-Norm and Regularity

https://doi.org/10.1007/s10884-020-09926-4

-Norm and Regularity

D. Treschev¹

Received: 12 August 2020 / Revised: 7 December 2020 / Accepted: 20 December 2020 / Published online: 9 January 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021

Abstract

In Treschev (Proc Steklov Math Inst 310:262–290, 2020) we introduce the concept of aμ- norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from Treschev (2020) and present further results on theμ-norm. More precisely, we specify three classes of unitary operators for which theμ-norm generates a bistochastic operator. We plan to use the latter in the construction of quantum entropy.

1 Introduction

LetX be a nonempty set and letBbe aσ-algebra of subsetsX⊂X. Consider the measure space(X,B, μ), whereμis a probability measure:μ(X)=1.

Consider the Hilbert spaceH=L²(X, μ)with the scalar product and the norm f,g =

X f g dμ, f = f, f. For any bounded operatorW onHletWbe itsL²norm:

W = sup

f=1W f.

We say thatχ = {Y1, . . . ,YJ}is a (finite, measurable) partition (ofX) if Yj ∈B, μ

X\ ∪1≤j≤JYj

=0, μ(Yj∩Yk)=0 for any j,k∈ {1, . . . ,J},k=j. We say thatκ = {X₁, . . . ,X_K} is a subpartition ofχ = {Y₁, . . . ,Y_J} if for anyk ∈ {1, . . . ,K}there exists j∈ {1, . . . ,J}such thatμ(X_k\Y_j)=0.

The research is supported by the RNF Grant 20-11-20141.

Handling editor: Yingfei yi.

B

1 Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

For anyX∈Bconsider the orthogonal projector

πX:H→H, H f →πXf =1X· f, (1.1)

where1_Xis the indicator ofX.

LetWbe a bounded operator onH. For any partitionχ = {Y1, . . . ,YJ}we define M_χ(W)=

J j=1

μ(Y_j)WπYj². (1.2)

In [15] we have introduced the definition of theμ-norm:¹ Wμ=inf

χ

M_χ(W). (1.3)

Recall that the operatorUis said to be an isometry if f,g = U f,U g, f,g∈H. If the isometryUis invertible thenU is a unitary operator.

For any boundedW, anyY ∈B, and any isometryU

WπY ≤ W, U W = W, πY =1 (ifμ(Y) >0).

This implies the following obvious properties of theμ-norm:

idμ =1, Wμ ≤ W, (1.4)

W1W2_μ ≤ W1W2_μ, (1.5)

λWμ = |λ| Wμ for anyλ∈C, (1.6) Wμ = U Wμ for any isometryU. (1.7) Theμ-norm is motivated by the problem of the extension of the measure entropy²to the case of quantum systems. Now we describe briefly the idea.

LetF : X →X be an endomorphism of the probability space(X,B, μ). This means that for any X ∈ B the set F⁻¹(X)(the complete preimage) also lies inBandμ(X) = μ(F⁻¹(X)). Invertible endomorphisms are called automorphisms. Let End(X)denote the semigroup of all endomorphisms of(X,B, μ). There are two standard constructions.

(1) AnyF ∈End(X, μ)generates the isometry (a unitary operator if Fis an automot- phism)U_FonH(the Koopman operator):

L²(X, μ) f →UFf = f ◦F, UF=:Koop(F).

(2) For anyF∈End(X, μ)it is possible to compute the measure entropyh(F). Our question is as follows. Is it possible to determine in some “natural way” a real non- negative functionhon the semigroup of isometries Iso(H)so that the diagram

End(X, μ)

h Koop

R+ h

←− Iso(H) is commutative?

1In fact, a seminorm

2Known also as the Kolmogorov-Sinai entropy

Recall the construction of the measure entropy of an endomorphism. LetJ_Nbe the set of multiindices j=(j₀, . . . ,j_N), where any component j_ntakes values in the set{0, . . . ,K}. For any partitionχ = {X₀, . . . ,X_K}andj ∈J_Nwe define

X_j=F^−N(X_jN)∩. . .∩F⁻¹(X_j1)∩X_j0. We defineh_F(χ,N +1)by

h_F(χ,N+1)= −

j∈J_N

μ(X_j)logμ(X_j).

