CUBO A Mathematical Journal
Vol.21, No¯ 01, (37–48). April 2019

http: // dx. doi. org/ 10. 4067/ S0719-06462019000100037

Commutator criteria for strong mixing II.
More general and simpler

S. Richard
1

Graduate school of mathematics,

Nagoya University,

Chikusa-ku, Nagoya 464-8602, Japan

richard@math.nagoya-u.ac.jp

R. Tiedra de Aldecoa
2

Facultad de Matemáticas,

Pontificia Universidad Católica de Chile,

Av. Vicuña Mackenna 4860, Santiago, Chile

rtiedra@mat.puc.cl

ABSTRACT

We present a new criterion, based on commutator methods, for the strong mixing

property of unitary representations of topological groups equipped with a proper length

function. Our result generalises and unifies recent results on the strong mixing property

of discrete flows {UN}N∈Z and continuous flows {e
−itH}t∈R induced by unitary operators

U and self-adjoint operators H in a Hilbert space. As an application, we present a

short alternative proof (not using convolutions) of the strong mixing property of the

left regular representation of σ-compact locally compact groups.

1Supported by the grantTopological invariants through scattering theory and noncommutative geometry from
Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from
Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre
1918, F-69622 Villeurbanne cedex, France.

2Supported by the Chilean Fondecyt Grant 1170008.

http://dx.doi.org/10.4067/S0719-06462019000100037


38 S. Richard and R. Tiedra de Aldecoa CUBO
21, 1 (2019)

RESUMEN

Presentamos un nuevo criterio, basado en métodos de conmutadores, para la propiedad

de mezcla fuerte de representaciones unitarias de grupos topológicos dotados de una

función de longitud propia. Nuestro resultado generaliza y unifica resultados recientes

acerca de la propiedad de mezcla fuerte de flujos discretos {UN}N∈Z y flujos continuos

{e−itH}t∈R inducidos por operadores unitarios U y operadores autoadjuntos H en un

espacio de Hilbert. Como aplicación, presentamos una demostración corta alternativa

(sin usar convoluciones) de la propiedad de mezcla fuerte de la representación regular

de grupos localmente compactos σ-compactos.

Keywords and Phrases: Strong mixing, unitary representations, commutator methods.

2010 AMS Mathematics Subject Classification: 22D10, 37A25, 58J51, 81Q10.



CUBO
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Commutator criteria for strong mixing II. . . . 39

1 Introduction

In the recent paper [14], itself motivated by the previous papers [8, 12, 13, 15], it has been shown

that commutator methods for unitary and self-adjoint operators can be used to establish strong

mixing. The main results of [14] are the following two commutator criteria for strong mixing.

First, given a unitary operator U in a Hilbert space H, assume there exists an auxiliary self-

adjoint operator A in H such that the commutators [A,UN] exist and are bounded in some precise

sense, and such that the strong limit

D1 := s-lim
N→∞

1

N
[A,UN]U−N (1.1)

exists. Then, the discrete flow {UN}N∈Z is strongly mixing in ker(D1)
⊥. Second, given a self-

adjoint operator H in H, assume there exists an auxiliary self-adjoint operator A in H such that

the commutators [A,e−itH] exist and are bounded in some precise sense, and such that the strong

limit

D2 := s-lim
t→∞

1

t
[A,e−itH] eitH (1.2)

exists. Then, the continuous flow {e−itH}t∈R is strongly mixing in ker(D2)
⊥. These criteria were

then applied to skew products of compact Lie groups, Furstenberg-type transformations, time

changes of horocycle flows and adjacency operators on graphs.

The purpose of this note is to unify these two commutator criteria into a single, more general,

commutator criterion for strong mixing of unitary representations of topological groups, and also

to remove an unnecessary invariance assumption made in [14].

Our main result is the following. We consider a topological group X equipped with a proper

length function ℓ : X → R+, a unitary representation U : X → U (H), and a net {xj}j∈J in X

with xj → ∞ (see Section 2 for precise definitions). Also, we assume there exists an auxiliary

self-adjoint operator A in H such that the commutators [A,U(xj)] exist and are bounded in some

precise sense, and such that the strong limit

D := s-lim
j

1

ℓ(xj)
[A,U(xj)]U(xj)

−1 (1.3)

exists. Then, under these assumptions we show that the unitary representation U is strongly

mixing in ker(D)⊥ along the net {xj}j∈J (Theorem 2.3). As a corollary, we obtain criteria for strong

mixing in the cases of unitary representations of compactly generated locally compact Hausdorff

groups (Corollary 2.5) and the Euclidean group Rd (Corollary 2.7). These results generalise the

commutator criteria of [14] for the strong mixing of discrete and continuous flows, as well as

the strong limit (1.3) generalises the strong limits (1.1) and (1.2) (see Remarks 2.6 and 2.8). To

conclude, we present in Example 2.9 an application which was not possible to cover with the results

of [14]: a short alternative proof (not using convolutions) of the strong mixing property of the left

regular representation of σ-compact locally compact Hausdorff groups.



