# Thermal equilibrium of an ideal gas in a free-floating box

Scott Tremaine  Institute for Advanced Study, Princeton, NJ 08540, USA
Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St.
George Street, Toronto, ON M5S 3H8, Canada  Bence Kocsis  Rudolf Peierls
Centre for Theoretical Physics, Clarendon Laboratory, Parks Road, Oxford OX1
3PU, UK  Abraham Loeb  Astronomy Department, Harvard University, 60 Garden
St., Cambridge, MA 02138, USA

###### Abstract

The equilibrium and fluctuations of an ideal gas in a rigid container are
studied by every student of statistical mechanics. Here we study the less
well-known case when the box is floating freely; in particular we determine
the fluctuations of the box in velocity and position due to interactions with
the gas it contains. This system is a toy model for the fluctuations in
velocity and position of a black hole surrounded by stars at the center of a
galaxy. These fluctuations may be observable in nearby galaxies.

##  1 Introduction

The determination of the equilibrium state of a dilute gas in a box is perhaps
the most famous problem in statistical mechanics, and usually the first one to
which students are exposed (Landau & Lifshitz, 1980). The problem is usually
solved with the assumption that the box is fixed, or at least very massive
compared to the gas it contains. The purpose of this short paper is to
describe the equilibrium state when the box is free to move and the mass of
the box is comparable to the mass of the gas.

This system offers both a simple analytic problem in statistical mechanics and
a toy model for a more complicated, and still poorly understood, problem from
astrophysics: the Brownian motion of the supermassive black holes found at the
centers of galaxies due to the gravitational forces from the surrounding stars
(see §4 for more detail).

We consider an isolated rigid box of mass $M$ containing $N$ particles of mass
$m$, so the total gas mass is $M_{g}=Nm$. The particles collide via short-
range forces and we neglect the self-gravity of the gas. The box is freely
floating and there is a heater attached to the box that maintains its
temperature at a fixed value $T$ (thus we are working with a canonical
ensemble in which the “heat bath” exchanges energy but not momentum with the
system; we could instead have used an insulating box and worked with the
microcanonical ensemble, with very similar results). The center of mass of the
system (box plus gas) is at rest. We can assume that the box is cubical, with
its symmetry axes parallel to the coordinate axes $(x,y,z)$. Then the dynamics
along each axis is independent. What is the root-mean-square velocity of the
box along each axis?

There are two simple arguments leading to different answers, and both turn out
to be correct only in limiting cases.

First, we can argue that the box is in thermal equilibrium with the gas. The
mean-square velocity of the gas particles in one dimension—say along the
$x$-axis—is $\langle v_{x}^{2}\rangle=k_{B}T/m$ where $k_{B}$ is Boltzmann’s
constant. Then equipartition implies that the mean-square velocity of the box
along the $x$-axis should be

$\langle V_{x}^{2}\rangle=\frac{k_{B}T}{M}.$ (1)

The second argument starts with the observation that the center of mass of the
system is at rest so momentum conservation implies that
$MV_{x}=-m\sum_{i=1}^{N}v_{xi}$ where $v_{xi}$ is the $x$-velocity of particle
$i$ and the sum is over all the particles. From the central limit theorem, the
mean-square value of the right-hand side should be $m^{2}N\langle
v_{x}^{2}\rangle=mNk_{B}T=M_{g}k_{B}T$ so

$\langle V_{x}^{2}\rangle=\frac{M_{g}}{M^{2}}k_{B}T,$ (2)

obviously not the same as (1).

##  2 The mean-square velocity of the free-floating box

Each microstate of the system we are examining is specified by the $N$
particle velocities $v_{xi}$ and the box velocity $V_{x}$, both measured along
the $x$-axis. Momentum conservation restricts the phase space to the
hyperplane $mv_{x1}+\cdots+mv_{xN}+MV_{x}=0$. The canonical ensemble assigns
to each microstate a probability proportional to $\exp[-E/(k_{B}T)]$, where
the kinetic energy
$E=\tfrac{1}{2}m(v_{x1}^{2}+\cdots+v_{xN}^{2})+\tfrac{1}{2}MV_{x}^{2}$.
Therefore the probability distribution of the velocities is given by

$\displaystyle p(V_{x},v_{x1},\ldots,v_{xN})dv_{x1}\cdots
dv_{xN}dV_{x}\propto$
$\displaystyle\exp\left(-\frac{mv_{x1}^{2}+\cdots+mv_{xN}^{2}+MV_{x}^{2}}{2k_{B}T}\right)$
$\displaystyle\qquad\times\delta(mv_{x1}+\cdots+mv_{xN}+MV_{x})dv_{x1}\cdots
dv_{xN}dV_{x}$ (3)

where $\delta(\cdot)$ is the Dirac delta function. The probability
distribution of $V_{x}$ is then obtained by integrating over the $N$ particle
velocities,

$p(V_{x})\propto\int_{-\infty}^{\infty}dv_{x1}\cdots\int_{-\infty}^{\infty}dv_{xN}\,\exp\left(-\frac{mv_{x1}^{2}+\cdots+mv_{xN}^{2}+MV_{x}^{2}}{2k_{B}T}\right)\delta(mv_{x1}+\cdots+mv_{xN}+MV_{x}).$
(4)

To evaluate this integral we use the relation

$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\kappa\,\exp(i\kappa x).$ (5)

Then

$p(V_{x})\propto\exp\left(-\frac{MV_{x}^{2}}{2k_{B}T}\right)\int_{-\infty}^{\infty}d\kappa\exp(i\kappa
MV_{x})\left[\int_{-\infty}^{\infty}dv\,\exp\left(-\frac{mv^{2}}{2k_{B}T}+i\kappa
mv\right)\right]^{N}.$ (6)

We use the result

$\int_{-\infty}^{\infty}du\exp(i\alpha u-\beta
u^{2})=2\int_{0}^{\infty}\cos(\alpha u)\exp(-\beta
u^{2})=\left(\frac{\pi}{\beta}\right)^{1/2}\exp\left(-\frac{\alpha^{2}}{4\beta}\right),\quad\beta>0.$
(7)

The integral inside the square brackets is then

$\int_{-\infty}^{\infty}dv\,\exp\left(-\frac{mv^{2}}{2k_{B}T}+i\kappa
mv\right)=\left(\frac{2\pi
k_{B}T}{m}\right)^{1/2}\exp\left(-\tfrac{1}{2}mk_{B}T\kappa^{2}\right).$ (8)

Thus

$p(V_{x})\propto\exp\left(-\frac{MV_{x}^{2}}{2k_{B}T}\right)\int_{-\infty}^{\infty}d\kappa\,\exp\left(i\kappa
MV_{x}-\tfrac{1}{2}Nmk_{B}T\kappa^{2}\right).$ (9)

The integral can be evaluated using equation (7), and yields

$p(V_{x})\propto\exp\left[-\frac{MV_{x}^{2}}{2k_{B}T}\left(1+\frac{M}{M_{g}}\right)\right].$
(10)

The mean-square velocity of the box along the $x$-axis is thus

$\langle V_{x}^{2}\rangle=\frac{k_{B}T}{M}\frac{M_{g}}{M+M_{g}}.$ (11)

This result agrees with equation (1) when the gas mass is much larger than the
box mass ($M_{g}\gg M$) and agrees with (2) when the gas mass is much less
than the box mass. When $M_{g}\ll M$, the argument leading to equation (1) is
wrong because it neglects the constraint imposed by momentum conservation.
When $M_{g}\gg M$ the argument leading to equation (2) is wrong because it
neglects energy equipartition between the gas and the box.

Although equation (11) was derived for a cubical box it should hold for a box
of arbitrary shape.

There is a simple heuristic derivation of this result (Lin & Tremaine, 1980).
Approximate the system as consisting of two bodies, the box with mass $M$ and
the gas with mass $M_{g}$. The reduced mass of the two-body system is
$\mu=MM_{g}/(M+M_{g})$. In thermal equilibrium the kinetic energy per degree
of freedom is $\tfrac{1}{2}k_{B}T$ so the relative velocity
$V_{x,\mathrm{rel}}$ between the two bodies satisfies

$\langle V_{x,\mathrm{rel}}^{2}\rangle=\frac{k_{B}T}{\mu}.$ (12)

Since the center of mass is fixed, the one-dimensional velocity of the box is
$V_{x}=M_{g}V_{x,\mathrm{rel}}/(M+M_{g})$ so

$\langle V_{x}^{2}\rangle=\frac{M_{g}^{2}}{(M+M_{g})^{2}}\langle
V_{x,\mathrm{rel}}^{2}\rangle=\frac{k_{B}T}{M}\frac{M_{g}}{M+M_{g}},$ (13)

the same as equation (11).

##  3 The mean-square displacement of the free-floating box

We can use similar methods to calculate the mean-square displacement of the
box. Assume that the center of mass of the system is at the origin so
$MX=-m\sum_{i=1}^{N}x_{i}$ where $x_{i}$ is the position of particle $i$ and
$X$ is the position of the center of mass of the box. We assume that the box
is cubical, with thin walls of length $L$ in each dimension.