The functionhF, as a function of the second argument, is subadditive:hF(χ,n+m) ≤ hF(χ,n)+hF(χ,m). This implies existence of the limit

h_F(χ)= lim

n→∞

1

nh_F(χ,n).

Finally, the measure entropy is defined by h(F)=sup

χ h_F(χ).

A rough idea is to construct the entropy of a unitary operatorU analogously with the following difference. Instead ofX_j we take

Xj=πX_jNUπX_jN−1U. . .UπXj1UπXj0. (1.8) We define

hU(χ,N +1)= −

j∈JN

Xj²_μlogXj²_μ. (1.9) Other details are the same:

hU(χ)= lim

n→∞

1

nh(χ,n), h(U)=sup

χ hU(χ) (1.10)

(provided the limit (1.10) exists).

In [15] we prove that for any automorphismF

h(U_F)=h(F). (1.11)

In the literature there exist several attempts to extend the concept of the measure entropy to quantum systems, see [2,3,5,10–12] and many others. In [1] several mutual relations between these approaches are given. Some works (for example, [2,4,9,13,14]) deal with the finite- dimensional case(#X <∞). In [6,7] a construction for the measure entropy is proposed for doubly stochastic (bistochastic) operators on various spaces of functions on a measure space. It remains unclear, which approach to quantum generalization of the measure entropy is “more physical”. This issue may become more clear after computation of the entropyhor its analogs in examples. We plan to do this in forthcoming papers.

The above definition ofh(U)meets several technical problems, including the question on subadditivity³ofhU(χ,n)and the inequalityhU(χ)≤hU(κ)ifκis a subpartition ofχ⁴. We plan to change slightly the definition by replacingXj²_μin (1.9) by another quantity similar to it to satisfy these two properties. To this end we associate withUa bistochastic operator

3This subadditivity is important for the existence of the limit (1.10).

4This inequality is necessary if we want to approach the supremum (1.10) on fine partitionsχ.

onL¹(X, μ). Then the entropy ofU may be defined analogously to [6,7]. We will present details in another paper, but seems, this bistochastic operator may turn to be interesting by itself. As we will see below in this paper, the concept of theμ-norm remains central in all our constructions.

We use theμ-norm to construct a bistochastic operator in three cases: Koopman operators, the case #X <∞and regular operators (a special class of operators defined in Sect.4) for X =T. It is more or less clear that the circleTmay be replaced by the torusT^d, but this will be a subject of another paper.

Before systematic attempts to compute entropy of various unitary operators and to study its proprties we have to study theμ-norm · _μ. We have started this in [15]. The present paper is a continuation of this program.

2 Notation and Previous Results

Here we collect basic results from [15]. We will refer to some of them below.

(1).πX²_μ=μ(X)for anyX∈B.

(2). Ifχis a subpartition ofχthen_Mχ(W)≤Mχ(W). Hence the quantities_Mχ(W) approach the infinum (1.3) on fine (small scale) partitions.

(3). For any two bounded operatorsW1andW2

W₁+W₂μ≤ W₁μ+ W₂μ.

This triangle inequality combined with (1.6) imply that · μis a seminorm on the space of bounded operators onH.

(4). LetFbe an automorphism of(X,B, μ)and letUF=Koop(F). Then

U_FπX=πF⁻¹(X)U_F for anyX∈B. (2.1) For any bounded operatorW

W UF_μ= W_μ. (2.2)

This impliesU_F⁻¹W U_Fμ= Wμ. Informally speaking, this means that measure preserv- ing coordinate changes onXpreserve theμ-norm.

(5). · μis a continuous function in theL²(X, μ)operator topology.