40 S. Richard and R. Tiedra de Aldecoa CUBO
21, 1 (2019)

We refer the reader to [4, 6, 9, 10, 11, 16] for references on strong mixing properties of unitary

representations of groups.

2 Commutator criteria for strong mixing

We start with a short review of basic facts on commutators of operators and regularity classes

associated with them. We refer to [1, Chap. 5-6] for more details.

Let H be an arbitrary Hilbert space with scalar product 〈·, ·〉 antilinear in the first argument,

denote by B(H) the set of bounded linear operators on H, and write ‖ · ‖ both for the norm on

H and the norm on B(H). Let A be a self-adjoint operator in H with domain D(A), and take

S ∈ B(H). For any k ∈ N, we say that S belongs to Ck(A), with notation S ∈ Ck(A), if the map

R ∋ t 7→ e−itA SeitA ∈ B(H) (2.1)

is strongly of class Ck. In the case k = 1, one has S ∈ C1(A) if and only if the quadratic form

D(A) ∋ ϕ 7→
〈
ϕ,iSAϕ

〉
−
〈
Aϕ,iSϕ

〉
∈ C

is continuous for the topology induced by H on D(A). We denote by [iS,A] the bounded operator

associated with the continuous extension of this form, or equivalently the strong derivative of the

map (2.1) at t = 0. Moreover, if we set Aε := (iε)
−1(eiεA −1) for ε ∈ R \ {0}, we have (see [1,

Lemma 6.2.3(a)]):

s-lim
εց0

[iS,Aε] = [iS,A]. (2.2)

Now, if H is a self-adjoint operator in H with domain D(H) and spectrum σ(H), we say that

H is of class Ck(A) if (H −z)−1 ∈ Ck(A) for some z ∈ C \σ(H). In particular, H is of class C1(A)

if and only if the quadratic form

D(A) ∋ ϕ 7→
〈
ϕ,(H − z)−1Aϕ

〉
−
〈
Aϕ,(H − z)−1ϕ

〉
∈ C

extends continuously to a bounded form with corresponding operator denoted by [(H − z)−1,A] ∈

B(H). In such a case, the set D(H) ∩ D(A) is a core for H and the quadratic form

D(H) ∩ D(A) ∋ ϕ 7→
〈
Hϕ,Aϕ

〉
−
〈
Aϕ,Hϕ

〉
∈ C

is continuous in the topology of D(H) (see [1, Thm. 6.2.10(b)]). This form then extends uniquely

to a continuous quadratic form on D(H) which can be identified with a continuous operator [H,A]

from D(H) to the adjoint space D(H)∗. In addition, the following relation holds in B(H) (see [1,

Thm. 6.2.10(b)]):

[(H − z)−1,A] = −(H − z)−1[H,A](H − z)−1. (2.3)



CUBO
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Commutator criteria for strong mixing II. . . . 41

With this, we can now present our first result, which is at the root of the new commutator

criterion for strong mixing. For it, we recall that a net {xj}j∈J in a topological space X diverges

to infinity, with notation xj → ∞, if {xj}j∈J has no limit point in X. This implies that for each

compact set K ⊂ X, there exists jK ∈ J such that xj /∈ K for j ≥ jK. In particular, X is not compact.

We also fix the notations U (H) for the set of unitary operators on H and R+ := [0,∞).

Proposition 2.1. Let {Uj}j∈J be a net in U (H), let {ℓj}j∈J ⊂ R+ satisfy ℓj → ∞, assume there

exists a self-adjoint operator A in H such that Uj ∈ C
1(A) for each j ∈ J, and suppose that the

strong limit

D := s-lim
j

1

ℓj
[A,Uj]U

−1
j

exists. Then, limj
〈
ϕ,Ujψ

〉
= 0 for all ϕ ∈ ker(D)⊥ and ψ ∈ H.