Since we are only concerned with the position of the box, we can integrate the
probability distribution in phase space over the velocities, so

$p(X,x_{1},\ldots,x_{N})\propto\delta(mx_{1}+\cdots+x_{N}+MX)\prod_{i=1}^{N}\Theta(\tfrac{1}{2}L+X-x_{i})\Theta(\tfrac{1}{2}L-X+x_{i}),$
(14)

where $\Theta(x)$ is the step function, equal to 1 if $x>0$ and 0 if $x<0$.
The delta function constrains the center of mass to remain at the origin and
the step functions constrain the particles to remain inside the box. Using the
identity (5),

$\displaystyle p(X,x_{1},\ldots,x_{N})$
$\displaystyle\propto\int_{-\infty}^{\infty}d\kappa\,\exp(i\kappa
MX)\prod_{i=1}^{N}\int_{-L/2+X}^{L/2+X}dx_{i}\,\exp(im\kappa x_{i})$
$\displaystyle\propto\int_{-\infty}^{\infty}d\kappa\,\exp[i\kappa
X(M+M_{g})]\left(\frac{\sin\tfrac{1}{2}mL\kappa}{\kappa}\right)^{N}.$ (15)

To evaluate the integral we assume that the gas mass $M_{g}=Nm$ is fixed and
let $N\to\infty$ or $m\to 0$. We replace the dummy variable $\kappa$ by
$u\equiv\kappa/\surd N$. Then

$p(X,x_{1},\ldots,x_{N})\propto\int_{-\infty}^{\infty}du\,\exp[iuN^{1/2}(M+M_{g})]\left(\frac{\sin\tfrac{1}{2}M_{g}LN^{-1/2}u}{u}\right)^{N}.$
(16)

Since $|\sin x/x|$ is less than unity except near $x=0$, $(\sin x/x)^{N}$ is
negligible as $N\to\infty$ except near $x=0$. Here it can be approximated by
the first two terms in its Taylor series, $\sin x/x=1-\frac{1}{6}x^{2}$, and
we have

$\left(\frac{\sin\tfrac{1}{2}M_{g}LN^{-1/2}u}{u}\right)^{N}\propto\left(1-\frac{M_{g}^{2}L^{2}u^{2}}{24N}\right)^{N}\to\exp\left(-\frac{M_{g}^{2}L^{2}u^{2}}{24}\right),$
(17)

in which we have used the identity $(1-z/N)^{N}\to\exp(-z)$ as $N\to\infty$.
The integral in equation (16) can now be evaluated using equation (7), and we
find

$p(X)\propto\exp\left[-\frac{6NX^{2}(M+M_{g})^{2}}{L^{2}M_{g}^{2}}\right].$
(18)

Note that as $N\to\infty$ this derivation converges to the correct answer at a
fixed value of $XN^{1/2}$ but not necessarily at a fixed value of $X$
(obviously, $|X|$ cannot exceed $\tfrac{1}{2}LM_{g}/(M+M_{g})$, the value it
would have if all of the gas were on one wall of the box). The mean-square
displacement of the box from the original center of mass is

$\langle
X^{2}\rangle=\frac{L^{2}}{12N}\left(\frac{M_{g}}{M+M_{g}}\right)^{2}.$ (19)

The factor $L^{2}/12$ is simply the variance of a uniform distribution between
$-\tfrac{1}{2}L$ and $\tfrac{1}{2}L$, given by $\int_{-L/2}^{L/2}x^{2}dx/L$.
For a box of general shape the factor $L^{2}/12$ should be replaced by
$\frac{1}{3}$ of the mean-square distance of the box wall from its center of
mass.

##  4 Nuclear star clusters

“Supermassive” black holes with masses between $M_{\bullet}=10^{6}$ and
$M_{\bullet}=10^{10}$ solar masses are found in the centers of most galaxies
(Kormendy & Ho, 2013). The black holes are typically surrounded by nuclear
star clusters (Neumayer et al., 2020), systems of stars orbiting the black
hole. These are the densest stellar systems in the universe, exceeding
$10^{7}$ times the stellar number density in the solar neighborhood. The mass
of stars in the nuclear cluster can be comparable to or even exceed the mass
of the black hole. Because the star cluster is usually much denser than the
surrounding galaxy, it is reasonable to approximate the cluster plus black
hole as an isolated system.

The dynamics and statistical mechanics of stars in a nuclear star cluster
differ from those of a gas in a box in several important respects: (i) the
stars are bound to the black hole because of their own gravity and the gravity
from the black hole, rather than being confined in a box; (ii) relaxation
occurs through gravitational encounters, rather than collisions, similar to
relaxation through Coulomb encounters in a plasma; (iii) the mean free path is
much longer than the size of the system; (iv) even the oldest, densest
clusters have only lived for a few tens of relaxation times; (v) the cluster
cannot approach thermodynamic equilibrium, since it has negative heat capacity
(Binney & Tremaine, 2008; Bahcall & Wolf, 1976).

The velocity and displacement of the black hole from the center of the nuclear
star cluster are accessible to observations, at least in the nearest galaxies.
The supermassive black hole at the center of the Milky Way is nearly
stationary: its velocity components perpendicular to the line of sight are
only $0.5\pm 2.2\,{\rm km\ s}^{-1}$ in the Galactic plane and $0.9\pm
0.8\,{\rm km\ s}^{-1}$ normal to the plane (Reid & Brunthaler, 2020), and the
component along the line of sight (Gravity Collaboration et al., 2019) is
$3\pm 2\,{\rm km\ s}^{-1}$. The supermassive black hole in the nearest giant
galaxy, M31, is displaced from the center of the large-scale galaxy (Kormendy
& Bender, 1999) by $0.07\pm 0.01$ arcseconds or $0.23\pm 0.03$ parsecs,
although this offset is probably due to the gravitational influence of the
eccentric stellar disk that orbits the black hole. The black hole in the giant
elliptical galaxy M87, which has been imaged by the Event Horizon Telescope,
appears to be offset from the photo-center of the galaxy by $7\pm 1$ pc
(Batcheldor et al., 2010; Broderick et al., 2011).

The free-floating box provides a toy model that captures some but not all of
the dynamics that determines the Brownian motion of the black hole due to its
interactions with the surrounding stars, both within the cluster and outside
it. The nuclear star cluster is not in thermal equilibrium, so the
“temperature” or mean-square velocity of the stars varies with radius, given
roughly by the Kepler relation $\langle v^{2}\rangle\simeq GM_{\bullet}/r$. If
the number of stars in a small radius range $dr$ is $dN$, and the mass of the
star cluster is small compared to the mass of the black hole, then equation
(11) suggests that these contribute $\langle V^{2}\rangle=\langle v^{2}\rangle
m^{2}dN/M_{\bullet}^{2}$ where $m$ is now the typical stellar mass. The mean-
square velocity of the black hole due to interactions with the cluster stars
is then

$\langle V^{2}\rangle\simeq\frac{Gm^{2}}{M_{\bullet}}\int\frac{dN}{r}.$ (20)

Similarly the mean-square displacement should be roughly

$\langle R^{2}\rangle\sim\frac{m^{2}}{M_{\bullet}^{2}}\int r^{2}\,dN,\quad
Nm\lesssim M_{\bullet}.$ (21)

For the usual Bahcall–Wolf model of an equilibrium cluster (Bahcall & Wolf,
1976), $dN\sim r^{1/4}dr$, both the velocity and the displacement are
dominated by the effects of stars in the outer parts of the cluster, and the
mean-square velocity and radius of the black hole are smaller than those of
the stars in this region by a factor of order $m^{2}N/M_{\bullet}^{2}$. A more
quantitatively reliable approach, of course, is to use N-body simulations (Lin
& Tremaine, 1980; Chatterjee et al., 2002; Reid & Brunthaler, 2020).

##  5 Summary

We have described aspects of the Brownian motion of a free-floating box
containing an isothermal gas. The mean-square velocity and displacement of the
box are given by equations (11) and (19); the mean-square velocity is
independent of the size and shape of the box, and the mean-square displacement
is independent of the temperature. This system provides a toy model that
describes some aspects of the Brownian motion of a supermassive black hole in
a nuclear star cluster.

We thank Walter Dehnen and Jean-Baptiste Fouvry for the discussions that
stimulated us to write this paper.

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada (NSERC), funding reference number RGPIN-2020-03885;
by the Black Hole Initiative at Harvard University, which is funded by grants
from JTF and GBMF; and by the European Research Council (ERC) under the
European Union’s Horizon 2020 Programme for Research and Innovation
ERC-2014-STG under grant agreement No. 638435 (GalNUC).

## References

  * Landau & Lifshitz (1980) L. D. Landau and E. M. Lifshitz 1980, Statistical Physics, Part I, 3rd ed. (Oxford: Pergamon)
  * Lin & Tremaine (1980) D. N. C. Lin and Scott Tremaine 1980, “A reinvestigation of the standard model for the dynamics of a massive black hole in a globular cluster”, The Astrophysical Journal, 242, 789
  * Kormendy & Ho (2013) John Kormendy and Luis C. Ho 2013, “Coevolution (or not) of supermassive black holes and host galaxies”, Annual Review of Astronomy and Astrophysics, 51, 511-–653
  * Neumayer et al. (2020) Nadine Neumayer, Anil Seth, and Torsten Böker 2020, “Nuclear star clusters”, Astronomy and Astrophysics Review, 28, 4
  * Binney & Tremaine (2008) James Binney and Scott Tremaine 2008, Galactic Dynamics, 2nd ed. (Princeton: Princeton University Press)
  * Bahcall & Wolf (1976) J. N. Bahcall and R. A. Wolf 1976, “Star distribution around a massive black hole in a globular cluster”, The Astrophysical Journal, 209, 214
  * Reid & Brunthaler (2020) M. J. Reid and A. Brunthaler 2020, “The proper motion of Sagittarius A*: III. The case for a supermassive black hole”, The Astrophysical Journal, 892, 39
  * Gravity Collaboration et al. (2019) Gravity Collaboration, R. Abuter et al. 2019, “A geometric distance measurement to the Galactic center black hole with 0.3% uncertainty”, Astronomy and Astrophysics, 625, L10
  * Kormendy & Bender (1999) John Kormendy and Ralf Bender 1999, “The double nucleus and central black hole of M31”, The Astrophysical Journal, 522, 77
  * Batcheldor et al. (2010) D. Batcheldor, A. Robinson, D. J. Axon, E. S. Perlman, and D. Merritt 2010, “A displaced supermassive black hole in M87”, The Astrophysical Journal, 717, L6
  * Chatterjee et al. (2002) Pinaki Chatterjee, Lars Hernquist, and Abraham Loeb 2002, “Dynamics of a massive black hole at the center of a dense stellar system”, The Astrophysical Journal, 572, 371
  * Di Cintio et al. (2020) Pierfrancesco Di Cintio, Luca Ciotti, and Carlo Nipoti 2020, “Brownian motion of supermassive black holes in galaxy cores”, in A. Bragaglia, M. B. Davies, A. Sills & E. Vesperini, eds., Star Clusters: From the Milky Way to the Early Universe, IAU Symposium 351, 93 (Cambridge: Cambridge University Press)
  * Broderick et al. (2011) Avery Broderick, Abraham Loeb, and Mark J. Reid 2011, “Localizing Sagittarius A* and M87 on microarcsecond scales with millimeter very long baseline interferometry”, The Astrophysical Journal, 735, 57


#  $\mu$-norm and regularity

D.Treschev
Steklov Mathematical Institute of Russian Academy of Sciences

###### Abstract

In [22] we introduce the concept of a $\mu$-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
[22] and present further results on the $\mu$-norm. More precisely, we specify
three classes of unitary operators for which the $\mu$-norm generates a
bistochastic operator. We plan to use the latter in the construction of
quantum entropy.

##  1 Introduction

Let ${\cal X}$ be a nonempty set and let ${\cal B}$ be a $\sigma$-algebra of
subsets $X\subset{\cal X}$. Consider the measure space $({\cal X},{\cal
B},\mu)$, where $\mu$ is a probability measure: $\mu({\cal X})=1$.