(6). If the measureμhas no atoms thenW+W0μ = Wμfor any boundedW and compactW0. In particular,μ-norm of any compact operator vanishes. In fact, there exist non-compact operators with zeroμ-norm.

(7). Giveng∈L^∞(X, μ)letgbe the multiplication operator defined by f →g f =g f. Thengμ= g.

(8). SupposeX = {1, . . . ,J}is finite and the measure of any element equals 1/J. Then His isomorphic toC^J with the Hermitian productf,gJ = ¹_JJ

j=1 f(j)g(j). Let f →W f, (W f)(k)=

J j=1

W(k,j)f(j) be an operator onH. Then

W²_μ= 1 J

J j,k=1

|W(k,j)|². (2.3)

(9). For any partition{X₁, . . . ,X_K}ofX W²_μ =

K k=1

WπXk²_μ, W²_μ≤ K k=1

πXkW²_μ.

(10). LetX be a compact metric space andμa Borel measure w.r.t. the corresponding topology. LetBr(x)⊂X denote the open ball with center atxand radiusr. Then for any x ∈Xthe limit

ϑ(x)=lim

ε0WπB_ε(x)²

exists, the functionϑ is measurable andW²_μ ≤ _Xϑdμ. There exists an example which shows that in general this inequality is strict. HoweverW²_μ = _Xϑdμprovided two additional conditions C1 and C2 hold:

C1. The functionϑis continuous.

C2. There existsc>0 such that for any openO⊂X, diam(O)≤εand anyx ∈Othere exists a function f =π_Of satisfying

W f²−ϑ(x)f²≤γ (ε)f², c<f|_O<c⁻¹, whereγ (ε)→0 asε→0. As usual, f|_Odenotes restriction of f to the setO.

The further results from [15] we mention here concern the caseX =T=R/2πZwith the Lebesgue measureμ. We expect that they can be extended to the caseX=T^d,d>1.

(11). LetX =Tbe a circle with the Lebesgue measuredμ= _2π¹ d x. For any bounded sequence{λk}k∈Zwe consider the distributionλ(x)=

λke^{i kx}. The convolution operator f →Conv_λ f =λ∗ f :=

Tλ(y)f(· −y)d y is bounded:Conv_λ =sup_k∈Z|λk|. Then

Conv_λ²_μ =ρ(λ), ρ(λ)=lim sup

#I→∞ ρI(λ), ρI(λ)= 1

#I

k∈I

|λk|², whereI ⊂Zare integer intervals.

(12). One of the main technical tools in the analysis of theμ-norm in the caseX =Tis the following lemma on Fourier coefficients of localized functions on the circle.

Lemma 2.1 Let Y = [a−ε,a+ε]and f = πY f =

k∈Z

fke^{i kx} ∈ L²(T). Then for any integer interval J and any m∈Z

f −e^{i m(x−a)}f ≤ |m|εf, (2.4)

|fm−e^ilafm+l| ≤ ε^3/2

√π|l|f, (2.5)

k∈Z

e^{−i ma}fkf_k+m− f²≤ |m|εf². (2.6) (13). Consider the operatorW =(Wj,k)_j,k∈Z^d onH=L²(T):

f =

k∈Z

f_ke^{i kx} →W f =

j,k∈Z

W_j,kf_ke^{i j x}.

We say thatWis of diagonal type (W∈DT(T)) if

supj∈Z|Wj+k,j| =ck<∞, k∈Z and

k∈Z

ck =c<∞.

The sequencec_sis said to be the majorating sequence forW ∈DT(T). We define the norm W_DT =c.

Operators fromDT(T)are bounded. As simple examples we have the following operators of diagonal type.

(a) Bounded convolution operators.

(b) Operators of multiplication by functions with absolutely converging Fourier series.

(c) The conjugated operatorW^∗ifW ∈DT(T).

(d) Linear combinations and products of operators of diagonal type. Moreover, WW_DT ≤ W_DTW_DT for allW,W∈DT(T).

(14). We prove that the normed space

DT(T), · _DT

is closed. As a corollary we obtain that

DT(T), · DT

is aC^∗-algebra.