Before the proof, we note that for j ∈ J large enough (so that ℓj 6= 0) the operators
1
ℓj
[A,Uj]U

−1
j

are well-defined, bounded and self-adjoint. Therefore, their strong limit D is also bounded and

self-adjoint.

Proof. Let ϕ = Dϕ̃ ∈ DD(A) and ψ ∈ D(A), take j ∈ J large enough, and set

Dj :=
1

ℓj
[A,Uj]U

−1
j .

Since Uj and U
−1
j belong to C

1(A) (see [1, Prop. 5.1.6(a)]), both Ujψ and U
−1
j ϕ̃ belong to D(A).

Thus,

∣∣〈ϕ,Ujψ
〉∣∣

=
∣∣〈(D − Dj)ϕ̃,Ujψ

〉
+
〈
Djϕ̃,Ujψ

〉∣∣

≤
∥∥(D − Dj)ϕ̃

∥∥‖ψ‖ +
1

ℓj

∣∣〈[A,Uj
]
U−1j ϕ̃,Ujψ

〉∣∣

≤
∥∥(D − Dj)ϕ̃

∥∥‖ψ‖ +
1

ℓj

∣∣〈Aϕ̃,Ujψ
〉∣∣ +

1

ℓj

∣∣〈UjAU−1j ϕ̃,Ujψ
〉∣∣

≤
∥∥(D − Dj)ϕ̃

∥∥‖ψ‖ +
1

ℓj

∥∥Aϕ̃
∥∥‖ψ‖ +

1

ℓj

∥∥ϕ̃
∥∥‖Aψ‖.

Since D = s-limj Dj and ℓj → ∞, we infer that limj
〈
ϕ,Ujψ

〉
= 0, and thus the claim follows by

the density of DD(A) in DH = ker(D)⊥ and the density of D(A) in H.

In the sequel, we assume that the unitary operators Uj are given by a unitary representation

of a topological group X. We also assume that the scalars ℓj are given by a proper length function

on X, that is, a function ℓ : X → R+ satisfying the following properties (e denotes the identity of

X):

(L1) ℓ(e) = 0,



42 S. Richard and R. Tiedra de Aldecoa CUBO
21, 1 (2019)

(L2) ℓ(x−1) = ℓ(x) for all x ∈ X,

(L3) ℓ(xy) ≤ ℓ(x) + ℓ(y) for all x,y ∈ X,

(L4) if K ⊂ R+ is compact, then ℓ
−1(K) ⊂ X is relatively compact.

Remark 2.2 (Topological groups with a proper left-invariant pseudo-metric). Let X be a Haus-

dorff topological group equipped with a proper left-invariant pseudo-metric d : X × X → R+ (see

[7, Def. 2.A.5 & 2.A.7]). Then, simple calculations show that the associated length function

ℓ : X → R+ given by ℓ(x) := d(e,x) satisfies the properties (L1)-(L4) above. Examples of groups

admitting a proper left-invariant pseudo-metric are σ-compact locally compact Hausdorff groups [7,

Prop. 4.A.2], as for instance compactly generated locally compact Hausdorff groups with the word

metric [7, Prop. 4.B.4(2)].

The next theorem provides a general commutator criterion for the strong mixing property of

a unitary representation of a topological group. Before stating it, we recall that if a topological

group X is equipped with a proper length function ℓ, and if {xj}j∈J is a net in X with xj → ∞, then

ℓ(xj) → ∞ (this can be shown by absurd using the property (L4) above).

Theorem 2.3 (Topological groups). Let X be a topological group equipped with a proper length

function ℓ, let U : X → U (H) be a unitary representation of X, let {xj}j∈J be a net in X with

xj → ∞, assume there exists a self-adjoint operator A in H such that U(xj) ∈ C
1(A) for each

j ∈ J, and suppose that the strong limit

D := s-lim
j

1

ℓ(xj)
[A,U(xj)]U(xj)

−1 (2.4)

exists. Then,

(a) limj
〈
ϕ,U(xj)ψ

〉
= 0 for all ϕ ∈ ker(D)⊥ and ψ ∈ H,

(b) U has no nontrivial finite-dimensional unitary subrepresentation in ker(D)⊥.

Proof. The claim (a) follows from Proposition 2.1 and the fact that ℓ(xj) → ∞. The claim (b)

follows from (a) and the fact that matrix coefficients of finite-dimensional unitary representations

of a group do not vanish at infinity (see for instance [3, Rem. 2.15(iii)]).

