Note on isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

C O L L O Q U I U M

M A T H E M A T I C U M

VOL.110 2008 NO.1

NOTE ON THEISOMORPHISM PROBLEM FOR WEIGHTED UNITARY OPERATORS ASSOCIATEDWITH A

NONSINGULARAUTOMORPHISM

BY

K.FRCZEK(Toru«andWarszawa)andM.WYSOKI‹SKA(Toru«)

Abstra t. Wegivea negativeanswerto aquestion putby Nadkarni: Let

S

be an ergodi , onservativeandnonsingularautomorphismon

( e

X, B

e

X

, m)

.Considerthe asso i-atedunitaryoperatorson

L

2

( e

X, B

X

e

, m)

givenby

U

e

S

f =

p

d(m ◦ S)/dm·(f ◦S)

and

ϕ· e

U

S

, where

ϕ

isa o y leofmodulus one.Doesspe tralisomorphismof thesetwooperators implythat

ϕ

isa oboundary?Toansweritnegatively,wegiveanexamplewhi harises fromaninnitemeasure-preservingtransformationwith ountableLebesguespe trum.

1.Question. Let

( e

X, B

e

X

, m)

beastandardprobabilityBorelspa eand let

S : ( e

X, B

e

X

, m) → ( e

X, B

X

e

, m)

be an automorphism nonsingular with respe t to themeasure

m

(i.e.

m ◦ S ≡ m

).Given a o y le

ϕ : e

X → S

1

we

dene two unitaryoperatorsa ting on

L

2

_{( e}

_{X, B}

e

X

, m)

:

e

U

S

f =

p

d(m ◦ S)/dm · (f ◦ S),

V

e

ϕ,S

f = ϕ · e

U

S

f.

If

ϕ

is a oboundary then

U

e

S

and

V

e

ϕ,S

are spe trally isomorphi . Con-versely,if

S

isadditionallyergodi andpreservesthemeasure

m

thenspe tral isomorphismof

U

e

S

and

V

e

ϕ,S

impliesthat

ϕ

isa oboundary.

ThefollowingquestionwasputbyM.G.Nadkarniin[4 ℄:Let

S

bean er-godi and onservativetransformation,nonsingularwithrespe tto

m

.Does spe tralisomorphismof

U

e

S

and

V

e

ϕ,S

imply that

ϕ

isa oboundary?

Theanswertothisquestionisnegative.Firstnoti ethatitissu ientto givea ounterexampleinthefamilyofinnitemeasure-preserving automor-phisms. Namely, take an ergodi onservative innite measure-preserving automorphism

S : ( e

X, B

e

X

, ̺) → ( e

X, B

X

e

, ̺)

(

̺

is

σ

-nite) with ountable Lebesguespe trumanda o y le

ϕ : e

X → S

1

whi hisnota oboundaryand forwhi htheKoopmanoperator

U

S

f = f ◦S

,

f ∈ L

2

_{( e}

_{X, B}

e

X

, ̺)

,andthe

or-2000Mathemati sSubje tClassi ation:37A40,37A30.

Keywords and phrases:weighted unitary operators, ountable Lebesgue spe trum, ylindri altransformations.

Resear h partially supported by Marie Curie Transfer of Knowledge program, proje tMTKD-CT-2005-030042(TODEQ).

[201℄

202

K.FRCZEKANDM.WYSOKI‹SKA

respondingweightedoperator

V

ϕ,S

= ϕ · U

S

arespe trallyisomorphi .Then takeanarbitrarynitemeasure

m

on

B

e

X

equivalentto

̺

,i.e.

dm = F d̺

fora positivefun tion

F ∈ L

1

_{( e}

_{X, B}

e

X

, ̺)

.Then

S

isergodi , onservativeand non-singularwith respe tto

m

andobviously

ϕ

isstill not a oboundary. More-over, spe tral isomorphismof

U

S

and

V

ϕ,S

implies spe tralisomorphism of

e

U

S

and

V

e

ϕ,S

.Indeed,the operator

W : L

2

_{( e}

_{X, B}

e

X

, m) → L

2

_{( e}

_{X, B}

e

X

, ̺)

given by

W f =

√

F · f

establishes spe tralisomorphism between

U

e

S

and

U

S

, as well as between

V

e

ϕ,S

and

V

ϕ,S

. Hen e, any

S

and

ϕ

satisfying the above assumptions give anegative answer tothequestion onsidered.

Re allthatsomeexamplesofergodi , onservative andinnite measure-preservingtransformations with ountable Lebesguespe trum areprovided byinnite

K

-automorphisms(see[5 ℄).Anexampleofsu hanautomorphism was given in [1 ℄ (see Example 1). We next show that for any ergodi and onservative

S

with ountable Lebesgue spe trum we an nd a o y le

ϕ

withtherequired properties.

Wewillneedsomefa tsabout

L

∞

-eigenvaluesofanonsingular automor-phism. Re all that a omplex number

γ

is said to be an

L

∞

-eigenvalue of an ergodi and nonsingular automorphism

S

if there exists a nonzero fun -tion

g ∈ L

∞

_{( e}

_{X, B}

e

X

, m)

su h that

g(Sx) = γg(x) m

-a.e. The group of all

L

∞

-eigenvalues,denotedby

e(S)

,isasubgroupofthe ir legroup

S

1

. More-over,itwasprovedin[2℄thatif

S

is onservativeandergodi then

e(S) ( S

1

.