(15). We have the following inequalities between the norms:

Wμ≤ W ≤ W_DT for anyW ∈DT(T), (2.7)

f∞≤ fDT for any f with absolutely converging Fourier series, (2.8) where f is the operator of multiplication by f.

(16). We associate withW∈DT(T)and any pointa∈Tthe distributionLa, La=

j∈Z

wj(a)e^{i j x}, wj(a)=

k∈Z

Wj,ke^i(j−k)a. (2.9) For anyl∈Zanda∈Twe have the estimate

|wl(a)| ≤c= W_DT. (2.10)

We prove that for anyW ∈DT(T)the functiona → ρ(L_a) = lim sup_#I→∞ρI(L_a) (I⊂Zare intervals,ρIis defined in item (11)) is continuous and

W²_μ= 1 2π

Tρ(La)da. (2.11)

(17). For any operatorW∈DT(T)we introduce the average trace ofW^∗W by T(W)=lim sup

#I→∞

1

#I

j∈Z,l∈I

|W_l,j|²,

whereI ⊂Zare intervals. Then

T(W)≤ W²_μ. (2.12)

(18). We also prove that ifU ∈DT(T)is a unitary operator then

T(W)=T(W U)=T(U W). (2.13)

3 Main Results

In this section we collect main results of the present paper. In short, these results concern (1) properties of theμ-norm on _R(T), a special class of operators on L²(T), and (2) a construction of a bistochastic operatorWonL¹(X, μ)associated with an operatorWunder some conditions, imposed onW.

• We say that an operatorW ∈DT(T)is regular (the notation isW ∈R(T)) if for any m,n∈Zthere exists the limit

ωm,n = lim

#I→∞ωI,m,n, ωI,m,n= 1

#I

j∈Z,l∈I

W_l+m,jW_l,j+n.

• By Lemma4.2_R(T)is a cone closed with respect to the norm · _DT.

• By Lemma4.3W²_μ=T(W)for anyW∈R(T)(compare with (2.12)).

• LetACF(T)be the space of functions onTwith absolutely converging Fourier series.

Then the space of multiplication operatorsg by functions g ∈ ACF(T)form a C^∗- subalgebra in theC^∗-algebraDT(T).

• In Sect.5we present several examples of regular operators.

LetW ∈DT(T)be an operator with periodic matrix i.e., there existsτ ∈Nsuch that Wj+τ,k+τ =Wj,kfor any j,k∈Z. Any such operator is regular (Lemma5.1).

Another example of a regular operator is Conv_λ, whereλk =e^iτk2.

• By Proposition6.1for anyW ∈ R(T)andg₁,g₂ ∈ ACF(T)the operatorg₁Wg₂ is regular.

• In Sect.6we associate with operatorsWfrom the following three classes (1) Koopman operators Koop(F),F∈Aut(X, μ),

(2) operators in the case #X <∞, (3) operators fromR(T)

a measuredμW(x,x)=ν(x,x)dμ(x)dμ(x)onX×Xsuch that for any “suffi- ciently regular” functionsg,g:X→C

gWg²_μ=

X²|g(x)|²|g(x)|²dμW(x,x).

• In Sect.7we introduce the operator L¹(X, μ) f →Wf =

Xν(·,a)f(a)dμ(a) and prove (Lemma7.1) thatWis a bistochastic operator onL¹(X, μ).

4 Regular Operators 4.1 Definition of!m,n

Definition 4.1 We say thatW ∈DT(T)is regular (W ∈R(T)) if for anym,n ∈Zthere exists the limit

#I→∞lim ω_I,m,n =ω_m,n, ω_I,m,n = 1

#I

j∈Z,l∈I

W_l+m,jW_l,j+n, (4.1)

whereIare integer intervals.

Note thatω0,0coincides with the average trace ofW^∗W:

ω0,0=T(W) ifW ∈R(T). (4.2)

For any integer intervalIwe put vI,m(a)=

l∈I

wl+m(a)wl(a)

#I , (4.3)

where the functionswl(a)are defined in (2.9).