Consider the Hilbert space ${\cal H}=L^{2}({\cal X},\mu)$ with the scalar
product and the norm

$\langle f,g\rangle=\int_{\cal X}f\overline{g}\,d\mu,\quad\|f\|=\sqrt{\langle
f,f\rangle}.$

For any bounded operator $W$ on ${\cal H}$ let $\|W\|$ be its $L^{2}$ norm:

$\|W\|=\sup_{\|f\|=1}\|Wf\|.$

We say that $\chi=\\{Y_{1},\ldots,Y_{J}\\}$ is a (finite, measurable)
partition (of ${\cal X}$) if

$Y_{j}\in{\cal B},\quad\mu\big{(}{\cal X}\setminus\cup_{1\leq j\leq
J}Y_{j}\big{)}=0,\quad\mu(Y_{j}\cap Y_{k})=0\quad\mbox{ for any
$j,k\in\\{1,\ldots,J\\}$, $k\neq j$}.$

We say that $\kappa=\\{X_{1},\ldots,X_{K}\\}$ is a subpartition of
$\chi=\\{Y_{1},\ldots,Y_{J}\\}$ if for any $k\in\\{1,\ldots,K\\}$ there exists
$j\in\\{1,\ldots,J\\}$ such that $\mu(X_{k}\setminus Y_{j})=0$.

For any $X\in{\cal B}$ consider the orthogonal projector

$\pi_{X}:{\cal H}\to{\cal H},\qquad{\cal H}\ni f\mapsto\pi_{X}f={\bf
1}_{X}\cdot f,$ (1.1)

where ${\bf 1}_{X}$ is the indicator of $X$.

Let $W$ be a bounded operator on ${\cal H}$. For any partition
$\chi=\\{Y_{1},\ldots,Y_{J}\\}$ we define

${\cal M}_{\chi}(W)=\sum_{j=1}^{J}\mu(Y_{j})\|W\pi_{Y_{j}}\|^{2}.$ (1.2)

In [22] we have introduced the definition of the $\mu$-norm:111in fact, a
seminorm

$\|W\|_{\mu}=\inf_{\chi}\sqrt{{\cal M}_{\chi}(W)}.$ (1.3)

Recall that the operator $U$ is said to be an isometry if

$\langle f,g\rangle=\langle Uf,Ug\rangle,\qquad f,g\in{\cal H}.$

If the isometry $U$ is invertible then $U$ is a unitary operator.

For any bounded $W$, any $Y\in{\cal B}$, and any isometry $U$

$\|W\pi_{Y}\|\leq\|W\|,\quad\|UW\|=\|W\|,\quad\|\pi_{Y}\|=1\;\mbox{(if
$\mu(Y)>0$)}.$

This implies the following obvious properties of the $\mu$-norm:

$\displaystyle\|\operatorname{id}\|_{\mu}$ $\displaystyle=$ $\displaystyle
1,\qquad\|W\|_{\mu}\;\leq\;\|W\|,$ (1.4) $\displaystyle\|W_{1}W_{2}\|_{\mu}$
$\displaystyle\leq$ $\displaystyle\|W_{1}\|\|W_{2}\|_{\mu},$ (1.5)
$\displaystyle\|\lambda W\|_{\mu}$ $\displaystyle=$
$\displaystyle|\lambda|\,\|W\|_{\mu}\quad\mbox{ for any
$\lambda\in\mathbb{C}$},$ (1.6) $\displaystyle\|W\|_{\mu}$ $\displaystyle=$
$\displaystyle\|UW\|_{\mu}\quad\mbox{ for any isometry $U$}.$ (1.7)

The $\mu$-norm is motivated by the problem of the extension of the measure
entropy222known also as the Kolmogorov-Sinai entropy to the case of quantum
systems. Now we describe briefly the idea.

Let $F:{\cal X}\to{\cal X}$ be an endomorphism of the probability space
$({\cal X},{\cal B},\mu)$. This means that for any $X\in{\cal B}$ the set
$F^{-1}(X)$ (the complete preimage) also lies in ${\cal B}$ and
$\mu(X)=\mu(F^{-1}(X))$. Invertible endomorphisms are called automorphisms.
Let $\operatorname{End}({\cal X})$ denote the semigroup of all endomorphisms
of $({\cal X},{\cal B},\mu)$. There are two standard constructions.

(1) Any $F\in\operatorname{End}({\cal X},\mu)$ generates the isometry (a
unitary operator if $F$ is an automotphism) $U_{F}$ on ${\cal H}$ (the Koopman
operator):

$L^{2}({\cal X},\mu)\ni f\mapsto U_{F}f=f\circ F,\qquad
U_{F}=:\operatorname{Koop}(F).$

(2) For any $F\in\operatorname{End}({\cal X},\mu)$ it is possible to compute
the measure entropy $h(F)$.

Our question is as follows. Is it possible to determine in some ‘‘natural
way’’ a real nonnegative function ${\mathfrak{h}}$ on the semigroup of
isometries $\operatorname{Iso}({\cal H})$ so that the diagram

$\begin{array}[]{rcl}&\operatorname{End}({\cal X},\mu)&\\\
h\swarrow&&\searrow\operatorname{Koop}\\\
\mathbb{R}_{+}&\stackrel{{\scriptstyle{\mathfrak{h}}}}{{\longleftarrow}}&\operatorname{Iso}({\cal
H})\end{array}$

is commutative?

Recall the construction of the measure entropy of an endomorphism. Let $J_{N}$
be the set of multiindices $j=(j_{0},\ldots,j_{N})$, where any component
$j_{n}$ takes values in the set $\\{0,\ldots,K\\}$. For any partition
$\chi=\\{X_{0},\ldots,X_{K}\\}$ and $j\in J_{N}$ we define

${\bf X}_{j}=F^{-N}(X_{j_{N}})\cap\ldots\cap F^{-1}(X_{j_{1}})\cap X_{j_{0}}.$

We define $h_{F}(\chi,N+1)$ by

$h_{F}(\chi,N+1)=-\sum_{j\in J_{N}}\mu({\bf X}_{j})\log\mu({\bf X}_{j}).$

The function $h_{F}$, as a function of the second argument, is subadditive:
$h_{F}(\chi,n+m)\leq h_{F}(\chi,n)+h_{F}(\chi,m)$. This implies existence of
the limit

$h_{F}(\chi)=\lim_{n\to\infty}\frac{1}{n}h_{F}(\chi,n).$

Finally, the measure entropy is defined by

$h(F)=\sup_{\chi}h_{F}(\chi).$

A rough idea is to construct the entropy of a unitary operator $U$ analogously
with the following difference. Instead of ${\bf X}_{j}$ we take

${\mathfrak{X}}_{j}=\pi_{X_{j_{N}}}U\pi_{X_{j_{N-1}}}U\ldots
U\pi_{X_{j_{1}}}U\pi_{X_{j_{0}}}.$ (1.8)

We define

${\mathfrak{h}}_{U}(\chi,N+1)=-\sum_{j\in
J_{N}}\|{\mathfrak{X}}_{j}\|_{\mu}^{2}\log\|{\mathfrak{X}}_{j}\|_{\mu}^{2}.$
(1.9)

Other details are the same:

${\mathfrak{h}}_{U}(\chi)=\lim_{n\to\infty}\frac{1}{n}{\mathfrak{h}}(\chi,n),\qquad{\mathfrak{h}}(U)=\sup_{\chi}{\mathfrak{h}}_{U}(\chi)$
(1.10)

(provided the limit (1.10) exists).

In [22] we prove that for any automorphism $F$

${\mathfrak{h}}(U_{F})=h(F).$ (1.11)

In the literature there exist several attempts to extend the concept of the
measure entropy to quantum systems, see [6, 17, 18, 2, 3, 15, 13] and many
others. In [1] several mutual relations between these approaches are given.
Some works (for example, [2, 21, 19, 4, 14]) deal with the finite-dimensional
case $(\\#{\cal X}<\infty)$. In [8, 9] 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 entropy ${\mathfrak{h}}$ or its
analogs in examples. We plan to do this in forthcoming papers.

The above definition of ${\mathfrak{h}}(U)$ meets several technical problems,
including the question on subadditivity333This subadditivity is important for
the existence of the limit (1.10). of ${\mathfrak{h}}_{U}(\chi,n)$ and the
inequality ${\mathfrak{h}}_{U}(\chi)\leq{\mathfrak{h}}_{U}(\kappa)$ if
$\kappa$ is a subpartition of $\chi$444This inequality is necessary if we want
to approach the supremum (1.10) on fine partitions $\chi$.. We plan to change
slightly the definition by replacing $\|{\mathfrak{X}}_{j}\|_{\mu}^{2}$ in
(1.9) by another quantity similar to it to satisfy these two properties. To
this end we associate with $U$ a bistochastic operator on $L^{1}({\cal
X},\mu)$. Then the entropy of $U$ may be defined analogously to [8, 9]. 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 the $\mu$-norm remains central in all our constructions.

We use the $\mu$-norm to construct a bistochastic operator in three cases:
Koopman operators, the case $\\#{\cal X}<\infty$ and regular operators (a
special class of operators defined in Section 4) for ${\cal X}=\mathbb{T}$. It
is more or less clear that the circle $\mathbb{T}$ may be replaced by the
torus $\mathbb{T}^{d}$, 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 the $\mu$-norm $\|\cdot\|_{\mu}$. We
have started this in [22]. The present paper is a continuation of this
program.

##  2 Notation and previous results

Here we collect basic results from [22]. We will refer to some of them below.

(1). $\|\pi_{X}\|_{\mu}^{2}=\mu(X)$ for any $X\in{\cal B}$.

(2). If $\chi^{\prime}$ is a subpartition of $\chi$ then ${\cal
M}_{\chi^{\prime}}(W)\leq{\cal M}_{\chi}(W)$. Hence the quantities ${\cal
M}_{\chi}(W)$ approach the infinum (1.3) on fine (small scale) partitions.

(3). For any two bounded operators $W_{1}$ and $W_{2}$

$\|W_{1}+W_{2}\|_{\mu}\leq\|W_{1}\|_{\mu}+\|W_{2}\|_{\mu}.$

This triangle inequality combined with (1.6) imply that $\|\cdot\|_{\mu}$ is a
seminorm on the space of bounded operators on ${\cal H}$.

