bounded operator and set
S=
∞
X
k=0
ck T k .
Operators constructed by subordination I
Let X be a Banach space
Let B(X ) be the algebra of bounded operators on X .
An operator T ∈ B(X ) is called power bounded if there exists C > 1
such that
∀ n > 0,
kT n k 6 C .
Construction I.
P
Let c = (ck )k>0 be a sequence of R+ with ∞
k=0 ck = 1 .
Let T ∈ B(X ) be a power bounded operator and set
S=
∞
X
ck T k .
k=0
Then S ∈ B(X ) is a power bounded operator.
Operators constructed by subordination I
Let X be a Banach space
Let B(X ) be the algebra of bounded operators on X .
An operator T ∈ B(X ) is called power bounded if there exists C > 1
such that
∀ n > 0,
kT n k 6 C .
Construction I.
P
Let c = (ck )k>0 be a sequence of R+ with ∞
k=0 ck = 1 .
Let T ∈ B(X ) be a power bounded operator and set
S=
∞
X
ck T k .
k=0
Then S ∈ B(X ) is a power bounded operator.
Reference : N. Dungey, Subordinated discrete semigroups of operators,
Transactions AMS, 2011.
Operators constructed by subordination II
Construction II.
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
measure.
Operators constructed by subordination II
Construction II.
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
measure.
Let π : G → B(X ) be a representation, that is,
∀ s, t ∈ G , π(ts) = π(t)π(s)
and π(e) = IX .
Assume that π is strongly continuous : for any x ∈ X , t 7→ π(t)x is
continuous from G into X .
Assume that π is bounded : there exists C > 1 such that kπ(t)k 6 C for
any t ∈ G . Set
Z
π(t) dν(t) .
S=
G
Operators constructed by subordination II
Construction II.
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
measure.
Let π : G → B(X ) be a representation, that is,
∀ s, t ∈ G , π(ts) = π(t)π(s)
and π(e) = IX .
Assume that π is strongly continuous : for any x ∈ X , t 7→ π(t)x is
continuous from G into X .
Assume that π is bounded : there exists C > 1 such that kπ(t)k 6 C for
any t ∈ G . Set
Z
π(t) dν(t) .
S=
G
Then S ∈ B(X ) is a power bounded operator.
Operators constructed by subordination II
Construction II.
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
measure.
Let π : G → B(X ) be a representation, that is,
∀ s, t ∈ G , π(ts) = π(t)π(s)
and π(e) = IX .
Assume that π is strongly continuous : for any x ∈ X , t 7→ π(t)x is
continuous from G into X .
Assume that π is bounded : there exists C > 1 such that kπ(t)k 6 C for
any t ∈ G . Set
Z
π(t) dν(t) .
S=
G
Then S ∈ B(X ) is a power bounded operator.
Reference : G. Cohen, C. Cuny, M. Lin, Almost everywhere convergence of
powers of some positive Lp -contractions, J. Math. Anal. App. 2014.
Questions
Let S be a subordinated operator, either
S=
∞
X
k=0
ck T k
or
S=
Z
π(t) dν(t) .
G
Questions
Let S be a subordinated operator, either
S=
∞
X
ck T k
or
S=
k=0
Question 1. When is S a Ritt operator ?
Z
π(t) dν(t) .
G
Questions
Let S be a subordinated operator, either
S=
∞
X
ck T k
or
S=
k=0
Z
π(t) dν(t) .
G
Question 1. When is S a Ritt operator ?
Question 2. When does S admit a bounded H ∞ -functional calculus ?
Ritt operators
A Ritt operator S ∈ B(X ) is a power bounded operator such that
∃ C > 0 ∀ n > 1,
nkS n − S n−1 k 6 C .
Ritt operators
A Ritt operator S ∈ B(X ) is a power bounded operator such that
∃ C > 0 ∀ n > 1,
nkS n − S n−1 k 6 C .
Let D = {λ ∈ C : |λ| < 1}.