Remark 2.4. (i) The result of Theorem 2.3(a) amounts to the strong mixing property of the

unitary representation U in ker(D)⊥ along the net {xj}j∈J, as mentioned in the introduction. If the

strong limit (2.4) exists for all nets {xj}j∈J with xj → ∞, then Theorem 2.3(a) implies the usual

strong mixing property of the unitary representation U in ker(D)⊥.

(ii) One can easily see that Theorem 2.3 remains true if the scalars ℓ(xj) in (2.4) are replaced

by (f ◦ ℓ)(xj), with f : R+ → R+ any proper function. For simplicity, we decided to present only

the case f = idR+, but we note this additional freedom might be useful in applications.



CUBO
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Commutator criteria for strong mixing II. . . . 43

Theorem 2.3 and Remark 2.2 imply the following result in the particular case of a compactly

generated locally compact group X :

Corollary 2.5 (Compactly generated locally compact groups). Let X be a compactly generated

locally compact Hausdorff group with generating set Y and word length function ℓ, let U : X → U (H)

be a unitary representation of X, let {xj}j∈J be a net in X with xj → ∞, assume there exists a self-

adjoint operator A in H such that U(y) ∈ C1(A) for each y ∈ Y, and suppose that the strong

limit

D := s-lim
j

1

ℓ(xj)
[A,U(xj)]U(xj)

−1 (2.5)

exists. Then,

(a) limj
〈
ϕ,U(xj)ψ

〉
= 0 for all ϕ ∈ ker(D)⊥ and ψ ∈ H,

(b) U has no nontrivial finite-dimensional unitary subrepresentation in ker(D)⊥.

Proof. In order to apply Theorem 2.3, we first note from Remark 2.2 that the word length function

ℓ is a proper length function. Second, we note that X =
⋃

n≥1
(Y ∪ Y−1)n. Therefore, for each

x ∈ X there exist n ≥ 1, y1, . . . ,yn ∈ Y and m1, . . . ,mn ∈ {±1} such that x = y
m1
1 · · ·y

mn
n . Thus,

U(x) = U
(
y
m1
1

· · ·ymnn
)
= U(y1)

m1 · · ·U(yn)
mn,

and it follows from the inclusions U(y1), . . . ,U(yn) ∈ C
1(A) and standard results on commutator

methods [1, Prop. 5.1.5 & 5.1.6(a)] that U(x) ∈ C1(A). Thus, we have U(xj) ∈ C
1(A) for each

j ∈ J, and the commutators [A,U(xj)] appearing in (2.5) make sense. So, we can apply Theorem

2.3 to conclude.

Remark 2.6. Corollary 2.5 is a generalisation of [14, Thm. 3.1] to the case of unitary represen-

tations of compactly generated locally compact Hausdorff groups. Indeed, if we let X be the additive

group Z with generating element 1, take the trivial net {xj = j}j∈N∗ = {N | N ∈ N
∗}, and set

U := U(1) in Corollary 2.5, then the strong limit (2.5) reduces to

D = s-lim
N→∞

1

N

[
A,UN

]
U−N = s-lim

N→∞

1

N

N−1∑

n=0

Un
(
[A,U]U−1

)
U−n,

which is the strong limit appearing in [14, Thm. 3.1]. In Corollary 2.5 we also removed the

unnecessary invariance assumption η(D)D(A) ⊂ D(A) for each η ∈ C∞
c
(R \ {0}). So, the strong

mixing properties for skew products and Furstenberg-type transformations established in [14, Sec. 3]

and [5, Sec. 3] can be obtained more directly using Corollary 2.5.

In the next corollary we consider the case of a strongly continuous unitary representation

U : Rd → U (H) of the Euclidean group Rd, d ≥ 1. In such a case Stone’s theorem implies the



44 S. Richard and R. Tiedra de Aldecoa CUBO
21, 1 (2019)

existence of a family of mutually commuting self-adjoint operators H1, . . . ,Hd such that U(x) =

e−i
∑

d
k=1

xkHk for each x = (x1, . . . ,xd) ∈ R
d. Therefore, we give a criterion for strong mixing in

terms of the operators H1, . . . ,Hd. We use the shorthand notations

H := (H1, . . . ,Hd), Π(H) := (H1 + i)
−1 · · · (Hd + i)

−1 and x · H :=

d∑

k=1

xkHk.