(4). Let $F$ be an automorphism of $({\cal X},{\cal B},\mu)$ and let
$U_{F}=\operatorname{Koop}(F)$. Then

$U_{F}\pi_{X}=\pi_{F^{-1}(X)}U_{F}\quad\mbox{for any }X\in{\cal B}.$ (2.1)

For any bounded operator $W$

$\|WU_{F}\|_{\mu}=\|W\|_{\mu}.$ (2.2)

This implies $\|U_{F}^{-1}WU_{F}\|_{\mu}=\|W\|_{\mu}$. Informally speaking,
this means that measure preserving coordinate changes on ${\cal X}$ preserve
the $\mu$-norm.

(5). $\|\cdot\|_{\mu}$ is a continuous function in the $L^{2}({\cal X},\mu)$
operator topology.

(6). If the measure $\mu$ has no atoms then $\|W+W_{0}\|_{\mu}=\|W\|_{\mu}$
for any bounded $W$ and compact $W_{0}$. In particular, $\mu$-norm of any
compact operator vanishes. In fact, there exist non-compact operators with
zero $\mu$-norm.

(7). Given $g\in L^{\infty}({\cal X},\mu)$ let $\widehat{g}$ be the
multiplication operator defined by $f\mapsto\widehat{g}f=gf$. Then
$\|\widehat{g}\|_{\mu}=\|g\|$.

(8). Suppose ${\cal X}=\\{1,\ldots,J\\}$ is finite and the measure of any
element equals $1/J$. Then ${\cal H}$ is isomorphic to $\mathbb{C}^{J}$ with
the Hermitian product $\langle
f,g\rangle_{J}=\frac{1}{J}\sum_{j=1}^{J}f(j)\overline{g(j)}$. Let

$f\mapsto Wf,\quad(Wf)(k)=\sum_{j=1}^{J}W(k,j)f(j)$

be an operator on ${\cal H}$. Then

$\|W\|_{\mu}^{2}=\frac{1}{J}\sum_{j,k=1}^{J}|W(k,j)|^{2}.$ (2.3)

(9). For any partition $\\{X_{1},\ldots,X_{K}\\}$ of ${\cal X}$

$\|W\|_{\mu}^{2}=\sum_{k=1}^{K}\|W\pi_{X_{k}}\|_{\mu}^{2},\quad\|W\|_{\mu}^{2}\leq\sum_{k=1}^{K}\|\pi_{X_{k}}W\|_{\mu}^{2}.$

(10). Let ${\cal X}$ be a compact metric space and $\mu$ a Borel measure
w.r.t. the corresponding topology. Let $B_{r}(x)\subset{\cal X}$ denote the
open ball with center at $x$ and radius $r$. Then for any $x\in{\cal X}$ the
limit

$\vartheta(x)=\lim_{\varepsilon\searrow 0}\|W\pi_{B_{\varepsilon}(x)}\|^{2}$

exists, the function $\vartheta$ is measurable and
$\|W\|_{\mu}^{2}\leq\int_{\cal X}\vartheta\,d\mu$. There exists an example
which shows that in general this inequality is strict. However
$\|W\|_{\mu}^{2}=\int_{\cal X}\vartheta\,d\mu$ provided two additional
conditions C1 and C2 hold:

C1. The function $\vartheta$ is continuous.

C2. There exists $c>0$ such that for any open ${\cal O}\subset{\cal X}$,
$\operatorname{diam}({\cal O})\leq\varepsilon$ and any $x\in{\cal O}$ there
exists a function $f=\pi_{\cal O}f$ satisfying

$\Big{|}\|Wf\|^{2}-\vartheta(x)\|f\|^{2}\Big{|}\leq\gamma(\varepsilon)\|f\|^{2},\quad
c<\big{|}f|_{\cal O}\big{|}<c^{-1},$

where $\gamma(\varepsilon)\to 0$ as $\varepsilon\to 0$. As usual, $f|_{\cal
O}$ denotes restriction of $f$ to the set ${\cal O}$.

The further results from [22] we mention here concern the case ${\cal
X}=\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ with the Lebesgue measure $\mu$. We
expect that they can be extended to the case ${\cal X}=\mathbb{T}^{d}$, $d>1$.

(11). Let ${\cal X}=\mathbb{T}$ be a circle with the Lebesgue measure
$d\mu=\frac{1}{2\pi}dx$. For any bounded sequence
$\\{\lambda_{k}\\}_{k\in\mathbb{Z}}$ we consider the distribution
$\lambda(x)=\sum\lambda_{k}e^{ikx}$. The convolution operator

$f\mapsto\operatorname{Conv}_{\lambda}f=\lambda*f:=\int_{\mathbb{T}}\lambda(y)f(\cdot-y)\,dy$

is bounded:
$\|\operatorname{Conv}_{\lambda}\|=\sup_{k\in\mathbb{Z}}|\lambda_{k}|$. Then

$\|\operatorname{Conv}_{\lambda}\|_{\mu}^{2}=\rho(\lambda),\qquad\rho(\lambda)=\limsup_{\\#I\to\infty}\rho_{I}(\lambda),\quad\rho_{I}(\lambda)=\frac{1}{\\#I}\sum_{k\in
I}|\lambda_{k}|^{2},$

where $I\subset\mathbb{Z}$ are integer intervals.

(12). One of the main technical tools in the analysis of the $\mu$-norm in the
case ${\cal X}=\mathbb{T}$ is the following lemma on Fourier coefficients of
localized functions on the circle.

###### Lemma 2.1

Let $Y=[a-\varepsilon,a+\varepsilon]$ and $\displaystyle
f=\pi_{Y}f=\sum_{k\in\mathbb{Z}}f_{k}e^{ikx}\in L^{2}(\mathbb{T})$. Then for
any integer interval $J$ and any $m\in\mathbb{Z}$

$\displaystyle\|f-e^{im(x-a)}f\|$ $\displaystyle\leq$
$\displaystyle|m|\varepsilon\|f\|,$ (2.4)
$\displaystyle|f_{m}-e^{ila}f_{m+l}|$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon^{3/2}}{\sqrt{\pi}}|l|\|f\|,$ (2.5)
$\displaystyle\Big{|}\sum_{k\in\mathbb{Z}}e^{-ima}f_{k}\overline{f}_{k+m}-\|f\|^{2}\Big{|}$
$\displaystyle\leq$ $\displaystyle|m|\varepsilon\|f\|^{2}.$ (2.6)

(13). Consider the operator $W=(W_{j,k})_{j,k\in\mathbb{Z}^{d}}$ on ${\cal
H}=L^{2}(\mathbb{T})$:

$f=\sum_{k\in\mathbb{Z}}f_{k}e^{ikx}\mapsto
Wf=\sum_{j,k\in\mathbb{Z}}W_{j,k}f_{k}e^{ijx}.$

We say that $W$ is of diagonal type ($W\in{\cal DT}(\mathbb{T})$) if

$\sup_{j\in\mathbb{Z}}|W_{j+k,j}|=c_{k}<\infty,\quad
k\in\mathbb{Z}\quad\mbox{and}\quad\sum_{k\in\mathbb{Z}}c_{k}={\bf c}<\infty.$

The sequence $c_{s}$ is said to be the majorating sequence for $W\in{\cal
DT}(\mathbb{T})$. We define the norm $\|W\|_{\cal DT}={\bf c}$.

Operators from ${\cal DT}(\mathbb{T})$ 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 operator $W^{*}$ if $W\in{\cal DT}(\mathbb{T})$.

(d) Linear combinations and products of operators of diagonal type. Moreover,

$\|W^{\prime}W^{\prime\prime}\|_{\cal DT}\leq\|W^{\prime}\|_{\cal
DT}\|W^{\prime\prime}\|_{\cal DT}\quad\mbox{for all
$W^{\prime},W^{\prime\prime}\in{\cal DT}(\mathbb{T})$}.$

(14). We prove that the normed space $\big{(}{\cal
DT}(\mathbb{T}),\|\cdot\|_{\cal DT}\big{)}$ is closed. As a corollary we
obtain that $\big{(}{\cal DT}(\mathbb{T}),\|\cdot\|_{\cal DT}\big{)}$ is a
$C^{*}$-algebra.

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

$\displaystyle\|W\|_{\mu}\leq\|W\|\leq\|W\|_{\cal DT}\quad\mbox{for any
$W\in{\cal DT}(\mathbb{T})$},$ (2.7)
$\displaystyle\|f\|_{\infty}\leq\|\widehat{f}\|_{\cal DT}\quad\mbox{for any
$f$ with absolutely converging Fourier series},$ (2.8)

where $\widehat{f}$ is the operator of multiplication by $f$.

(16). We associate with $W\in{\cal DT}(\mathbb{T})$ and any point
$a\in\mathbb{T}$ the distribution $L_{a}$,

$L_{a}=\sum_{j\in\mathbb{Z}}w_{j}(a)e^{ijx},\qquad
w_{j}(a)=\sum_{k\in\mathbb{Z}}W_{j,k}e^{i(j-k)a}.$ (2.9)

For any $l\in\mathbb{Z}$ and $a\in\mathbb{T}$ we have the estimate

$|w_{l}(a)|\leq{\bf c}=\|W\|_{\cal DT}.$ (2.10)

We prove that for any $W\in{\cal DT}(\mathbb{T})$ the function
$a\mapsto\rho(L_{a})=\limsup_{\\#I\to\infty}\rho_{I}(L_{a})$
($I\subset\mathbb{Z}$ are intervals, $\rho_{I}$ is defined in item (11)) is
continuous and

$\|W\|_{\mu}^{2}=\frac{1}{2\pi}\int_{\mathbb{T}}\rho(L_{a})\,da.$ (2.11)

(17). For any operator $W\in{\cal DT}(\mathbb{T})$ we introduce the average
trace of $W^{*}W$ by

${\bf T}(W)=\limsup_{\\#I\to\infty}\frac{1}{\\#I}\sum_{j\in\mathbb{Z},\,l\in
I}|W_{l,j}|^{2},$

where $I\subset\mathbb{Z}$ are intervals. Then

${\bf T}(W)\leq\|W\|_{\mu}^{2}.$ (2.12)

(18). We also prove that if $U\in{\cal DT}(\mathbb{T})$ is a unitary operator
then

${\bf T}(W)={\bf T}(WU)={\bf T}(UW).$ (2.13)

##  3 Main results

In this section we collect main results of the present paper. In short, these
results concern (1) properties of the $\mu$-norm on ${\cal R}(\mathbb{T})$, a
special class of operators on $L^{2}(\mathbb{T})$, and (2) a construction of a
bistochastic operator ${\cal W}$ on $L^{1}({\cal X},\mu)$ associated with an
operator $W$ under some conditions, imposed on $W$.