This is equivalent to the so-called Ritt condition :
σ(S) ⊂ D
and
∀λ ∈
/ D,
kR(λ, S)k 6
K
.
|λ − 1|
Ritt operators
A Ritt operator S ∈ B(X ) is a power bounded operator such that
∃ C > 0 ∀ n > 1,
nkS n − S n−1 k 6 C .
Let D = {λ ∈ C : |λ| < 1}.
This is equivalent to the so-called Ritt condition :
σ(S) ⊂ D
and
∀λ ∈
/ D,
kR(λ, S)k 6
K
.
|λ − 1|
This is a discrete analogue of the notion of bounded analytic semigroup.
Ritt operators
A Ritt operator S ∈ B(X ) is a power bounded operator such that
∃ C > 0 ∀ n > 1,
nkS n − S n−1 k 6 C .
Let D = {λ ∈ C : |λ| < 1}.
This is equivalent to the so-called Ritt condition :
σ(S) ⊂ D
and
∀λ ∈
/ D,
kR(λ, S)k 6
K
.
|λ − 1|
This is a discrete analogue of the notion of bounded analytic semigroup.
Reference : Coulhon-Saloff Coste, Nevanlinna, Lyubich, Nagy-Zemanek,
Blunck, etc.
Spectral property of Ritt operators
For any angle 0 < γ < π2 , let Bγ be the convex hull of 1 and the disc of
center 0 and radius sin γ.
Bγ
0
γ
1
Spectral property of Ritt operators
For any angle 0 < γ < π2 , let Bγ be the convex hull of 1 and the disc of
center 0 and radius sin γ.
Bγ
0
If S ∈ B(X ) is a Ritt operator, then
π ∃0 < γ < 2
γ
1
σ(S) ⊂ Bγ .
H ∞ -calculus of Ritt operators
Let P be the algebra of complex polynomials.
Let S be a Ritt operator and let 0 < γ < π2 .
H ∞ -calculus of Ritt operators
Let P be the algebra of complex polynomials.
Let S be a Ritt operator and let 0 < γ < π2 .
We say that S has a bounded H ∞ (Bγ ) functional calculus if
σ(S) ⊂ Bγ
and there exists a constant K > 1 such that
∀ ϕ ∈ P,
kϕ(S)k 6 K sup |ϕ(z)| : z ∈ Bγ .
H ∞ -calculus of Ritt operators
Let P be the algebra of complex polynomials.
Let S be a Ritt operator and let 0 < γ < π2 .
We say that S has a bounded H ∞ (Bγ ) functional calculus if
σ(S) ⊂ Bγ
and there exists a constant K > 1 such that
∀ ϕ ∈ P,
kϕ(S)k 6 K sup |ϕ(z)| : z ∈ Bγ .
We simply say that S admits a bounded H ∞ -functional calculus if this
happens for some 0 < γ < π2 .
H ∞ -calculus of Ritt operators (continued)
In many contexts, a bounded H ∞ -functional calculus for a Ritt operator is
equivalent to certain square function estimates (Le Merdy, Haak-Haase).
H ∞ -calculus of Ritt operators (continued)
In many contexts, a bounded H ∞ -functional calculus for a Ritt operator is
equivalent to certain square function estimates (Le Merdy, Haak-Haase).
If a Ritt operator S ∈ B(X ) admits a bounded H ∞ -functional calculus,
then it is polynomially bounded, that is, there exists a constant K > 1
such that
∀ ϕ ∈ P,
kϕ(S)k 6 K sup |ϕ(z)| : z ∈ D .
H ∞ -calculus of Ritt operators (continued)
In many contexts, a bounded H ∞ -functional calculus for a Ritt operator is
equivalent to certain square function estimates (Le Merdy, Haak-Haase).
If a Ritt operator S ∈ B(X ) admits a bounded H ∞ -functional calculus,
then it is polynomially bounded, that is, there exists a constant K > 1
such that
∀ ϕ ∈ P,
kϕ(S)k 6 K sup |ϕ(z)| : z ∈ D .
The converse is not true (F. Lancien-Le Merdy, Proc. Edinburgh, 2015).