Corollary 2.7 (Euclidean group Rd). Let Rd, d ≥ 1, be the Euclidean group with Euclidean

length function ℓ, let U : Rd → U (H) be a strongly continuous unitary representation of Rd, let

{xj}j∈J be a net in R
d with xj → ∞, assume there exists a self-adjoint operator A in H such that

(Hk − i)
−1 ∈ C1(A) for each k ∈ {1, . . . ,d}, and suppose that the strong limit

D := s-lim
j

1

ℓ(xj)

∫1

0

ds e−is(xj·H) Π(H)
[
i(xj · H),A

]
Π(H)∗ eis(xj·H) (2.6)

exists. Then,

(a) limj
〈
ϕ,U(xj)ψ

〉
= 0 for all ϕ ∈ ker(D)⊥ and ψ ∈ H,

(b) U has no nontrivial finite-dimensional unitary subrepresentation in ker(D)⊥.

Proof. The proof consists in applying Theorem 2.3 with A replaced by a new operator à that we

now define.

The inclusions (H1 − i)
−1, . . . ,(Hd − i)

−1 ∈ C1(A) and the standard result on commutator

methods [1, Prop. 5.1.5] imply that Π(H)∗ ∈ C1(A). So, we have Π(H)∗ D(A) ⊂ D(A), and the

operator

Ãϕ := Π(H)AΠ(H)∗ϕ, ϕ ∈ D(A),

is essentially self-adjoint (see [1, Lemma 7.2.15]). Take ϕ ∈ D(A) and j0 ∈ J such that ℓ(xj) > 0

for all j ≥ j0, and define for ε ∈ R \ {0} the operator Aε := (iε)
−1(eiεA −1). Then, we have

〈
Ãϕ,U(xj)ϕ

〉
−
〈
ϕ,U(xj)Ãϕ

〉

= lim
εց0

(〈
ϕ,Π(H)AεΠ(H)

∗ e−i(xj·H) ϕ
〉
−
〈
ϕ,e−i(xj·H) Π(H)AεΠ(H)

∗ϕ
〉)

= lim
εց0

∫ℓ(xj)

0

dq
d

dq

〈
ϕ,ei(q−ℓ(xj))(xj·H)/ℓ(xj) Π(H)AεΠ(H)

∗ e−iq(xj·H)/ℓ(xj) ϕ
〉

=
1

ℓ(xj)
lim
εց0

∫ℓ(xj)

0

dq
〈
ϕ,ei(q−ℓ(xj))(xj·H)/ℓ(xj) Π(H)

[
i(xj · H),Aε

]
Π(H)∗ e−iq(xj·H)/ℓ(xj) ϕ

〉
.

(2.7)

But, (H1 − i)
−1, . . . ,(Hd − i)

−1 ∈ C1(A). Therefore, (2.2) and (2.3) imply that

s-lim
εց0

Π(H)
[
i(xj · H),Aε

]
Π(H)∗ = Π(H)

[
i(xj · H),A

]
Π(H)∗,



CUBO
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Commutator criteria for strong mixing II. . . . 45

and we can exchange the limit and the integral in (2.7) to obtain

〈
Ãϕ,U(xj)ϕ

〉
−
〈
ϕ,U(xj)Ãϕ

〉

=
1

ℓ(xj)

∫ℓ(xj)

0

dq
〈
ϕ,ei(q−ℓ(xj))(xj·H)/ℓ(xj) Π(H)

[
i(xj · H),A

]
Π(H)∗ e−iq(xj·H)/ℓ(xj) ϕ

〉

=
1

ℓ(xj)

∫ℓ(xj)

0

dr
〈
ϕ,e−ir(xj·H)/ℓ(xj) Π(H)

[
i(xj · H),A

]
Π(H)∗ ei(r−ℓ(xj))(xj·H)/ℓ(xj) ϕ

〉

=

∫1

0

ds
〈
ϕ,e−is(xj·H) Π(H)

[
i(xj · H),A

]
Π(H)∗ eis(xj·H) U(xj)ϕ

〉

=
〈
ϕ,ℓ(xj)DjU(xj)ϕ

〉

with

Dj :=
1

ℓ(xj)

∫1

0

ds e−is(xj·H) Π(H)
[
i(xj · H),A

]
Π(H)∗ eis(xj·H) .

Since D(A) is a core for Ã, this implies that U(xj) ∈ C
1(Ã) with

[
Ã,U(xj)

]
= ℓ(xj)DjU(xj).

