Ergodic properties of convolution operatorsbanach2019/pdf/Galindo.pdf · Jorge Galindo Joint work with Enrique Jord a, Universitat Polit ecnica de Val encia UPV (Spain). BANACH ALGEBRAS

Ergodic properties of convolution operators

Jorge Galindo

Joint work with Enrique Jorda,

Universitat Politecnica de Valencia UPV (Spain).

BANACH ALGEBRAS AND APPLICATIONS 2019,

University of Manitoba in Winnipeg, 11-18 July 2019.

Jorge Galindo Ergodic properties of convolution operators

Mean ergodic operators

Notation

Let X be a Banach space.

I If E is a Banach space, L(E) = {T : X → X : T is bounded and linear }.

I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).

I We say that T is a mean ergodic operator (ME, for short) if there is an operator

P ∈ L(E) such that limn→∞ T[n] = P in the strong operator topology.

I We say that T is a uniformly mean ergodic operator (UME, for short) if there is

an operator P ∈ L(E) such that limn→∞ T[n] = P in the operator norm.

I We say that T is a weakly mean ergodic operator (WME, for short) if there is an operator

P ∈ L(E) such that limn→∞ T[n] = P in the weak operator topology.



Notation



I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).









Notation



I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).









Notation



I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).









Notation



I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).









Notation



I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).









Notation



I If T ∈ L(E), T[n] =1

n

n∑k=1

T k .

Definition

Let T ∈ L(E).








Convolution operators

Let G be a locally compact group with left Haar measure dmG .

Definition

Let µ1, µ2 ∈ M(G). We define:

I The convolution of measures:

〈µ1 ∗ µ2, f 〉 =

∫ ∫f (xy)dµ1(x)dµ2(y), for every f ∈ C0(G).

I The convolution operator: λp(µ) : Lp(G)→ Lp(G) given by

λp(µ)(f )(s) = (µ ∗ f )(s) :=

∫f (x−1s)dµ(x), for all f ∈ Lp(G) and all s ∈ G .




Definition



〈µ1 ∗ µ2, f 〉 =



λp(µ)(f )(s) = (µ ∗ f )(s) :=





Definition



〈µ1 ∗ µ2, f 〉 =



λp(µ)(f )(s) = (µ ∗ f )(s) :=



Continuity of λp

Theorem

Let G be a locally compact group and consider the mapping λp : M(G)→ L(Lp(G)).

1 λp is vague-WOT continuous on norm bounded subsets of M(G), for 1 < p <∞.

2 If G is compact, then λp is vague-SOT sequentially continuous for 1 ≤ p <∞.

Vague topology: σ(M(G),C00(G))

Definition

A measure µ ∈ M(G) is vaguely ergodic if the sequence of means

µ[n] =µ+ µ2 + · · ·+ µn

nconverges to µc ∈ M(G) in the vague topology.

Theorem

Let µ ∈ M(G). Then:

1 If µ is vaguely ergodic, then λp(µ) is weakly mean ergodic .

2 If G is compact and µ is vaguely ergodic, then λp(µ) is mean ergodic .


Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem





Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition

A measure µ ∈ M(G) is said vaguely mean ergodic if the sequence of means

µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Continuity of λp

Theorem




Definition


µ[n] =µ+ µ2 + · · ·+ µn


Theorem





Multiplication operators

T[n] =1

n

n∑k=1

Tk . T is ME if T[n] is SOT convergent, T is UME if T[n] is norm convergent.

Example (Multiplication operators)

Let X be a compact space, µ a measure on X and ϕ ∈ L∞(X , µ) with ‖ϕ‖∞ ≤ 1.

Define Mϕ : L2(X , µ)→ L2(X , µ) by (Mϕf )(x) = ϕ(x) · f (x). Then, for x with

ϕ(x) 6= 1 and any f ∈ L2(X , µ):

(Mϕ)[n]f =ϕ

n

1− ϕn

1− ϕ=⇒ lim

n→∞‖(Mϕ)[n]f −M1

ϕ−1({1})‖2 = 0.

It follows that Mϕ is mean ergodic.

This immediately implies:

Theorem (von Neumann’s mean ergodic theorem)

Unitary operators, even normal operators of norm ≤ 1, on Hilbert spaces are mean

ergodic .

If G is Abelian, convolution operators λ2(µ) are multiplication operators, via

Fourier-Stieltjes transforms

λ2(µ)f = (µ ∗ f ) = f · µ = Mµ f ,

where µ ∈ CB(G), f ∈ L2(G) and G = {χ : G → T : χ a continuous homomorphism}.



T[n] =1

n

n∑k=1

Tk . T is ME if T[n] is SOT convergent, T is UME if T[n] is norm convergent.