Conditions on T
Theorem (Gomilko-Tomilov, 2015)
Let TP∈ B(X ) be a Ritt operator and let c = (ck )k>0 be a sequence of R+
with ∞
k=0 ck = 1 . Then
∞
X
ck T k
S=
k=0
is a Ritt operator.
Conditions on T
Theorem (Gomilko-Tomilov, 2015)
Let TP∈ B(X ) be a Ritt operator and let c = (ck )k>0 be a sequence of R+
with ∞
k=0 ck = 1 . Then
∞
X
ck T k
S=
k=0
is a Ritt operator.
Reference : A. Gomilko, Y. Tomilov, On discrete subordination of power
bounded and Ritt operators, to appear.
Conditions on T
Theorem (Gomilko-Tomilov, 2015)
Let TP∈ B(X ) be a Ritt operator and let c = (ck )k>0 be a sequence of R+
with ∞
k=0 ck = 1 . Then
∞
X
ck T k
S=
k=0
is a Ritt operator.
Reference : A. Gomilko, Y. Tomilov, On discrete subordination of power
bounded and Ritt operators, to appear.
Proposition/Remark
If further T ∈ B(X ) has a bounded H ∞ -functional calculus, then S has a
bounded H ∞ -functional calculus.
Conditions on T
Theorem (Gomilko-Tomilov, 2015)
Let TP∈ B(X ) be a Ritt operator and let c = (ck )k>0 be a sequence of R+
with ∞
k=0 ck = 1 . Then
∞
X
ck T k
S=
k=0
is a Ritt operator.
Reference : A. Gomilko, Y. Tomilov, On discrete subordination of power
bounded and Ritt operators, to appear.
Proposition/Remark
If further T ∈ B(X ) has a bounded H ∞ -functional calculus, then S has a
bounded H ∞ -functional calculus.
No possible analogues in the group case.
Necessary condition on c
P∞
Let c = (ck )k>0 be a sequence of R
P+∞with k k=0 ck = 1 .
Associate Fc : T → C by Fc (z) = k=0 ck z .
Necessary condition on c
P∞
Let c = (ck )k>0 be a sequence of R
P+∞with k k=0 ck = 1 .
Associate Fc : T → C by Fc (z) = k=0 ck z .
Definition. We say that c has BAR (for ‘bounded angular ratio’) if there
exists δ > 1 such that
∀ z ∈ T,
|1 − Fc (z)| 6 δ(1 − |Fc (z)|).
Necessary condition on c
P∞
Let c = (ck )k>0 be a sequence of R
P+∞with k k=0 ck = 1 .
Associate Fc : T → C by Fc (z) = k=0 ck z .
Definition. We say that c has BAR (for ‘bounded angular ratio’) if there
exists δ > 1 such that
∀ z ∈ T,
|1 − Fc (z)| 6 δ(1 − |Fc (z)|).
This is equivalent to the existence of 0 < γ <
π
2
such that Fc (T) ⊂ Bγ .
Necessary condition on c
P∞
Let c = (ck )k>0 be a sequence of R
P+∞with k k=0 ck = 1 .
Associate Fc : T → C by Fc (z) = k=0 ck z .
Definition. We say that c has BAR (for ‘bounded angular ratio’) if there
exists δ > 1 such that
∀ z ∈ T,
|1 − Fc (z)| 6 δ(1 − |Fc (z)|).
This is equivalent to the existence of 0 < γ <
π
2
such that Fc (T) ⊂ Bγ .
Let U ∈ B(ℓ2Z ) be the shift operator and consider Mc :=
Then
Mc = c∗ · : ℓ2Z → ℓ2Z ,
this operator is normal and σ(Mc ) = Fc (T).
P∞
k=0 ck U
k
.
Necessary condition on c
P∞
Let c = (ck )k>0 be a sequence of R
P+∞with k k=0 ck = 1 .
Associate Fc : T → C by Fc (z) = k=0 ck z .