Therefore, we have

Dj =
1

ℓ(xj)

[
Ã,U(xj)

]
U(xj)

−1,

and all the assumptions of Theorem 2.3 are satisfied with A replaced by Ã.

Remark 2.8. Corollary 2.7 is a generalisation of [14, Thm. 4.1] to the case of strongly continuous

unitary representations of Rd for an arbitrary d ≥ 1. Indeed, if we set d = 1, write H for H1, and

take the trivial net {xj = j}j∈(0,∞) = {t | t > 0} in Corollary 2.7, then the strong limit (2.6) reduces

to

D = s-lim
t→∞

1

t

∫1

0

ds e−is(t·H)(H + i)−1
[
itH,A

]
(H − i)−1 eis(t·H)

= s-lim
t→∞

1

t

∫t

0

ds e−isH(H + i)−1
[
iH,A

]
(H − i)−1 eisH,

which is (up to a sign) the strong limit appearing in [14, Thm. 4.1]. In Corollary 2.7, we also

removed the unnecessary invariance assumption η(D)D(A) ⊂ D(A) for each η ∈ C∞
c
(R \ {0}).

So, the strong mixing properties for adjacency operators, time changes of horocycle flows, etc.,

established in [14, Sec. 4] can be obtained more directly using Corollary 2.7.

To conclude, we add to the list of examples presented in [14] an application which was not

possible to cover with the results of [14]. It is a short alternative proof, not using convolutions,

of the strong mixing property of the left regular representation of σ-compact locally compact

Hausdorff groups (see for instance [2, Sec. C.4] for the proof using convolutions):



46 S. Richard and R. Tiedra de Aldecoa CUBO
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Example 2.9 (Left regular representation). Let X be a σ-compact locally compact Hausdorff group

with left Haar measure µ and proper length function ℓ (see Remark 2.2). Let D ⊂ H be the set of

functions X → C with compact support, and let U : X → U (H) be the left regular representation of

X on H := L2(X,µ) given by

U(x)ϕ := ϕ(x−1 ·), x ∈ X, ϕ ∈ H.

Let finally A be the maximal multiplication operator in H given by

Aϕ := ℓϕ ≡ ℓ(·)ϕ, ϕ ∈ D(A) :=
{
ϕ ∈ H | ‖ℓϕ‖ < ∞

}
.

For ϕ ∈ D and x ∈ X, one has

AU(x)ϕ − U(x)Aϕ =
(
ℓ(·) − ℓ(x−1 ·)

)
U(x)ϕ.

Furthermore, the properties (L2)-(L3) of a length function imply that

∣∣(ℓ(·) − ℓ(x−1 ·)
)∣∣ ≤ ℓ(x). (2.8)

Therefore, since D is dense in D(A), it follows that U(x) ∈ C1(A) with

[A,U(x)]U(x)−1 = ℓ(·) − ℓ(x−1 ·).

Now, we take {xj}j∈J a net in X with xj → ∞, and show that

D := s-lim
j

1

ℓ(xj)
[A,U(xj)]U(xj)

−1 = −1. (2.9)

For this, we first note that for ϕ ∈ H we have

(
1

ℓ(xj)
[A,U(xj)]U(xj)

−1 + 1

)
ϕ =

ℓ(·) − ℓ(x−1
j

·) + ℓ(xj)

ℓ(xj)
ϕ.

Next, we note that (2.8) implies that

∣∣∣∣∣
ℓ(·) − ℓ(x−1

j
·) + ℓ(xj)

ℓ(xj)
ϕ

∣∣∣∣∣

2

≤ 4|ϕ|2 ∈ L1(X,µ),

and that the properties (L2)-(L3) imply that

lim
j

∣∣∣∣∣
ℓ(·) − ℓ(x−1

j
·) + ℓ(xj)

ℓ(xj)
ϕ

∣∣∣∣∣

2

≤ lim
j

∣∣∣∣
2ℓ(·)

ℓ(xj)
ϕ

∣∣∣∣
2

= 0 µ-almost everywhere.

Therefore, we can apply Lebesgue dominated convergence theorem to get the equality

s-lim
j

(
1

ℓ(xj)
[A,U(xj)]U(xj)

−1 + 1

)
ϕ = 0,

which proves (2.9). So, Theorem 2.3 applies with D = −1, and thus limj
〈
ϕ,U(xj)ψ

〉
= 0 for all

ϕ,ψ ∈ H.



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Commutator criteria for strong mixing II. . . . 47

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	Introduction
	Commutator criteria for strong mixing