Lemma 4.1 Suppose W is regular. Then for any m∈Zthere exists the limit

#Ilim→∞vI,m(a)=vm(a), vm(a)=

n∈Z

ωm,ne^i(m+n)a (4.4)

uniformly in a∈T. The Fourier series of the functionvmabsolutely converges.

Proof By (2.9)

vm(a)= lim

#I→∞

1

#I

j,k∈Z,l∈I

W_l+m,jW_l,ke^i(m−j+k)a

= lim

#I→∞

n∈Z

ω_I,m,ne^i(m+n)a =

n∈Z

ω_m,ne^i(m+n)a.

By (10.1) the limit is uniform ina. By (10.2) this Fourier series absolutely converges.

Note that by (2.10) for any intervalI⊂Z, anym∈Z, and anya∈T

|vI,m(a)| ≤c², |vm(a)| ≤c². (4.5) 4.2 Closeness with Respect to · _DT

IfW ∈R(T)then for anyλ∈Cthe operatorλWis also regular. Hence, regular operators form a coneR(T)⊂DT(T).

Lemma 4.2 The cone_R(T)is closed with respect to the norm · _DT.

Proof Suppose{W_p}p∈N,W_p ∈ R(T) is a Cauchy sequence. By (14) there existsW = lim_p→∞W_p, where the limit is taken with respect to the norm · _DT.

For anyε >0 there exists positiveNsuch that

for any integer p,q>N we have: W_p−W_qDT < ε. (4.6) We define(ωp)I,m,nand(ωp)m,nby (4.1), whereWis replaced byWp. Then

(ωp)I,m,n−(ωq)I,m,n= 1

#I

j∈Z,l∈I

(Wp)l+m,j(Wp)l,j+n−(Wq)l+m,j(Wq)l,j+n

≤ 1

#I

1+2

,

1=

j∈Z,l∈I

(Wp)_l+m,j

(Wp)_l,j+n−(Wq)_l,j+n ,

2=

j∈Z,l∈I

(Wp)l+m,j−(Wq)l+m,j

(Wq)l,j+n

. To estimate the sums1and2, we put

(c_p)k =sup

j∈Z|(W_p)k+j,j|, (c_q)k=sup

j∈Z|(W_q)k+j,j|, d_k=sup

j∈Z|(W_p)k+j,j−(W_q)k+j,j|.

The sums

k(cp)kare uniformly bounded:

k∈Z

(cp)k≤ ˜c,

k∈Z

(cq)k≤ ˜c for some constantc.˜ (4.7) Moreover, by (4.6)

k∈Z

dk< ε. (4.8)

By (4.7) and (4.8)

1≤

j∈Z,l∈I

(cp)l+m−jdl−j−n ≤#Icε.˜ Analogously2≤#I˜cε. This implies that for any intervalI

(ωp)I,m,n−(ωq)I,m,n≤2˜cε.

Hence,(ωp)m,n−(ωq)m,n≤2˜cεi.e., for any integerm,nthe sequence(ωp)m,n,p∈Nis

a Cauchy sequence.

4.3-Norm of a Regular Operator

Lemma 4.3 If W ∈R(T)then (compare with(2.12))

W²_μ=T(W). (4.9)

Proof By (2.9), (2.11) and (4.3) W²_μ= 1

2π

T lim

#I→∞vI,0(a)da. (4.10)

By Lemma4.1 the limitv0(a) = lim_#I→∞vI,0(a) exists for any a ∈ T and by (4.5)

|vI,0(a)| ≤c²for allI anda. Therefore by the Lebesgue theorem on bounded convergence we may exchange the integration and the limit:

W²_μ= lim

#I→∞

1 2π

TvI,0(a)da= lim

#I→∞

1 2π

T

j,k∈Z,l∈I

1

#IW_l,jW_l,ke^i(k−j)ada

= lim

#I→∞

j∈Z,l∈I

1

#IW_l,jW_l,j = ω0,0.