Definition. We say that c has BAR (for ‘bounded angular ratio’) if there
exists δ > 1 such that
∀ z ∈ T,
|1 − Fc (z)| 6 δ(1 − |Fc (z)|).
This is equivalent to the existence of 0 < γ <
π
2
such that Fc (T) ⊂ Bγ .
Let U ∈ B(ℓ2Z ) be the shift operator and consider Mc :=
Then
Mc = c∗ · : ℓ2Z → ℓ2Z ,
this operator is normal and σ(Mc ) = Fc (T).
Hence Mc is Ritt if and only if c has BAR.
P∞
k=0 ck U
k
.
Necessary condition on c
P∞
Let c = (ck )k>0 be a sequence of R
P+∞with k k=0 ck = 1 .
Associate Fc : T → C by Fc (z) = k=0 ck z .
Definition. We say that c has BAR (for ‘bounded angular ratio’) if there
exists δ > 1 such that
∀ z ∈ T,
|1 − Fc (z)| 6 δ(1 − |Fc (z)|).
This is equivalent to the existence of 0 < γ <
π
2
such that Fc (T) ⊂ Bγ .
Let U ∈ B(ℓ2Z ) be the shift operator and consider Mc :=
Then
Mc = c∗ · : ℓ2Z → ℓ2Z ,
P∞
k=0 ck U
k
.
this operator is normal and σ(Mc ) = Fc (T).
Hence Mc is Ritt if and only if c has BAR.
In this case, it automatically admits a bounded H ∞ -functional calculus.
Necessary condition on ν
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
b → C.
measure and consider its Fourier transform νb : G
Necessary condition on ν
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
b → C.
measure and consider its Fourier transform νb : G
Definition. We say that ν has BAR if there exists δ > 1 such that
b,
∀γ ∈ G
|1 − νb(γ)| 6 δ(1 − |b
ν (γ)|).
Necessary condition on ν
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
b → C.
measure and consider its Fourier transform νb : G
Definition. We say that ν has BAR if there exists δ > 1 such that
b,
∀γ ∈ G
|1 − νb(γ)| 6 δ(1 − |b
ν (γ)|).
Let λ ∈ B(L2 (G )) be defined by λ(t)f = f (· −t) for any f ∈ L2 (G ). Then
Z
λ(t) dν(t) = ν∗ · : L2 (G ) → L2 (G ),
Mν :=
G
b ).
this is normal operator and σ(Mν ) = νb(G
Necessary condition on ν
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
b → C.
measure and consider its Fourier transform νb : G
Definition. We say that ν has BAR if there exists δ > 1 such that
b,
∀γ ∈ G
|1 − νb(γ)| 6 δ(1 − |b
ν (γ)|).
Let λ ∈ B(L2 (G )) be defined by λ(t)f = f (· −t) for any f ∈ L2 (G ). Then
Z
λ(t) dν(t) = ν∗ · : L2 (G ) → L2 (G ),
Mν :=
G
b ).
this is normal operator and σ(Mν ) = νb(G
Hence Mν is Ritt if and only if ν has BAR.
Necessary condition on ν
Let G be an abelian locally compact group. Let ν ∈ M(G ) be a probability
b → C.
measure and consider its Fourier transform νb : G
Definition. We say that ν has BAR if there exists δ > 1 such that
b,
∀γ ∈ G
|1 − νb(γ)| 6 δ(1 − |b
ν (γ)|).
Let λ ∈ B(L2 (G )) be defined by λ(t)f = f (· −t) for any f ∈ L2 (G ). Then
Z
λ(t) dν(t) = ν∗ · : L2 (G ) → L2 (G ),
Mν :=
G
b ).
this is normal operator and σ(Mν ) = νb(G
Hence Mν is Ritt if and only if ν has BAR.
In this case, it automatically admits a bounded H ∞ -functional calculus.
Refined questions
• Assume that c has BAR.
P
(i) Is S = k ck T k a Ritt operator for any power bounded T ?
(ii) Which additional conditions imply this property ?
Refined questions
• Assume that c has BAR.