ϕ(x) 6= 1 and any f ∈ L2(X , µ):

(Mϕ)[n]f =ϕ

n

1− ϕn

1− ϕ=⇒ lim

n→∞‖(Mϕ)[n]f −M1

ϕ−1({1})‖2 = 0.





ergodic .



λ2(µ)f = (µ ∗ f ) = f · µ = Mµ f ,







ϕ(x) 6= 1 and any f ∈ L2(X , µ):

(Mϕ)[n]f =ϕ

n

1− ϕn

1− ϕ=⇒ lim

n→∞‖(Mϕ)[n]f −M1

ϕ−1({1})‖2 = 0.





ergodic .



λ2(µ)f = (µ ∗ f ) = f · µ = Mµ f ,







ϕ(x) 6= 1 and any f ∈ L2(X , µ):

(Mϕ)[n]f =ϕ

n

1− ϕn

1− ϕ=⇒ lim

n→∞‖(Mϕ)[n]f −M1

ϕ−1({1})‖2 = 0.





ergodic .



λ2(µ)f = (µ ∗ f ) = f · µ = Mµ f ,







ϕ(x) 6= 1 and any f ∈ L2(X , µ):

(Mϕ)[n]f =ϕ

n

1− ϕn

1− ϕ=⇒ lim

n→∞‖(Mϕ)[n]f −M1

ϕ−1({1})‖2 = 0.





ergodic .



λ2(µ)f = (µ ∗ f ) = f · µ = Mµ f ,

where µ ∈ CB(G), f ∈ L2(G) and G = {χ : G → T : χ a continuous homomorphism}.Jorge Galindo Ergodic properties of convolution operators

General properties of mean ergodic operators

Theorem

If T ∈ L(E) is weakly mean ergodic , then:

I ‖T[n]‖n is a bounded sequence.

I WOT− limn→∞T n

n= 0.

I r(T ) ≤ 1.

Theorem

If T ∈ L(E) is mean ergodic , then:

I SOT− limn→∞T n

n= 0.

I E = Fix(T)⊕ (I − T ) (E).

Theorem (Yosida’s mean ergodic theorem)

If E is reflexive and ‖T[n]‖ is bounded, then T ∈ L(E) is mean ergodic if and only if

SOT− limn→∞T n

n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem




n= 0.

I r(T ) ≤ 1.

Theorem



n= 0.

I E = Fix(T)⊕ (I − T ) (E).




n= 0.



Theorem (Dunford 1943, Lin 1974)

T ∈ L(E) with SOT− limn→∞T n

n= 0. TFAE:

I T is uniformly mean Ergodic .

I (I − T ) (E) is closed.

I E = Fix(T)⊕ (I − T ) (E).

I Either 1 /∈ σ(T ) or 1 is a pole of order 1 of the resolvent of T .

Corollary

If r(T ) < 1, then T is uniformly mean Ergodic .





n= 0. TFAE:



I E = Fix(T)⊕ (I − T ) (E).


Corollary






n= 0. TFAE:



I E = Fix(T)⊕ (I − T ) (E).


Corollary






n= 0. TFAE:



I E = Fix(T)⊕ (I − T ) (E).


Corollary






n= 0. TFAE:



I E = Fix(T)⊕ (I − T ) (E).


Corollary






n= 0. TFAE:



I E = Fix(T)⊕ (I − T ) (E).


Corollary



A not so simple reduction

Theorem (Herz 1973, Derighetti 1974)

If H is a closed subgroup of G , then the restriction mapping R : Ap(G)→ Ap(H) is

almost an isometry. Its adjoint T∗PMp(H)→ PMp(G) is a multiplicative linear

isometry.

Theorem

If H is a closed subgroup of G and µ ∈ M(G) is supported in H, then λpH(µ) is mean

ergodic (resp. uniformly mean Ergodic ) if and only if λpG (µ) is mean ergodic (resp.

uniformly mean Ergodic ).


A not so simple reduction

Theorem (Herz 1973, Derighetti 1974)

If H is a closed subgroup of G , then the restriction mapping R : Ap(G)→ Ap(H) is

almost an isometry. Its adjoint T∗PMp(H)→ PMp(G) is a multiplicative linear

isometry.

Theorem

If H is a closed subgroup of G and µ ∈ M(G) is supported in H, then λpH(µ) is mean

ergodic (resp. uniformly mean Ergodic ) if and only if λpG (µ) is mean ergodic (resp.

uniformly mean Ergodic ).


A simple case, normal measures

Definition

If µ ∈ M(G), µ∗ is defined so that 〈µ∗, h 〉 =⟨µ, h

⟩, with f (s) = f (s−1).

We say that µ isoperator normal if µ ∗ µ∗ = µ∗ ∗ µ.