By (4.1) this implies (4.9).

Assertion (18) and Lemma4.3imply the following.

Corollary 4.1 Suppose W,U ∈DT(T), where U is unitary and both W and U W U⁻¹are regular. Then by (18) and(1.7)

W_μ = U W U⁻¹_μ= W U⁻¹_μ.

5 Regular Operators: Examples

Definition 5.1 We say that the matrix (W_k,j)of the operatorW ∈ DT(T)is τ-periodic, τ ∈N, if

W_k+τ,j+τ =W_k,j for anyk,j∈Z.

In particular, for anyg∈ACF(T)matrix of the operatorg∈DT(T)is 1-periodic.

Lemma 5.1 Suppose W ∈ DT(T) is an operator with τ-periodic matrix,τ ∈ N. Then W ∈R(T)andωm,n = ˘ωm,n,

˘ ωm,n = 1

τ

j∈Z,l∈J

W_l+m,jW_l,j+n, where J ⊂Zis any interval with#J =τ.

Proof Byτ-periodicityω˘_m,n does not depend onJ. For anyI = {s,s+1, . . . ,s+K}let I⊂I be the maximal integer interval of the formI= {s,s+1, . . . ,s+qτ−1},q∈Z. We put I = {s+qτ,s+qτ +1, . . . ,s+K}. Then #I < τ,ωI,m,n = ˘ωm,n while ωI,m,n−ωI,m,n =A1+A2,

A₁= 1

#I

l∈I\I

W_l+m,jW_l,j+n, A₂= 1

#I − 1

#I

l∈I

W_l+m,jW_l,j+n. The inequalities

|A1| ≤τc²

#I , |A2| ≤ τc²

#I−τ

imply the existence of the limit (4.1) and the equationω_m,n= ˘ω_m,n. Corollary 5.1 Consider the operatorg, where g ∈ACF(T). After simple calculations we obtain:

ωm,n(g)=g_m+n, g(x)g(x)=

k∈Z

g_ke^{i kx}, (5.1)

where the last equation is the definition ofg_k.

Lemma 5.2 Let{λk}_k∈Zbe defined byλk=e^iτk2. Then the operatorConv_λis regular:

ωm,n=δ0,m+nδτm,πZe^iτm2, δτm,πZ=

1ifτm/π ∈Z, 0ifτm/π /∈Z.

Proof In this caseWk,j=δk je^iτk2 and ωI,m,n = 1

#I

j∈Z,l∈I

δl+m,jδl,j+ne^{iτ((l+m)2−l2}

= 1

#I

l∈I

δl,l+m+ne^iτm2+2iτml = 1

#Iδ0,m+ne^iτm2

l∈I

e^2iτml.

Hence, ifτm/π ∈Zthenωm,n =δ0,m+ne^iτm2. Ifτm/π /∈Zthenωm,n=0.

6 Measure Associated with an Operator 6.1 Koopman Operator

Let(X,B, μ)be a probability space.

Lemma 6.1 F∈Aut(X, μ), g0, . . . ,g_K ∈L^∞(X, μ). Then gKUFg_K−1. . .UFg0²

μ=

X|gK ◦F^K|²|gK−1◦F^K−1|². . .|g0|²dμ (6.1) Proof Take a small constantσ >0 and consider the partition{X1, . . . ,XJ}such that⁵

g_k−ϕk∞< σg_k∞, ϕk= J

j=1

g_k,j1_Xj, ϕk∞≤ g_k∞, k=0, . . . ,K. (6.2) Heregk,j ∈ Care some constants. We put S = ϕKUFϕK−1. . .UFϕ0. By the triangle inequality the quantity

gKUFgK−1. . .UFg0_μ− S_μ does not exceed

gKUFgK−1. . .UFg0_μ− ϕKUFgK−1. . .UFg0_μ +ϕKU_Fg_K−1. . .U_Fg₀μ− ϕKU_Fg_K−1. . .U_Fg₀μ

+. . . + ϕKUFϕK−1. . .ϕ1UFg0_μ− ϕKUFϕK−1. . .ϕ1UFϕ0_μ

≤ (gK−ϕK)UFgK−1. . .UFg0_μ+ ϕKUF(gK−1−ϕK−1) . . .UFg0_μ +. . . + ϕKU_FϕK−1. . .ϕ1U_F(g₀−ϕ0)μ

≤(K+1)σ

0≤k≤K

gk∞.