P
(i) Is S = k ck T k a Ritt operator for any power bounded T ?
(ii) Which additional conditions imply this property ?
• Assume that ν has BAR.
R
(i) Is S = G π(t) dν(t) a Ritt operator for any strongly continuous
bounded representation π ?
(ii) Which additional conditions imply this property ?
Refined questions
• Assume that c has BAR.
P
(i) Is S = k ck T k a Ritt operator for any power bounded T ?
(ii) Which additional conditions imply this property ?
• Assume that ν has BAR.
R
(i) Is S = G π(t) dν(t) a Ritt operator for any strongly continuous
bounded representation π ?
(ii) Which additional conditions imply this property ?
• What about H ∞ -functional calculus in these two cases ?
Refined questions
• Assume that c has BAR.
P
(i) Is S = k ck T k a Ritt operator for any power bounded T ?
(ii) Which additional conditions imply this property ?
• Assume that ν has BAR.
R
(i) Is S = G π(t) dν(t) a Ritt operator for any strongly continuous
bounded representation π ?
(ii) Which additional conditions imply this property ?
• What about H ∞ -functional calculus in these two cases ?
Proposition
There exist a sequence c = (ck )k>0 satisfying BAR
P and a power bounded
operator T on Hilbert space H such that S = k ck T k is Ritt but does
not have a bounded H ∞ -functional calculus.
A positive result on UMD Banach lattices
A positive result on UMD Banach lattices
Theorem
Let G be an abelian locally compact group, assume that G is σ-compact.
Let X be a UMD Banach lattice and let π : G → B(X ) be a strongly
continuous bounded representation. Let ν ∈ M(G ) be a probability
measure with BAR. Then
Z
π(t) dν(t)
S=
G
is a Ritt operator which admits a bounded H ∞ -functional calculus.
A positive result on UMD Banach lattices
Theorem
Let G be an abelian locally compact group, assume that G is σ-compact.
Let X be a UMD Banach lattice and let π : G → B(X ) be a strongly
continuous bounded representation. Let ν ∈ M(G ) be a probability
measure with BAR. Then
Z
π(t) dν(t)
S=
G
is a Ritt operator which admits a bounded H ∞ -functional calculus.
Elements of proof.
• Consider λX ∈ B(L2 (G ; X )) be defined by λ(t)f = f (· −t) for any
f ∈ L2 (G : X ). Then we have
Z
X
λX (t) dν(t) = ν∗ · : L2 (G ; X ) → L2 (G ; X ).
Mν :=
G
A positive result on UMD Banach lattices (continued)
• By transference arguments, one shows that it suffices to consider the
case when π = λX ; the space X is replaced by L2 (G ; X ) and S is replaced
by MνX .
A positive result on UMD Banach lattices (continued)
• By transference arguments, one shows that it suffices to consider the
case when π = λX ; the space X is replaced by L2 (G ; X ) and S is replaced
by MνX .
• Since X is a UMD Banach lattice, there exists a Hilbert space H, a
UMD Banach space Y and θ ∈ (0, 1) such that
X = [Y , H]θ
in the sense of complex interpolation (Rubio de Francia, 1986).
A positive result on UMD Banach lattices (continued)
• By transference arguments, one shows that it suffices to consider the
case when π = λX ; the space X is replaced by L2 (G ; X ) and S is replaced
by MνX .
• Since X is a UMD Banach lattice, there exists a Hilbert space H, a
UMD Banach space Y and θ ∈ (0, 1) such that
X = [Y , H]θ
in the sense of complex interpolation (Rubio de Francia, 1986).
• Then L2 (G ; X ) = [L2 (G ; Y ), L2 (G ; H)]θ and MνX interpolates betwen
MνH and MνY .
A positive result on UMD Banach lattices (continued)
• By transference arguments, one shows that it suffices to consider the
case when π = λX ; the space X is replaced by L2 (G ; X ) and S is replaced
by MνX .