λ2(µ)∗ = λ2(µ)∗, so µ is operator normal precisely when λ2 is a normal operator.

Theorem

Let µ ∈ M(G) be operator normal. TFAE:

I λ2(µ) is mean ergodic .

I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem


I λ2(µ) is uniformly mean Ergodic .

I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1 and 1 is not an accumulation point of σ(λ2(µ)).



Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem






Definition


⟩, with f (s) = f (s−1).



Theorem



I r(λ2(µ)) = ‖λ2(µ)‖ ≤ 1.

Theorem





Amenable groups

• Notation: Hµ ≡ closed subgroup generated by supp(µ).

Amenable groups satisfy property Pp for 1 ≤ p <∞: for any ε > 0 and any compact K ⊆ G

there is f ∈ C00(G) with ‖f ‖p = 1 such that ‖x−1 f − f ‖p < ε for x ∈ K .

Theorem (Well-known)

If G is amenable and µ ∈ M(G), then µ(G) is an approximate eigenvalue of λp(µ).

Corollary

Let µ ∈ M+(G) so that Hµ is amenable and 1 < p <∞. TFAE:

I λp(µ) is mean ergodic .

I λp(µ) is weakly mean ergodic .

I ‖µ‖ ≤ 1.

I µ is vaguely ergodic

Theorem

Let µ ∈ M+(G) be operator normal so that Hµ is amenable and let 1 < p <∞. Then λp(µ) is

uniformly mean Ergodic if and only if ‖µ‖ ≤ 1 and 1 is not an accumulation point of σ(λp(µ)).


Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Amenable groups






Corollary




I ‖µ‖ ≤ 1.


Theorem




Fixed points and uniform mean ergodicity

(Dunford) iF T ∈ L(E) satisfies SOT− limn→∞Tn

n= 0. T is uniformly mean Ergodic if and only if either

1 /∈ σ(T ) or 1 is a pole of order 1 of the resolvent of T .

Notation: P(G) ≡ Probability measures on G .

Corollary

If µ ∈ M(G) satisfies SOT− limn→∞λp(µ)

n= 0 and Fix(λp(µ)) = {0}, then λp(µ)

is uniformly mean Ergodic if and only if 1 /∈ σ(λp(µ).

Theorem (Choquet-Deny for Abelian G , Chu for . . . )

Let µ ∈ P(G) . If f ∈ Lp(G), 1 < p <∞ and µ ∗ f = f , then t−1|f| = |f|, for each

t ∈ supp(µ).

Corollary

Let µ ∈ P(G) with noncompact Hµ. If f ∈ Lp(G) and µ ∗ f = f , then f = 0.

Theorem

Let µ ∈ P(G) with noncompact Hµ. Then λp(µ) is uniformly mean Ergodic if and

only if 1 /∈ σ(λp(µ)).







Corollary






t ∈ supp(µ).

Corollary


Theorem









Corollary






t ∈ supp(µ).

Corollary


Theorem









Corollary






t ∈ supp(µ).

Corollary


Theorem









Corollary






t ∈ supp(µ).

Corollary


Theorem




Mise-en-scene of the spectral radius

Lemma (Bekka 2017 for p = 2 and µ ∈ P(G))

Let µ ∈ M+(G) with ‖λp(µ)‖ ≤ 1, 1 < p <∞. If r(λp(µ)) = 1, then 1 ∈ σap(λp(µ)).

Theorem

Let µ ∈ P(G) with noncompact Hµ and let 1 < p <∞. Then λp(µ) is uniformly

mean Ergodic if and only if r(λp(µ)) < 1.

Corollary

Let µ ∈ P(G) with Hµ noncompact but amenable, 1 < p <∞. Then λp(µ) is

uniformly mean Ergodic if and only if ‖µ‖ < 1.





Theorem



Corollary







Theorem



Corollary




Summary

Consider the following properties of µ ∈ M(G), µ > 0 , µ 6= δe :

1 µ is vaguely ergodic.

2 λp(µ) is mean ergodic , 1 < p <∞.

3 λ1(µ) is mean ergodic .

4 λp(µ) is uniformly mean Ergodic , 1 < p <∞.

If Hµ is compact and (‖µ[n]‖)n is bounded, then properties 1–3 are equivalent. If

µ = f dmG , for some f ∈ L1(G), then properties 1–4 are equivalent.

If G is amenable, then properties 1 and 2 are both equivalent to ‖µ‖ ≤ 1. If Hµ is

not compact, (e.g., if G = R for any µ), then 3 never happens and 4 is equivalent to

‖µ‖ = r(λp(µ)) < 1.

Examples


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples

I If G is not amenable, 4 may hold and yet 1 fail. See next slide.