Therefore if

S : ( e

X, B

e

X

, m) → ( e

X, B

X

e

, m)

is onservative and ergodi then the onstant o y le

ϕ : e

X → S

1

given by

ϕ ≡ a ∈ S

1

_{\ e(S)}

is not a

oboundary.Indeed,if

a = ξ(Sx)/ξ(x)

forameasurablefun tion

ξ : e

X → S

1

then

a ∈ e(S)

(sin e

ξ ∈ L

∞

_{( e}

_{X, B}

e

X

, m)

).Moreover, if the operator

U

S

has ountable Lebesgue spe trum then the spe trum of

V

a,S

is also ountable Lebesgue, hen e

U

S

and

V

a,S

arespe trallyisomorphi . Indeed, foran arbi-trary

g ∈ L

2

_{( e}

_{X, B}

e

X

, m)

wegetthefollowing onne tionbetween itsspe tral measures withrespe tto both operators onsidered:

σ

g,V

a,S

= δ

a

∗ σ

g,S

,and the y li subspa esgeneratedby

g

withrespe ttobothoperators oin ide. Now the question is whether it is possible to nd a non onstant, for instan ewithintegral zero, o y le

ϕ

withtherequiredproperties.

2. Remarks on examples arising from ylindri al transforma-tions. We now show another way of nding innite measure-preserving transformations with ountable Lebesgue spe trum. This approa h gives some examples of su h systems whi h have zero entropy, unlike

K

-auto-morphisms.Namely,wewill studythe so- alled ylindri al transformations. For agivenautomorphism

T

ofastandard probabilityBorelspa e

(X, B, µ)

and a real o y le

f : X → R

we dene a ylindri al automorphism of

ISOMORPHISMPROBLEMFORWEIGHTEDOPERATORS

203

(X × R, B ⊗ B

R

, µ ⊗ λ

R

)

(

λ

R

stands for Lebesguemeasure) by

T

f

(x, r) = (T x, r + f (x)).

Su h an automorphism preserves the (innite) measure

µ ⊗ λ

R

. Moreover, sin e

µ⊗λ

R

isnonatomi ,theergodi ityof

T

f

impliesits onservativity.The maximalspe traltypeof

T

f

isstri tly onne ted withthemaximalspe tral types

σ

V

e

2

_{πicf ,T}

ofthe operators

V

e

2

πicf

,T

: L

2

_{(X, B, µ) → L}

2

_{(X, B, µ)}

,

c 6= 0

, and with the maximal spe tral type

σ

Σ

of the translation

Σ = (Σ

t

)

t∈R

on

(R, B

R

, λ

R

)

given by

Σ

t

r = r + t

.Re all that the

R

-a tion

Σ

has Lebesgue spe trum on

L

2

_{(R, B}

R

, λ

R

)

. Moreover, similarly to [3 ℄ (see also Lemmas 1 and 2in[6 ℄), thefollowing result holds.

Lemma1. Themaximalspe traltypeof

T

f

on

L

2

_{(X ×R, B⊗B}

R

, µ⊗λ

R

)

isgiven by

σ

T

f

≡

\

R

σ

V

_e

2

_{πicf ,T}

dσ

Σ

(c),

and for an arbitrary

h ∈ L

2

_{(R, λ}

R

)

the maximal spe tral type of

T

f

on

L

2

_{(X, µ) ⊗ Z(h)}

,where

Z(h) = span{h ◦ Σ

t

: t ∈ R}

, isgiven by

σ

T

f

_|L

2

(X,µ)⊗Z(h)

≡

\

R

σ

V

_e

2

_{πicf ,T}

dσ

h,Σ

(c).

Now,ifwe ouldnd

f

and

T

su hthatea h

V

e

2πicf

_,T

,

c 6= 0

,hasLebesgue spe trumon

L

2

_{(X, B, µ)}

thenbyLemma1,

T

f

has ountableLebesgue spe -trum on

L

2

_{(X × R, B ⊗ B}

R

, µ ⊗ λ

R

)

.

Suppose that

T x = x + α

is an irrational rotation on

(R/Z, µ)

, where

µ

stands for Lebesgue measure on

R/Z

and the sequen e of arithmeti al means of partialquotients of

α

isbounded.Consider o y les

f : R/Z → R

with pie ewise ontinuous se ond derivative su h that

f

′

has nitely many dis ontinuity points for whi h the one-sided limits exist with at least one equal to innity and

f

′

_{(x) > 0}

for all

x ∈ R/Z

. It was shown in [6 ℄ that for su hfun tions

f

the operators

V

e

2

πicf

_,T

,

c 6= 0

,have Lebesguespe trum. However,we do not knowif

T

f

isergodi for this lassof o y les.

Nevertheless,weareabletogiveanexampleofergodi ylindri al trans-formationwith ountableLebesguespe trum.Namely,takeas

T

aGaussian automorphismon

(R

Z

_{, B}

R

Z

, µ

G

)

.Assumethatthepro ess

(Π

n

)

n∈Z

of proje -tionson the

n

th oordinateisindependentand thedistribution

ν

of

Π

0

isa enteredGaussiandistributionon

R

(soinfa t

T

isaBernoullisystem).Put

f = Π

0

: R

Z

→ R

.Then

T

Π

0

preservesthemeasure

µ

G

⊗ λ

R

andis ergodi , be ause

T

Π

0

is a random walk on

R

with transition probability determined by

ν

.Moreover,forea h

c 6= 0

theoperator

V

e

2

πicΠ0

_,T

hasLebesguespe trum on

L

2

_(R

Z

_{, B}

204

K.FRCZEKANDM.WYSOKI‹SKA

WethankProfessorM.Lema« zykfor manyremarksanddis ussionson thesubje t.

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E-mail:fra zekmat.uni.torun.pl mwysokinmat.uni.torun.pl

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