• Since X is a UMD Banach lattice, there exists a Hilbert space H, a
UMD Banach space Y and θ ∈ (0, 1) such that
X = [Y , H]θ
in the sense of complex interpolation (Rubio de Francia, 1986).
• Then L2 (G ; X ) = [L2 (G ; Y ), L2 (G ; H)]θ and MνX interpolates betwen
MνH and MνY .
• Since H is a Hilbert space, the operator MνH behaves like Mν : this is a
Ritt operator with a bounded H ∞ -functional calculus.
A positive result on UMD Banach lattices (continued)
• By transference arguments, one shows that it suffices to consider the
case when π = λX ; the space X is replaced by L2 (G ; X ) and S is replaced
by MνX .
• Since X is a UMD Banach lattice, there exists a Hilbert space H, a
UMD Banach space Y and θ ∈ (0, 1) such that
X = [Y , H]θ
in the sense of complex interpolation (Rubio de Francia, 1986).
• Then L2 (G ; X ) = [L2 (G ; Y ), L2 (G ; H)]θ and MνX interpolates betwen
MνH and MνY .
• Since H is a Hilbert space, the operator MνH behaves like Mν : this is a
Ritt operator with a bounded H ∞ -functional calculus.
• By interpolation (Blunck, 2001), this implies that MνX is a Ritt operator.
A positive result on UMD Banach lattices (continued)
• By transference arguments, one shows that it suffices to consider the
case when π = λX ; the space X is replaced by L2 (G ; X ) and S is replaced
by MνX .
• Since X is a UMD Banach lattice, there exists a Hilbert space H, a
UMD Banach space Y and θ ∈ (0, 1) such that
X = [Y , H]θ
in the sense of complex interpolation (Rubio de Francia, 1986).
• Then L2 (G ; X ) = [L2 (G ; Y ), L2 (G ; H)]θ and MνX interpolates betwen
MνH and MνY .
• Since H is a Hilbert space, the operator MνH behaves like Mν : this is a
Ritt operator with a bounded H ∞ -functional calculus.
• By interpolation (Blunck, 2001), this implies that MνX is a Ritt operator.
• Further interpolation arguments show that MνX admits a bounded
H ∞ -functional calculus. (Here the UMD property of Y plays a role.)
A positive result on UMD Banach lattices (continued)
Corollary
Let 1 < p < ∞, let (Ω, µ) be a measure space and let T ∈ B(Lp (Ω)) be a
positive contraction. Let c = (ck )k>1 be a sequence satisfying BAR. Then
S=
∞
X
ck T k
k=0
is a Ritt operator which admits a bounded H ∞ -functional calculus.
A positive result on UMD Banach lattices (continued)
Corollary
Let 1 < p < ∞, let (Ω, µ) be a measure space and let T ∈ B(Lp (Ω)) be a
positive contraction. Let c = (ck )k>1 be a sequence satisfying BAR. Then
S=
∞
X
ck T k
k=0
is a Ritt operator which admits a bounded H ∞ -functional calculus.
Proof. By Akcoglu’s dilation Theorem, there exist another measure space
(Ω′ , µ′ ), an isometric invertible U ∈ B(Lp (Ω′ )) and two contractions
J : Lp (Ω) → Lp (Ω′ ) and Q : Lp (Ω′ ) → Lp (Ω) such that
∀ n > 0,
T n = QU n J.
A positive result on UMD Banach lattices (continued)
Corollary
Let 1 < p < ∞, let (Ω, µ) be a measure space and let T ∈ B(Lp (Ω)) be a
positive contraction. Let c = (ck )k>1 be a sequence satisfying BAR. Then
S=
∞
X
ck T k
k=0
is a Ritt operator which admits a bounded H ∞ -functional calculus.
Proof. By Akcoglu’s dilation Theorem, there exist another measure space
(Ω′ , µ′ ), an isometric invertible U ∈ B(Lp (Ω′ )) and two contractions
J : Lp (Ω) → Lp (Ω′ ) and Q : Lp (Ω′ ) → Lp (Ω) such that
∀ n > 0,
Then one introduces
V =
T n = QU n J.