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples

I If (‖µ[n]‖)n is not bounded, then 2 may hold but 1 and 3 fail, even if G is

compact. Take µ = δs − δs2 ∈ M(T), where s3 = 1, s 6= 1.


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples

I If Hµ is compact, then 4 is not equivalent to ‖µ‖ = 1. Let G = T, put x1 = e2πiq

and x2 = e2πit with q ∈ Q and t ∈ R \ Q. Then λ2(δx1 ) is uniformly mean

Ergodic while λ2(δx2 ) is not uniformly mean Ergodic .


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples

I If µ is not positive, λp(µ) may be uniformly mean Ergodic with r(λp(µ)) = 1.

Take µ = δ1 − δ2 ∈ M(Z), then σ(λ2(µ)) ∩ T = {−1} and λ2(µ) is uniformly

mean Ergodic .


Summary










‖µ‖ = r(λp(µ)) < 1.

Examples

I If µ is not positive, it is possible for λq to be uniformly mean Ergodic while λp is

not uniformly mean Ergodic for 1 < p < q <∞. This needs a result of Igari on

functions that operate on B(G).


Next slide. Measures on free groups.

Theorem

Let F (X ) be a free discrete group and let S ⊆ F (X ). Consider µ =∑

s∈S δs ∈ M+(F (X )).

1 (Akemann and Ostrand, 1976) If S is free, then ‖λ(µ)‖ = 2√|S| − 1.

2 (strong Haagerup inequality. Kemp and Speicher, 2007) If S consists of words of

length n in the semigroup generated by X : ‖λ(µ)‖ ≤ e√n + 1‖µ‖2.

Example (Mean ergodic convolution operators of large norm.)

Let G = F (x1, x2, x3), a free group. Put µ = (δx1 + δx2 + δx3 ), and µr = rµ, r > 0.

I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√

2r = ‖λ2(µr )‖. Hence,

I λ2(µ) is uniformly mean Ergodic even if ‖λ2(µ)‖L (L2(G)) > 1.

I Since limn‖µn‖

n=∞, µr is not vaguely ergodic, either.



Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√







Theorem


s∈S δs ∈ M+(F (X )).






I ‖λ2(µnr )‖ ≤(√

3α)n

e√n + 1.

I ‖µnr ‖ = (3r)n.

I ‖λ2(µr )‖ = 2√

2r .

Choose 1√8< α < 1√

3. Then:

I r(λ2(µr )) ≤√

3r < 1 < 2√






Open ends

Concerning mean ergodicity, we don’t know much when G is not amenable or µ is not

positive:

Question

Is it true that λp(µ) is mean ergodic if and only if λp(µ) is power bounded.

This is true for Abelian G and p = 2 and for amenable G with µ > 0.

If G = F (x1, x2, x3), a free group, we don’t know if α(δx1 + δx2 + δx3 ) ∈ M(G), with1√8< α < 1√

3is power bounded.

Question

Suppose that λp(µ) is mean ergodic for some 1 < p <∞, is λq(µ) mean

ergodic for all 1 < q <∞?

For weak mean ergodicity, yes. If µ ∈ P(G) and Hµ is compact, yes.


Open ends


positive:

Question




3is power bounded.

Question





Open ends


positive:

Question




3is power bounded.

Question





Open ends


positive:

Question




3is power bounded.

Question





Open ends


positive:

Question




3is power bounded.

Question





Open ends


positive:

Question




3is power bounded.

Question



For weak mean ergodicity, yes.

If µ ∈ P(G) and Hµ is compact, yes.


Open ends


positive:

Question




3is power bounded.

Question





Open ends


positive:

Question




3is power bounded.

Question





Open ends (ctd)

Consider the following conditions:

1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .

3 1 is not an accumulation point of σ(λp(µ)).

Clearly,

(1) =⇒ (2) =⇒ uniformly mean Ergodic =⇒ (3).

Question

Let µ ∈ P(G). Under which conditions is it true that conditions (1), (2) or (3)

characterize uniform mean ergodicity?

I (3) works when G is Abelian and p = 2 or when Hµ is amenable.

I (2) works when Hµ is not compact.

I If Hµ is amenable and noncompact, then (1) works.


Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







Open ends (ctd)


1 ‖µ‖ < 1.

2 r(λ(µ)) < 1 =⇒ .


Clearly,


Question







THANK YOU FOR YOUR ATTENTION. . .

. . . AND PATTIENCE


THANK YOU FOR YOUR ATTENTION. . .

. . . AND PATTIENCE


Documents

Ergodic properties of convolution operatorsbanach2019/pdf/Galindo.pdf · Jorge Galindo Joint work with Enrique Jord a, Universitat Polit ecnica de Val encia UPV (Spain). BANACH ALGEBRAS