  * •

We say that an operator $W\in{\cal DT}(\mathbb{T})$ is regular (the notation
is $W\in{\cal R}(\mathbb{T})$) if for any $m,n\in\mathbb{Z}$ there exists the
limit

$\omega_{m,n}=\lim_{\\#I\to\infty}\omega_{I,m,n},\qquad\omega_{I,m,n}=\frac{1}{\\#I}\sum_{j\in\mathbb{Z},\,l\in
I}W_{l+m,j}\overline{W}_{l,j+n}.$

  * •

By Lemma 4.2 ${\cal R}(\mathbb{T})$ is a cone closed with respect to the norm
$\|\cdot\|_{\cal DT}$.

  * •

By Lemma 4.3 $\|W\|_{\mu}^{2}={\bf T}(W)$ for any $W\in{\cal R}(\mathbb{T})$
(compare with (2.12)).

  * •

Let ${\cal ACF}(\mathbb{T})$ be the space of functions on $\mathbb{T}$ with
absolutely converging Fourier series. Then the space of multiplication
operators $\widehat{g}$ by functions $g\in{\cal ACF}(\mathbb{T})$ form a
$C^{*}$-subalgebra in the $C^{*}$-algebra ${\cal DT}(\mathbb{T})$.

  * •

In Section 5 we present several examples of regular operators.

Let $W\in{\cal DT}(\mathbb{T})$ be an operator with periodic matrix i.e.,
there exists $\tau\in\mathbb{N}$ such that $W_{j+\tau,k+\tau}=W_{j,k}$ for any
$j,k\in\mathbb{Z}$. Any such operator is regular (Lemma 5.1).

Another example of a regular operator is $\operatorname{Conv}_{\lambda}$,
where $\lambda_{k}=e^{i\tau k^{2}}$.

  * •

By Proposition 6.1 for any $W\in{\cal R}(\mathbb{T})$ and $g_{1},g_{2}\in{\cal
ACF}(\mathbb{T})$ the operator $\widehat{g}_{1}W\widehat{g}_{2}$ is regular.

  * •

In Section 6 we associate with operators $W$ from the following three classes

(1) Koopman operators $\operatorname{Koop}(F)$, $F\in\operatorname{Aut}({\cal
X},\mu)$,

(2) operators in the case $\\#{\cal X}<\infty$,

(3) operators from ${\cal R}(\mathbb{T})$

a measure
$d\mu_{W}(x^{\prime},x^{\prime\prime})=\nu(x^{\prime},x^{\prime\prime})\,d\mu(x^{\prime})d\mu(x^{\prime\prime})$
on ${\cal X}\times{\cal X}$ such that for any ‘‘sufficiently regular’’
functions $g^{\prime},g^{\prime\prime}:{\cal X}\to\mathbb{C}$

$\|\widehat{g}^{\prime}W\widehat{g}^{\prime\prime}\|_{\mu}^{2}=\int_{{\cal
X}^{2}}|g^{\prime}(x^{\prime})|^{2}|g^{\prime\prime}(x^{\prime\prime})|^{2}\,d\mu_{W}(x^{\prime},x^{\prime\prime}).$

  * •

In Section 7 we introduce the operator

$L^{1}({\cal X},\mu)\ni f\mapsto{\cal W}f=\int_{\cal
X}\nu(\cdot,a)f(a)\,d\mu(a)$

and prove (Lemma 7.1) that ${\cal W}$ is a bistochastic operator on
$L^{1}({\cal X},\mu)$.

##  4 Regular operators

###  4.1 Definition of $\omega_{m,n}$

###### Definition 4.1

We say that $W\in{\cal DT}(\mathbb{T})$ is regular ($W\in{\cal
R}(\mathbb{T})$) if for any $m,n\in\mathbb{Z}$ there exists the limit

$\lim_{\\#I\to\infty}\omega_{I,m,n}=\omega_{m,n},\qquad\omega_{I,m,n}=\frac{1}{\\#I}\sum_{j\in\mathbb{Z},\,l\in
I}W_{l+m,j}\overline{W}_{l,j+n},$ (4.1)

where $I$ are integer intervals.

Note that $\omega_{0,0}$ coincides with the average trace of $W^{*}W$:

$\omega_{0,0}={\bf T}(W)\quad\mbox{if $W\in{\cal R}(\mathbb{T})$}.$ (4.2)

For any integer interval $I$ we put

$v_{I,m}(a)=\sum_{l\in I}\frac{w_{l+m}(a)\overline{w}_{l}(a)}{\\#I},$ (4.3)

where the functions $w_{l}(a)$ are defined in (2.9).

###### Lemma 4.1

Suppose $W$ is regular. Then for any $m\in\mathbb{Z}$ there exists the limit

$\lim_{\\#I\to\infty}v_{I,m}(a)=v_{m}(a),\quad
v_{m}(a)=\sum_{n\in\mathbb{Z}}\omega_{m,n}e^{i(m+n)a}$ (4.4)

uniformly in $a\in\mathbb{T}$. The Fourier series of the function $v_{m}$
absolutely converges.

Proof. By (2.9)

$\displaystyle v_{m}(a)$ $\displaystyle=$
$\displaystyle\lim_{\\#I\to\infty}\frac{1}{\\#I}\sum_{j,k\in\mathbb{Z},\,l\in
I}W_{l+m,j}\overline{W}_{l,k}e^{i(m-j+k)a}$ $\displaystyle=$
$\displaystyle\lim_{\\#I\to\infty}\sum_{n\in\mathbb{Z}}\omega_{I,m,n}e^{i(m+n)a}\;=\;\sum_{n\in\mathbb{Z}}\omega_{m,n}e^{i(m+n)a}.$

By (10.1) the limit is uniform in $a$. By (10.2) this Fourier series
absolutely converges.

Note that by (2.10) for any interval $I\subset\mathbb{Z}$, any
$m\in\mathbb{Z}$, and any $a\in\mathbb{T}$

$|v_{I,m}(a)|\leq{\bf c}^{2},\quad|v_{m}(a)|\leq{\bf c}^{2}.$ (4.5)

###  4.2 Closeness with respect to $\|\cdot\|_{{\cal DT}}$

If $W\in{\cal R}(\mathbb{T})$ then for any $\lambda\in\mathbb{C}$ the operator
$\lambda W$ is also regular. Hence, regular operators form a cone ${\cal
R}(\mathbb{T})\subset{\cal DT}(\mathbb{T})$.

###### Lemma 4.2

The cone ${\cal R}(\mathbb{T})$ is closed with respect to the norm
$\|\cdot\|_{{\cal DT}}$.

Proof. Suppose $\\{W_{p}\\}_{p\in\mathbb{N}}$, $W_{p}\in{\cal R}(\mathbb{T})$
is a Cauchy sequence. By (14) there exists $W=\lim_{p\to\infty}W_{p}$, where
the limit is taken with respect to the norm $\|\cdot\|_{{\cal DT}}$.

For any $\varepsilon>0$ there exists positive $N$ such that

$\mbox{for any integer $p,q>N$ we have:}\quad\|W_{p}-W_{q}\|_{\cal
DT}<\varepsilon.$ (4.6)

We define $(\omega_{p})_{I,m,n}$ and $(\omega_{p})_{m,n}$ by (4.1), where $W$
is replaced by $W_{p}$. Then

$\displaystyle\big{|}(\omega_{p})_{I,m,n}-(\omega_{q})_{I,m,n}\big{|}$
$\displaystyle=$ $\displaystyle\frac{1}{\\#I}\bigg{|}\sum_{j\in\mathbb{Z},l\in
I}\bigg{(}(W_{p})_{l+m,j}(\overline{W}_{p})_{l,j+n}-(W_{q})_{l+m,j}(\overline{W}_{q})_{l,j+n}\bigg{)}\bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\\#I}\big{(}\Sigma_{1}+\Sigma_{2}\big{)},$
$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z},l\in
I}\bigg{|}(W_{p})_{l+m,j}\bigg{(}(\overline{W}_{p})_{l,j+n}-(\overline{W}_{q})_{l,j+n}\bigg{)}\bigg{|},$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z},l\in
I}\bigg{|}\bigg{(}(W_{p})_{l+m,j}-(W_{q})_{l+m,j}\bigg{)}(\overline{W}_{q})_{l,j+n}\bigg{|}.$

To estimate the sums $\Sigma_{1}$ and $\Sigma_{2}$, we put

$(c_{p})_{k}=\sup_{j\in\mathbb{Z}}|(W_{p})_{k+j,j}|,\quad(c_{q})_{k}=\sup_{j\in\mathbb{Z}}|(W_{q})_{k+j,j}|,\quad
d_{k}=\sup_{j\in\mathbb{Z}}|(W_{p})_{k+j,j}-(W_{q})_{k+j,j}|.$

The sums $\sum_{k}(c_{p})_{k}$ are uniformly bounded:

$\sum_{k\in\mathbb{Z}}(c_{p})_{k}\leq\tilde{\bf
c},\quad\sum_{k\in\mathbb{Z}}(c_{q})_{k}\leq\tilde{\bf c}\quad\mbox{for some
constant $\tilde{\bf c}$}.$ (4.7)

Moreover, by (4.6)

$\sum_{k\in\mathbb{Z}}d_{k}<\varepsilon.$ (4.8)

By (4.7) and (4.8)

$\Sigma_{1}\leq\sum_{j\in\mathbb{Z},\,l\in
I}(c_{p})_{l+m-j}d_{l-j-n}\leq\\#I\tilde{\bf c}\varepsilon.$

Analogously $\Sigma_{2}\leq\\#I\tilde{\bf c}\varepsilon$. This implies that
for any interval $I$

$\big{|}(\omega_{p})_{I,m,n}-(\omega_{q})_{I,m,n}\big{|}\leq 2\tilde{\bf
c}\varepsilon.$

Hence, $\big{|}(\omega_{p})_{m,n}-(\omega_{q})_{m,n}\big{|}\leq 2\tilde{\bf
c}\varepsilon$ i.e., for any integer $m,n$ the sequence $(\omega_{p})_{m,n}$,
$p\in\mathbb{N}$ is a Cauchy sequence.