∞
X
k=0
ck U k .
A positive result on UMD Banach lattices (continued)
One shows that for any ϕ ∈ P,
ϕ(S) = Qϕ(V )J.
A positive result on UMD Banach lattices (continued)
One shows that for any ϕ ∈ P,
ϕ(S) = Qϕ(V )J.
By the previous theorem, V is a Ritt operator which admits a bounded
H ∞ -functional calculus.
A positive result on UMD Banach lattices (continued)
One shows that for any ϕ ∈ P,
ϕ(S) = Qϕ(V )J.
By the previous theorem, V is a Ritt operator which admits a bounded
H ∞ -functional calculus.
The result follows at once.
K -convex Banach spaces
• Let X be a Banach space. We say that X contains ℓ1n ’s uniformly if
there is a constant K > 1 such that for any integer n > 1, there is an
K
n-dimensional subspace En ⊂ X such that ℓn1 ≃ En . That is, for any n > 1,
there exist x1 , . . . , xn in X such that for any α1 , · · · , αn ∈ C,
n
X
j=1
n
n
X
X
α j xj 6 K
|αj | 6 |αj |
j=1
j=1
K -convex Banach spaces
• Let X be a Banach space. We say that X contains ℓ1n ’s uniformly if
there is a constant K > 1 such that for any integer n > 1, there is an
K
n-dimensional subspace En ⊂ X such that ℓn1 ≃ En . That is, for any n > 1,
there exist x1 , . . . , xn in X such that for any α1 , · · · , αn ∈ C,
n
X
j=1
n
n
X
X
α j xj 6 K
|αj | 6 |αj |
j=1
j=1
Pisier’s Theorem, 1982
The following are equivalent.
(i) X does not contains ℓ1n ’s uniformly.
(ii) X has Rademacher type > 1.
(iii) The Rademacher projection is bounded on L2 (Ω; X ).
K -convex Banach spaces
• Let X be a Banach space. We say that X contains ℓ1n ’s uniformly if
there is a constant K > 1 such that for any integer n > 1, there is an
K
n-dimensional subspace En ⊂ X such that ℓn1 ≃ En . That is, for any n > 1,
there exist x1 , . . . , xn in X such that for any α1 , · · · , αn ∈ C,
n
X
j=1
n
n
X
X
α j xj 6 K
|αj | 6 |αj |
j=1
j=1
Pisier’s Theorem, 1982
The following are equivalent.
(i) X does not contains ℓ1n ’s uniformly.
(ii) X has Rademacher type > 1.
(iii) The Rademacher projection is bounded on L2 (Ω; X ).
Such spaces are called K -convex.
K -convex Banach spaces
• Let X be a Banach space. We say that X contains ℓ1n ’s uniformly if
there is a constant K > 1 such that for any integer n > 1, there is an
K
n-dimensional subspace En ⊂ X such that ℓn1 ≃ En . That is, for any n > 1,
there exist x1 , . . . , xn in X such that for any α1 , · · · , αn ∈ C,
n
X
j=1
n
n
X
X
α j xj 6 K
|αj | 6 |αj |
j=1
j=1
Pisier’s Theorem, 1982
The following are equivalent.
(i) X does not contains ℓ1n ’s uniformly.
(ii) X has Rademacher type > 1.
(iii) The Rademacher projection is bounded on L2 (Ω; X ).
Such spaces are called K -convex.
UMD implies K -convex but there exist non reflexive K -convex spaces.
Symmetric measures
Let G be a locally compact abelian group and let ν ∈ M(G ) be a
probability. We say that ν is symmetric when ν(A) = ν(−A) for any
measurable A ⊂ G . The following is easy to check :
Symmetric measures
Let G be a locally compact abelian group and let ν ∈ M(G ) be a
probability. We say that ν is symmetric when ν(A) = ν(−A) for any
measurable A ⊂ G . The following is easy to check :
ν is symetric ⇔ Mν : L2 (G ) → L2 (G ) is selfadjoint
⇔ νb is real valued
⇔ νb is valued in [−1, 1].