By (2.1)

S=

j0,...,jK

g_K,jKg_K−1,jK−1. . .g_0,j0πX_jKπF⁻¹(X_jK−1). . . πF^−K(Xj0)U_F^K.

We put X_jK,...,j0 = X_jK ∩ F⁻¹(X_jK−1)∩. . .∩ F^−K(X_j0). Note that for any two sets X= X_j

K,...,j₀ andX= X_j

K,...,j₀we have:μ(X∩X)=0 if the collections of indices j_K, . . . ,j₀andj_K, . . . ,j₀do not coincide. Then by (2.2) and (1) we obtain:

S=

j0,...,jK

|gK,j_KgK−1,j_K−1. . .g0,j₀|²πX_jK,...,j0²_μ

=

j0,...,jK

|gK,jKg_K−1,jK−1. . .g_0,j0|²μ(XjK,...,j0). (6.3)

5We can use the same partition for all the functionsgk.

Equation (6.2) implies

ϕk◦F^k= J

j=1

gk,j1_F−k(Xj). Hence (6.3) is an integral sum for the integral

X|ϕK ◦F^K|²|ϕK−1◦F^K−1|². . .|ϕ0|²dμ.

This integral differs from the integral (6.1) at most by 2σ

0≤k≤Kgk²_∞. Sinceσis arbi-

trarily small, we obtain Eq. (6.1).

Now suppose thatXis in addition a topological space andBis the corresponding Borel σ-algebra. Consider the distributionδ∈(C(X²))^∗(a measure onX²by the Riesz theorem) such that for any∈C(X²)

X²δ(x,x)(x,x)dμ(x)dμ(x)=

X(x,x)dμ(x). (6.4) Taking in (6.4)(x,x)=ϕ(x)ϕ(x), whereϕ, ϕ:X →Care arbitrary continuous functions, we obtain:

Xδ(x,x)ϕ(x)dμ(x)=ϕ(x),

Xδ(x,x)ϕ(x)dμ(x)=ϕ(x). (6.5) Recall also that ifF∈End(X, μ)

X f dμ=

X f ◦F dμ for any f ∈L¹(X, μ). (6.6) Then for any continuousF∈Aut(X, μ)

δ(x,x)=δ(F(x),F(x)). (6.7)

Indeed, for any∈C(X²)by (6.6)

X²δ(F(x),F(x))(x,x)dμ(x)dμ(x)

=

X²δ(x,x)(F⁻¹(x),F⁻¹(x))dμ(x)dμ(x) =

X(x,x)dμ(x).

Lemma 6.2 Let F∈Aut(X, μ)be continuous and g₀, . . . ,g_K ∈C(X). Then g_KU_F. . .g₁U_Fg₀²

μ

=

X^K+1|gK(xK)|²δ(xK,F(xK−1)) . . .|g1(x1)|²δ(x1,F(x0))|g0(x0)|²dμ^K+1,

dμ^K+1=dμ(xK) . . .dμ(x0). (6.8)

Proof LetIdenote the integral (6.8). We use in this integral the change of coordinates x0=x₀, x1=F(x₁), . . . xK =F^K(x_K), dμ^K+1=dμ(x_K) . . .dμ(x₀).

Then

I =

|g_K◦F^K(x_K )|²δ(F^K(x_K),F^K(x_K−1)) . . . |g1◦F(x₁)|²δ(F(x₁),F(x₀))|g0(x₀)|²dμ^K+1

=

X|gK◦F^K|². . .|g1◦F|²|g0|²dμ.