###  4.3 $\mu$-norm of a regular operator

###### Lemma 4.3

If $W\in{\cal R}(\mathbb{T})$ then (compare with (2.12))

$\|W\|_{\mu}^{2}={\bf T}(W).$ (4.9)

Proof. By (2.9), (2.11) and (4.3)

$\|W\|_{\mu}^{2}=\frac{1}{2\pi}\int_{\mathbb{T}}\lim_{\\#I\to\infty}v_{I,0}(a)\,da.$
(4.10)

By Lemma 4.1 the limit $v_{0}(a)=\lim_{\\#I\to\infty}v_{I,0}(a)$ exists for
any $a\in\mathbb{T}$ and by (4.5) $|v_{I,0}(a)|\leq{\bf c}^{2}$ for all $I$
and $a$. Therefore by the Lebesgue theorem on bounded convergence we may
exchange the integration and the limit:

$\displaystyle\|W\|_{\mu}^{2}$ $\displaystyle=$
$\displaystyle\lim_{\\#I\to\infty}\frac{1}{2\pi}\int_{\mathbb{T}}v_{I,0}(a)\,da=\lim_{\\#I\to\infty}\frac{1}{2\pi}\int_{\mathbb{T}}\sum_{j,k\in\mathbb{Z},\,l\in
I}\frac{1}{\\#I}W_{l,j}\overline{W}_{l,k}e^{i(k-j)a}\,da$ $\displaystyle=$
$\displaystyle\lim_{\\#I\to\infty}\sum_{j\in\mathbb{Z},\,l\in
I}\frac{1}{\\#I}W_{l,j}\overline{W}_{l,j}\;=\;\omega_{0,0}.$

By (4.1) this implies (4.9).

Assertion (18) and Lemma 4.3 imply the following.

###### Corollary 4.1

Suppose $W,U\in{\cal DT}(\mathbb{T})$, where $U$ is unitary and both $W$ and
$UWU^{-1}$ are regular. Then by (18) and (1.7)

$\|W\|_{\mu}=\|UWU^{-1}\|_{\mu}=\|WU^{-1}\|_{\mu}.$

##  5 Regular operators: examples

###### Definition 5.1

We say that the matrix $(W_{k,j})$ of the operator $W\in{\cal DT}(\mathbb{T})$
is $\tau$-periodic, $\tau\in\mathbb{N}$, if

$W_{k+\tau,j+\tau}=W_{k,j}\quad\mbox{for any }k,j\in\mathbb{Z}.$

In particular, for any $g\in{\cal ACF}(\mathbb{T})$ matrix of the operator
$\widehat{g}\in{\cal DT}(\mathbb{T})$ is 1-periodic.

###### Lemma 5.1

Suppose $W\in{\cal DT}(\mathbb{T})$ is an operator with $\tau$-periodic
matrix, $\tau\in\mathbb{N}$. Then $W\in{\cal R}(\mathbb{T})$ and
$\omega_{m,n}=\breve{\omega}_{m,n}$,

$\breve{\omega}_{m,n}=\frac{1}{\tau}\sum_{j\in\mathbb{Z},\,l\in
J}W_{l+m,j}\overline{W}_{l,j+n},$

where $J\subset\mathbb{Z}$ is any interval with $\\#J=\tau$.

Proof. By $\tau$-periodicity $\breve{\omega}_{m,n}$ does not depend on $J$.
For any $I=\\{s,s+1,\ldots,s+K\\}$ let $I^{\prime}\subset I$ be the maximal
integer interval of the form $I^{\prime}=\\{s,s+1,\ldots,s+q\tau-1\\}$,
$q\in\mathbb{Z}$. We put
$I^{\prime\prime}=\\{s+q\tau,s+q\tau+1,\ldots,s+K\\}$. Then
$\\#I^{\prime\prime}<\tau$, $\omega_{I^{\prime},m,n}=\breve{\omega}_{m,n}$
while $\omega_{I,m,n}-\omega_{I^{\prime},m,n}=A_{1}+A_{2}$,

$A_{1}=\frac{1}{\\#I}\sum_{l\in I^{\prime\prime}\setminus
I^{\prime}}W_{l+m,j}\overline{W}_{l,j+n},\quad
A_{2}=\Big{(}\frac{1}{\\#I}-\frac{1}{\\#I^{\prime}}\Big{)}\sum_{l\in
I^{\prime}}W_{l+m,j}\overline{W}_{l,j+n}.$

The inequalities

$|A_{1}|\leq\frac{\tau{\bf c}^{2}}{\\#I},\quad|A_{2}|\leq\frac{\tau{\bf
c}^{2}}{\\#I-\tau}$

imply the existence of the limit (4.1) and the equation
$\omega_{m,n}=\breve{\omega}_{m,n}$.

###### Corollary 5.1

Consider the operator $\widehat{g}$, where $g\in{\cal ACF}(\mathbb{T})$. After
simple calculations we obtain:

$\omega_{m,n}(\widehat{g})={\bf g}_{m+n},\qquad
g(x)\overline{g}(x)=\sum_{k\in\mathbb{Z}}{\bf g}_{k}e^{ikx},$ (5.1)

where the last equation is the definition of ${\bf g}_{k}$.

###### Lemma 5.2

Let $\\{\lambda_{k}\\}_{k\in\mathbb{Z}}$ be defined by $\lambda_{k}=e^{i\tau
k^{2}}$. Then the operator $\operatorname{Conv}_{\lambda}$ is regular:

$\omega_{m,n}=\delta_{0,m+n}\delta_{\tau m,\pi\mathbb{Z}}e^{i\tau
m^{2}},\qquad\delta_{\tau
m,\pi\mathbb{Z}}=\left\\{\begin{array}[]{cl}1&\mbox{if }\tau
m/\pi\in\mathbb{Z},\\\ 0&\mbox{if }\tau
m/\pi\not\in\mathbb{Z}.\end{array}\right.$

Proof. In this case $W_{k,j}=\delta_{kj}e^{i\tau k^{2}}$ and

$\displaystyle\omega_{I,m,n}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{j\in\mathbb{Z},\,l\in
I}\delta_{l+m,j}\delta_{l,j+n}e^{i\tau((l+m)^{2}-l^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{l\in I}\delta_{l,l+m+n}e^{i\tau m^{2}+2i\tau
ml}\;=\;\frac{1}{\\#I}\delta_{0,m+n}e^{i\tau m^{2}}\sum_{l\in I}e^{2i\tau
ml}.$

Hence, if $\tau m/\pi\in\mathbb{Z}$ then $\omega_{m,n}=\delta_{0,m+n}e^{i\tau
m^{2}}$. If $\tau m/\pi\not\in\mathbb{Z}$ then $\omega_{m,n}=0$.

##  6 Measure associated with an operator

###  6.1 Koopman operator

Let $({\cal X},{\cal B},\mu)$ be a probability space.

###### Lemma 6.1

$F\in\operatorname{Aut}({\cal X},\mu)$, $g_{0},\ldots,g_{K}\in
L^{\infty}({\cal X},\mu)$. Then

$\big{\|}\widehat{g}_{K}U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\big{\|}^{2}_{\mu}=\int_{\cal X}|g_{K}\circ
F^{K}|^{2}|g_{K-1}\circ F^{K-1}|^{2}\ldots|g_{0}|^{2}\,d\mu$ (6.1)

Proof. Take a small constant $\sigma>0$ and consider the partition
$\\{X_{1},\ldots,X_{J}\\}$ such that555We can use the same partition for all
the functions $g_{k}$.

$\|g_{k}-\varphi_{k}\|_{\infty}<\sigma\|g_{k}\|_{\infty},\quad\varphi_{k}=\sum_{j=1}^{J}g_{k,j}{\bf
1}_{X_{j}},\quad\|\varphi_{k}\|_{\infty}\leq\|g_{k}\|_{\infty},\qquad
k=0,\ldots,K.$ (6.2)

Here $g_{k,j}\in\mathbb{C}$ are some constants. We put
$S=\widehat{\varphi}_{K}U_{F}\widehat{\varphi}_{K-1}\ldots
U_{F}\widehat{\varphi}_{0}$. By the triangle inequality the quantity

$\Big{|}\|\widehat{g}_{K}U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\|_{\mu}-\|S\|_{\mu}\Big{|}$

does not exceed

$\displaystyle\Big{|}\|\widehat{g}_{K}U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\|_{\mu}-\|\widehat{\varphi}_{K}U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\|_{\mu}\Big{|}$ $\displaystyle+$
$\displaystyle\Big{|}\|\widehat{\varphi}_{K}U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\|_{\mu}-\|\widehat{\varphi}_{K}U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\|_{\mu}\Big{|}$ $\displaystyle+$
$\displaystyle\ldots\;+\;\Big{|}\|\widehat{\varphi}_{K}U_{F}\widehat{\varphi}_{K-1}\ldots\widehat{\varphi}_{1}U_{F}\widehat{g}_{0}\|_{\mu}-\|\widehat{\varphi}_{K}U_{F}\widehat{\varphi}_{K-1}\ldots\widehat{\varphi}_{1}U_{F}\widehat{\varphi}_{0}\|_{\mu}\Big{|}$
$\displaystyle\leq$
$\displaystyle\|(\widehat{g}_{K}-\widehat{\varphi}_{K})U_{F}\widehat{g}_{K-1}\ldots
U_{F}\widehat{g}_{0}\|_{\mu}+\|\widehat{\varphi}_{K}U_{F}(\widehat{g}_{K-1}-\widehat{\varphi}_{K-1})\ldots
U_{F}\widehat{g}_{0}\|_{\mu}$ $\displaystyle+$
$\displaystyle\ldots\;+\|\widehat{\varphi}_{K}U_{F}\widehat{\varphi}_{K-1}\ldots\widehat{\varphi}_{1}U_{F}(\widehat{g}_{0}-\widehat{\varphi}_{0})\|_{\mu}$
$\displaystyle\leq$ $\displaystyle(K+1)\sigma\prod_{0\leq k\leq
K}\|g_{k}\|_{\infty}.$

By (2.1)

$S=\sum_{j_{0},\ldots,j_{K}}g_{K,j_{K}}g_{K-1,j_{K-1}}\ldots
g_{0,j_{0}}\pi_{X_{j_{K}}}\pi_{F^{-1}(X_{j_{K-1}})}\ldots\pi_{F^{-K}(X_{j_{0}})}U_{F}^{K}.$

We put $X_{j_{K},\ldots,j_{0}}=X_{j_{K}}\cap F^{-1}(X_{j_{K-1}})\cap\ldots\cap
F^{-K}(X_{j_{0}})$. Note that for any two sets
$X^{\prime}=X_{j^{\prime}_{K},\ldots,j^{\prime}_{0}}$ and
$X^{\prime\prime}=X_{j^{\prime\prime}_{K},\ldots,j^{\prime\prime}_{0}}$ we
have: $\mu(X^{\prime}\cap X^{\prime\prime})=0$ if the collections of indices
$j^{\prime}_{K},\ldots,j^{\prime}_{0}$ and
$j^{\prime\prime}_{K},\ldots,j^{\prime\prime}_{0}$ do not coincide. Then by
(2.2) and (1) we obtain:

$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum_{j_{0},\ldots,j_{K}}|g_{K,j_{K}}g_{K-1,j_{K-1}}\ldots
g_{0,j_{0}}|^{2}\|\pi_{X_{j_{K},\ldots,j_{0}}}\|_{\mu}^{2}$ (6.3)
$\displaystyle=$
$\displaystyle\sum_{j_{0},\ldots,j_{K}}|g_{K,j_{K}}g_{K-1,j_{K-1}}\ldots
g_{0,j_{0}}|^{2}\mu(X_{j_{K},\ldots,j_{0}}).$

Equation (6.2) implies

$\varphi_{k}\circ F^{k}=\sum_{j=1}^{J}g_{k,j}{\bf 1}_{F^{-k}(X_{j})}.$

Hence (6.3) is an integral sum for the integral

$\int_{\cal X}|\varphi_{K}\circ F^{K}|^{2}|\varphi_{K-1}\circ
F^{K-1}|^{2}\ldots|\varphi_{0}|^{2}\,d\mu.$

This integral differs from the integral (6.1) at most by $2\sigma\prod_{0\leq
k\leq K}\|g_{k}\|_{\infty}^{2}$. Since $\sigma$ is arbitrarily small, we
obtain equation (6.1).