Symmetric measures
Let G be a locally compact abelian group and let ν ∈ M(G ) be a
probability. We say that ν is symmetric when ν(A) = ν(−A) for any
measurable A ⊂ G . The following is easy to check :
ν is symetric ⇔ Mν : L2 (G ) → L2 (G ) is selfadjoint
⇔ νb is real valued
⇔ νb is valued in [−1, 1].
In this case, ν 2 = ν ∗ ν is a probability measure satisfying BAR.
Main result
Theorem
Let ν be a symmetric probability measure on some σ-compact locally
compact abelian group. Let X be a K -convex Banach space. Then for any
bounded strongly continuous representation π : G → B(X ),
Z
π(t) dν 2 (t)
S=
G
is a Ritt operator.
Main result
Theorem
Let ν be a symmetric probability measure on some σ-compact locally
compact abelian group. Let X be a K -convex Banach space. Then for any
bounded strongly continuous representation π : G → B(X ),
Z
π(t) dν 2 (t)
S=
G
is a Ritt operator.
The proof relies on some techniques introduced by Pisier, in particular the
following result :
If Y is a K -convex Banach space, then there exists C > 1 such that
for any integer n > 1 and for any commuting contractive projections
in B(Y ), then
n
X
Y
Pk 6 C .
(IY − Pj )
j=1
16j6=k6n
Main result (continued)
Let ν, X and π as above and let S =
R
G
π(t) dν(t) .
b ).
If ν ∈ L1 (G ) (i.e. ν has a density), then σ(S) ⊂ νb(G
Main result (continued)
Let ν, X and π as above and let S =
R
G
π(t) dν(t) .
b ).
If ν ∈ L1 (G ) (i.e. ν has a density), then σ(S) ⊂ νb(G
If further ν has BAR (equivalently, there exists a ∈ (−1, 1] such that
b ), then 1 + S is invertible.
νb(γ) ∈ [a, 1] for any γ ∈ G
Main result (continued)
Let ν, X and π as above and let S =
R
G
π(t) dν(t) .
b ).
If ν ∈ L1 (G ) (i.e. ν has a density), then σ(S) ⊂ νb(G
If further ν has BAR (equivalently, there exists a ∈ (−1, 1] such that
b ), then 1 + S is invertible.
νb(γ) ∈ [a, 1] for any γ ∈ G
2
Since S is a Ritt operator, this implies that S is a Ritt operator.
Main result (continued)
Let ν, X and π as above and let S =
R
G
π(t) dν(t) .
b ).
If ν ∈ L1 (G ) (i.e. ν has a density), then σ(S) ⊂ νb(G
If further ν has BAR (equivalently, there exists a ∈ (−1, 1] such that
b ), then 1 + S is invertible.
νb(γ) ∈ [a, 1] for any γ ∈ G
2
Since S is a Ritt operator, this implies that S is a Ritt operator.
Corollary
P
Let
c
=
(c
)
be
a
nonnegative
sequence
with
k
k∈Z
k ck = 1. Assume that
P
ikt
> 0 for any t ∈ R. Let X be a K -convex space and let
k ck e
U ∈ B(X ) be an invertible operator with sup{kT k k : k ∈ Z} < ∞. Then
S=
∞
X
k=−∞
is a Ritt operator.
ck U k
Necessity of K -convexity
Theorem
Let G be a non-discrete locally compact abelian group. Let X be a Banach
space and assume that X in not K -convex. Then there exists a symmetric
probability measure ν ∈ M(G ) such that MνX2 is not a Ritt operator.
Necessity of K -convexity
Theorem
Let G be a non-discrete locally compact abelian group. Let X be a Banach
space and assume that X in not K -convex. Then there exists a symmetric
probability measure ν ∈ M(G ) such that MνX2 is not a Ritt operator.
The proof relies on the existence of a symmetric probability measure
ν ∈ M(G ) such that 1 + ν 2 is not invertible in M(G ) (Rudin, Fourier
analysis on groups).