It remains to use Lemma6.1.

We associate with anyU_F=Koop(F), whereF∈Aut(X, μ)is continuous,⁶the measure μUF onX²:

dμU_F(x,x)=δ(x,F(x))dμ(x)dμ(x). (6.9) 6.2 The Finite-Dimensional Case

LetX = {1, . . . ,J}be a finite set. We identifyXwithZJ =Z/JZ, the cyclic additive group withJ elements.⁷Let the measureμof any point be equal to 1/J. ThenH=L²(X, μ)∼= (C^J,,), where,equals the standard Hermitian product divided byJ.

We putη=e^2πi/J. Thenη^J =1 andη=η⁻¹. The space_Hmay be identified with the space of “discrete trigonometric polynomials”

f = f(x)=

j∈ZJ

f_jη^{j x}, f_j∈C, x ∈ZJ. (6.10) This polynomial representation generates on_Han operation of multiplication: for any two vectors f=

f_jη^{j x}and f= f_jη^{j x} ff= f =

k∈ZJ

fkη^kx, fk=

j∈ZJ

f_{k− j}f_j.

This product introduces onHthe structure of a commutative ring. The structure of a Hilbert space is determined by

f,g = 1 J

k∈ZJ

f(k)g(k).

The coefficients fkand f(k)are connected by the “discrete Fourier transform”

f(k)=

j∈ZJ

fjη^{k j}, fj= 1 J

k∈ZJ

f(k)η^{−k j}. (6.11)

For any f satisfying (6.10) and anyA⊂ZJwe put

A

f(x)dμ(x):= 1 J

k∈A

f(k)= 1 J

k∈A,j∈ZJ

f_jη^{j k}.

Then

X f dμ= f₀, f,g = 1 J

X f g dμ.

6We expect that the continuity is inessential here.

7Below we also use the structure of a commutative ring onZJ.

Consider a linear operator

f →W f, (W f)k =

j∈ZJ

Wk jfj. In another basis it takes the form

(W f)(k)=

j∈ZJ

W(k,j)f(j).

Equations (6.11) imply

W(m,n)= 1 J

j,k∈ZJ

η^mkW_{k j}η^{−j n}. (6.12) We define

ωm,n = 1 J

j,l∈ZJ

W_l+m,jW_l,j+n, ν(x,a)=

m,n∈ZJ

ωm,nη^mxη^na, x,a∈ZJ.(6.13)

Lemma 6.3

m,n∈ZJ|W(m,n)|²=

j,k∈ZJ|W_j,k|².

Proof Direct computation with the help of (6.12) and the identity

j∈ZJη^{j k}= Jδk,0. Corollary 6.1 By(2.3)andLemma6.3

W²_μ=ω0,0=

X²ν(x,a)dμ(x)dμ(a). (6.14) Lemma 6.4 For any g,g ∈Hthe operatorW=gWggenerates the coefficientsωm,n

such that

m,n∈Z

ωm,nη^mxη^na = |g(x)|²ν(x,a)|g(a)|². (6.15) The measure dμW(x,a)=ν(x,a)dμ(x)dμ(a)onX²satisfies

gWg²_μ=

X²|g(x)|²|g(a)|²dμW(x,a). (6.16) Proof We put

g(x)=

g_kη^kx, g(x)=

g_kη^kx, |g(x)|²=

g_kη^kx, |g(x)|²= g_kη^kx. The equationW_p,q=

α,βg_p−αW_α,βg_β−q implies

ωm,n = 1 J

l,j,α,β

g_l+m−αW_α,βg_β−jg_l−αW_α,βg_β−j−n

= 1 J

α,β

g_m−α+αW_α,βW_α,βg_β−β+n

= 1 J

p,q,α,β

g_m−pW_p+α,βW_α,β−qg_q+n =

p,q

g_m−pωp,−qg_q+n.

This implies (6.15). Equation (6.16) follows from (6.14).