Now suppose that ${\cal X}$ is in addition a topological space and ${\cal B}$
is the corresponding Borel $\sigma$-algebra. Consider the distribution
$\delta\in(C({\cal X}^{2}))^{*}$ (a measure on ${\cal X}^{2}$ by the Riesz
theorem) such that for any $\Phi\in C({\cal X}^{2})$

$\int_{{\cal
X}^{2}}\delta(x^{\prime},x^{\prime\prime})\Phi(x^{\prime},x^{\prime\prime})\,d\mu(x^{\prime})d\mu(x^{\prime\prime})=\int_{\cal
X}\Phi(x,x)\,d\mu(x).$ (6.4)

Taking in (6.4)
$\Phi(x^{\prime},x^{\prime\prime})=\varphi^{\prime}(x^{\prime})\varphi^{\prime\prime}(x^{\prime\prime})$,
where $\varphi^{\prime},\varphi^{\prime\prime}:{\cal X}\to\mathbb{C}$ are
arbitrary continuous functions, we obtain:

$\int_{\cal
X}\delta(x^{\prime},x^{\prime\prime})\varphi^{\prime}(x^{\prime})\,d\mu(x^{\prime})=\varphi^{\prime}(x^{\prime\prime}),\quad\int_{\cal
X}\delta(x^{\prime},x^{\prime\prime})\varphi^{\prime\prime}(x^{\prime\prime})\,d\mu(x^{\prime\prime})=\varphi^{\prime\prime}(x^{\prime}).$
(6.5)

Recall also that if $F\in\operatorname{End}({\cal X},\mu)$

$\int_{\cal X}f\,d\mu=\int_{\cal X}f\circ F\,d\mu\quad\mbox{for any }f\in
L^{1}({\cal X},\mu).$ (6.6)

Then for any continuous $F\in\operatorname{Aut}({\cal X},\mu)$

$\delta(x^{\prime},x^{\prime\prime})=\delta(F(x^{\prime}),F(x^{\prime\prime})).$
(6.7)

Indeed, for any $\Phi\in C({\cal X}^{2})$ by (6.6)

$\displaystyle\int_{{\cal
X}^{2}}\delta(F(x^{\prime}),F(x^{\prime\prime}))\Phi(x^{\prime},x^{\prime\prime})\,d\mu(x^{\prime})d\mu(x^{\prime\prime})$
$\displaystyle=$ $\displaystyle\int_{{\cal
X}^{2}}\delta(x^{\prime},x^{\prime\prime})\Phi(F^{-1}(x^{\prime}),F^{-1}(x^{\prime\prime}))\,d\mu(x^{\prime})d\mu(x^{\prime\prime})\;=\;\int_{\cal
X}\Phi(x,x)\,d\mu(x).$

###### Lemma 6.2

Let $F\in\operatorname{Aut}({\cal X},\mu)$ be continuous and
$g_{0},\ldots,g_{K}\in C({\cal X})$. Then

$\displaystyle\big{\|}\widehat{g}_{K}U_{F}\ldots\widehat{g}_{1}U_{F}\widehat{g}_{0}\big{\|}_{\mu}^{2}$
$\displaystyle=$ $\displaystyle\int_{{\cal
X}^{K+1}}|g_{K}(x_{K})|^{2}\delta(x_{K},F(x_{K-1}))\ldots|g_{1}(x_{1})|^{2}\delta(x_{1},F(x_{0}))|g_{0}(x_{0})|^{2}\,d\mu^{K+1},$
$\displaystyle\qquad d\mu^{K+1}=d\mu(x_{K})\ldots d\mu(x_{0}).$

Proof. Let $I$ denote the integral (LABEL:gU_F=int). We use in this integral
the change of coordinates

$x_{0}=x^{\prime}_{0},\quad x_{1}=F(x^{\prime}_{1}),\quad\ldots\quad
x_{K}=F^{K}(x^{\prime}_{K}),\quad d\mu^{\prime K+1}=d\mu(x^{\prime}_{K})\ldots
d\mu(x^{\prime}_{0}).$

Then

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int|g_{K}\circ
F^{K}(x^{\prime}_{K})|^{2}\delta(F^{K}(x^{\prime}_{K}),F^{K}(x^{\prime}_{K-1}))$
$\displaystyle\qquad\qquad\ldots\;|g_{1}\circ
F(x^{\prime}_{1})|^{2}\delta(F(x^{\prime}_{1}),F(x^{\prime}_{0}))|g_{0}(x^{\prime}_{0})|^{2}\,d\mu^{\prime
K+1}$ $\displaystyle=$ $\displaystyle\int_{\cal X}|g_{K}\circ
F^{K}|^{2}\ldots|g_{1}\circ F|^{2}|g_{0}|^{2}\,d\mu.$

It remains to use Lemma 6.1.

We associate with any $U_{F}=\operatorname{Koop}(F)$, where
$F\in\operatorname{Aut}({\cal X},\mu)$ is continuous,666We expect that the
continuity is inessential here. the measure $\mu_{U_{F}}$ on ${\cal X}^{2}$:

$d\mu_{U_{F}}(x^{\prime},x^{\prime\prime})=\delta(x^{\prime},F(x^{\prime\prime}))\,d\mu(x^{\prime})d\mu(x^{\prime\prime}).$
(6.9)

###  6.2 The finite-dimensional case

Let ${\cal X}=\\{1,\ldots,J\\}$ be a finite set. We identify ${\cal X}$ with
$\mathbb{Z}_{J}=\mathbb{Z}/J\mathbb{Z}$, the cyclic additive group with $J$
elements.777Below we also use the structure of a commutative ring on
$\mathbb{Z}_{J}$. Let the measure $\mu$ of any point be equal to $1/J$. Then
${\cal H}=L^{2}({\cal X},\mu)\cong(\mathbb{C}^{J},\langle\,,\rangle)$, where
$\langle\,,\rangle$ equals the standard Hermitian product divided by $J$.

We put $\eta=e^{2\pi i/J}$. Then $\eta^{J}=1$ and $\overline{\eta}=\eta^{-1}$.
The space ${\cal H}$ may be identified with the space of ‘‘discrete
trigonometric polynomials’’

$f=f(x)=\sum_{j\in\mathbb{Z}_{J}}f_{j}\eta^{jx},\qquad
f_{j}\in\mathbb{C},\quad x\in\mathbb{Z}_{J}.$ (6.10)

This polynomial representation generates on ${\cal H}$ an operation of
multiplication: for any two vectors $f^{\prime}=\sum f^{\prime}_{j}\eta^{jx}$
and $f^{\prime\prime}=\sum f^{\prime\prime}_{j}\eta^{jx}$

$f^{\prime}f^{\prime\prime}=f=\sum_{k\in\mathbb{Z}_{J}}f_{k}\eta^{kx},\qquad
f_{k}=\sum_{j\in\mathbb{Z}_{J}}f^{\prime}_{k-j}f^{\prime\prime}_{j}.$

This product introduces on ${\cal H}$ the structure of a commutative ring. The
structure of a Hilbert space is determined by

$\langle f,g\rangle=\frac{1}{J}\sum_{k\in\mathbb{Z}_{J}}f(k)\overline{g(k)}.$

The coefficients $f_{k}$ and $f(k)$ are connected by the ‘‘discrete Fourier
transform’’

$f(k)=\sum_{j\in\mathbb{Z}_{J}}f_{j}\eta^{kj},\quad
f_{j}=\frac{1}{J}\sum_{k\in\mathbb{Z}_{J}}f(k)\eta^{-kj}.$ (6.11)

For any $f$ satisfying (6.10) and any $A\subset\mathbb{Z}_{J}$ we put

$\int_{A}f(x)\,d\mu(x):=\frac{1}{J}\sum_{k\in A}f(k)=\frac{1}{J}\sum_{k\in
A,\,j\in\mathbb{Z}_{J}}f_{j}\eta^{jk}.$

Then

$\int_{\cal X}f\,d\mu=f_{0},\quad\langle f,g\rangle=\frac{1}{J}\int_{\cal
X}f\overline{g}\,d\mu.$

Consider a linear operator

$f\mapsto Wf,\qquad(Wf)_{k}=\sum_{j\in\mathbb{Z}_{J}}W_{kj}f_{j}.$

In another basis it takes the form

$(Wf)(k)=\sum_{j\in\mathbb{Z}_{J}}W(k,j)f(j).$

Equations (6.11) imply

$W(m,n)=\frac{1}{J}\sum_{j,k\in\mathbb{Z}_{J}}\eta^{mk}W_{kj}\eta^{-jn}.$
(6.12)

We define

$\omega_{m,n}=\frac{1}{J}\sum_{j,l\in\mathbb{Z}_{J}}W_{l+m,j}\overline{W}_{l,j+n},\quad\nu(x,a)=\sum_{m,n\in\mathbb{Z}_{J}}\omega_{m,n}\eta^{mx}\eta^{na},\qquad
x,a\in\mathbb{Z}_{J}.$ (6.13)

###### Lemma 6.3

$\sum_{m,n\in\mathbb{Z}_{J}}|W(m,n)|^{2}=\sum_{j,k\in\mathbb{Z}_{J}}|W_{j,k}|^{2}$.

Proof. Direct computation with the help of (6.12) and the identity
$\sum_{j\in\mathbb{Z}_{J}}\eta^{jk}=J\delta_{k,0}$.

###### Corollary 6.1

By (2.3) and Lemma 6.3

$\|W\|_{\mu}^{2}=\omega_{0,0}=\int_{{\cal X}^{2}}\nu(x,a)\,d\mu(x)d\mu(a).$
(6.14)

###### Lemma 6.4

For any $g^{\prime},g^{\prime\prime}\in{\cal H}$ the operator
$\widetilde{W}=\widehat{g}^{\prime}W\widehat{g}^{\prime\prime}$ generates the
coefficients $\widetilde{\omega}_{m,n}$ such that

$\sum_{m,n\in\mathbb{Z}}\widetilde{\omega}_{m,n}\eta^{mx}\eta^{na}=|g^{\prime}(x)|^{2}\nu(x,a)|g^{\prime\prime}(a)|^{2}.$
(6.15)

The measure $d\mu_{W}(x,a)=\nu(x,a)\,d\mu(x)d\mu(a)$ on ${\cal X}^{2}$
satisfies

$\|\widehat{g}^{\prime}W\widehat{g}^{\prime\prime}\|_{\mu}^{2}=\int_{{\cal
X}^{2}}|g^{\prime}(x)|^{2}|g^{\prime\prime}(a)|^{2}\,d\mu_{W}(x,a).$ (6.16)

Proof. We put

$g^{\prime}(x)=\sum g^{\prime}_{k}\eta^{kx},\quad g^{\prime\prime}(x)=\sum
g^{\prime\prime}_{k}\eta^{kx},\quad|g^{\prime}(x)|^{2}=\sum{\bf
g}^{\prime}_{k}\eta^{kx},\quad|g^{\prime\prime}(x)|^{2}=\sum{\bf
g}^{\prime\prime}_{k}\eta^{kx}.$

The equation
$\widetilde{W}_{p,q}=\sum_{\alpha,\beta}g^{\prime}_{p-\alpha}W_{\alpha,\beta}g^{\prime\prime}_{\beta-q}$
implies

$\displaystyle\widetilde{\omega}_{m,n}$ $\displaystyle=$
$\displaystyle\frac{1}{J}\sum_{l,j,\alpha,\beta}g^{\prime}_{l+m-\alpha^{\prime}}W_{\alpha^{\prime},\beta^{\prime}}g^{\prime\prime}_{\beta^{\prime}-j}\overline{g}^{\prime}_{l-\alpha^{\prime\prime}}\overline{W}_{\alpha^{\prime\prime},\beta^{\prime\prime}}\overline{g}^{\prime\prime}_{\beta^{\prime\prime}-j-n}$
$\displaystyle=$ $\displaystyle\frac{1}{J}\sum_{\alpha,\beta}{\bf
g}^{\prime}_{m-\alpha^{\prime}+\alpha^{\prime\prime}}W_{\alpha^{\prime},\beta^{\prime}}\overline{W}_{\alpha^{\prime\prime},\beta^{\prime\prime}}{\bf
g}^{\prime\prime}_{\beta^{\prime}-\beta^{\prime\prime}+n}$ $\displaystyle=$
$\displaystyle\frac{1}{J}\sum_{p,q,\alpha^{\prime\prime},\beta^{\prime}}{\bf
g}^{\prime}_{m-p}W_{p+\alpha^{\prime\prime},\beta^{\prime}}\overline{W}_{\alpha^{\prime\prime},\beta^{\prime}-q}{\bf
g}^{\prime\prime}_{q+n}\;=\;\sum_{p,q}{\bf g}^{\prime}_{m-p}\omega_{p,-q}{\bf
g}^{\prime\prime}_{q+n}.$

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

We put

$c_{k}=\max_{j\in\mathbb{Z}_{J}}|W_{k+j,j}|,\quad\|W\|_{{\cal
DT}}=\sum_{k\in\mathbb{Z}_{J}}c_{k}.$

Then

$\sum_{m\in\mathbb{Z}_{J}}|\omega_{m,n}|\leq\|W\|_{\cal
DT}^{2},\quad\sum_{n\in\mathbb{Z}_{J}}|\omega_{m,n}|\leq\|W\|_{\cal DT}^{2}.$
(6.17)

###### Definition 6.1

We say that $f=\sum_{j\in\mathbb{Z}_{J}}f_{j}\eta^{jx}$ is nonnegative ($f\geq
0$) if the numbers $f(x)$ are real and nonnegative for all
$x\in\mathbb{Z}_{J}$.

###### Lemma 6.5

If ${\cal X}=\mathbb{Z}_{J}$ then the following three statements are
equivalent.

(1) $f\geq 0$,

(2) $f=g\overline{g}$ for some $g\in{\cal H}$,

(3) $f_{k}=\sum_{s\in\mathbb{Z}_{J}}g_{k+s}\overline{g}_{s}$.

Proof. Equivalence of statements (2) and (3) is obvious. Equivalence of (1)
and (2) follows from two simple facts.

(a) For any $g^{\prime},g^{\prime\prime}\in{\cal H}$ and any
$x\in\mathbb{Z}_{J}$
$(g^{\prime}g^{\prime\prime})(x)=g^{\prime}(x)g^{\prime\prime}(x)$.

(b) The numbers $f(x)$, $x\in\mathbb{Z}_{J}$ determine uniquely $f\in{\cal
H}$.

###### Lemma 6.6

Suppose $f\geq 0$ and $\displaystyle
F(x)=\sum_{m,n\in\mathbb{Z}_{J}}\omega_{m,-n}f_{n}\eta^{mx}$. Then $F\geq 0$.

Proof. By Lemma 6.5 $f=g\overline{g}$ for some $g\in{\cal H}$. Then

$F(x)=\frac{1}{J}\sum_{j,l,n,s\in\mathbb{Z}_{J}}W_{l+m,j}\overline{W}_{l,j-n}g_{n+s}\overline{g}_{s}\eta^{mx}.$

By using the change of summation indices $(n,s)\mapsto(k,q)$, $k=j-n$,
$q=s-k$, we obtain:

$F(x)=\frac{1}{J}\sum_{j,l,k,q\in\mathbb{Z}_{J}}W_{l+m,j}g_{j+q}\overline{W}_{l,k}\overline{g}_{k+q}\eta^{mx}=\sum_{q\in\mathbb{Z}}F^{(q)}(x)\overline{F}^{(q)}(x),$

where

$F^{(q)}(x)=\sum_{j,l\in\mathbb{Z}_{J}}W_{l,j}g_{j+q}\eta^{lx}.$

Hence, $F\geq 0$.

Now suppose $W$ is a unitary operator:

$\frac{1}{J}\sum_{l\in\mathbb{Z}_{J}}W_{l,j}\overline{W}_{l,k}=\delta_{j,k},\qquad
j,k\in\mathbb{Z}_{J}.$ (6.18)

We put ${\bf 1}_{\cal X}=\sqrt{J}\in{\cal H}$. Hence,

${\bf 1}_{{\cal X}}(x)=\sum_{k\in\mathbb{Z}_{J}}({\bf 1}_{\cal
X})_{k}\eta^{kx},\qquad({\bf 1}_{\cal
X})_{k}=\sqrt{J}\delta_{k,0},\quad\langle{\bf 1}_{\cal X},{\bf 1}_{\cal
X}\rangle=1.$

###### Lemma 6.7

Suppose $W$ is unitary. Then

$\sum_{n\in\mathbb{Z}_{J}}\omega_{m,-n}({\bf 1}_{\cal X})_{n}=({\bf 1}_{\cal
X})_{m},\quad\sum_{m\in\mathbb{Z}_{J}}({\bf 1}_{\cal
X})_{m}\omega_{-m,n}=({\bf 1}_{\cal X})_{n}.$ (6.19)

Proof. By (6.13) and (6.18)

$\sum_{n\in\mathbb{Z}_{J}}\omega_{m,-n}({\bf 1}_{\cal
X})_{n}=\sum_{j,l,n\in\mathbb{Z}_{J}}\frac{W_{l+m,j}\overline{W}_{l,j-n}}{J}\sqrt{J}\delta_{n,0}=\sum_{j,l\in\mathbb{Z}_{J}}\frac{W_{l+m,j}\overline{W}_{l,j}}{\sqrt{J}}=({\bf
1}_{\cal X})_{m}$

The second equation (6.19) can be obtained analogously.

###  6.3 Measure associated with a regular operator

We start from a simple remark. Take any $f\in L^{2}(\mathbb{T})$. Then $f\in
L^{1}(\mathbb{T})$ and we may associate with $f$ the Radon measure $fd\mu$ on
$\mathbb{T}$ i.e., a linear functional on $C(\mathbb{T})$:

$C(\mathbb{T})\ni\varphi\mapsto\int_{\mathbb{T}}f\varphi\,d\mu.$

We use this observation in the following lemma.

###### Lemma 6.8

Suppose $W\in{\cal R}(\mathbb{T})$ and for any small $\varepsilon>0$ the
function $f_{\varepsilon}=f_{\varepsilon}(x)$ satisfies

$f_{\varepsilon}=\pi_{Y}f_{\varepsilon}\in
L^{2}(\mathbb{T}),\quad\|f_{\varepsilon}\|^{2}=1,\quad
Y=(a-\varepsilon,a+\varepsilon).$ (6.20)

Let $L_{a}$ be determined by (2.9). Then there exists the weak limit888i.e.,
we regard here $|L_{a}*f_{\varepsilon}|$ as functionals on $C(\mathbb{T})$

$\lim_{\varepsilon\searrow
0}|L_{a}*f_{\varepsilon}|^{2}=\nu,\qquad\nu=\nu(x,a)=\sum_{m\in\mathbb{Z}}v_{m}(a)e^{im(x-a)}.$
(6.21)

The limit (6.21) is independent of the choice of the family $f_{\varepsilon}$,
satisfying (6.20). For any $a\in\mathbb{T}$

$d\widetilde{\mu}_{W,a}=\frac{1}{2\pi}\nu(x,a)\,dx$ (6.22)

is a (non-negative) measure on $\mathbb{T}$ and its norm as a functional on
$C(\mathbb{T})$ satisfies

$\|\widetilde{\mu}_{W,a}\|_{C^{*}}=\int_{\mathbb{T}}d\widetilde{\mu}_{W,a}\leq{\bf
c}^{2}=\|W\|^{2}_{{\cal DT}}.$ (6.23)

###### Remark 6.1

By using (4.4) we obtain the equation

$\nu(x,a)=\sum_{m,n\in\mathbb{Z}}\omega_{m,n}e^{imx+ina}.$ (6.24)

###### Corollary 6.2

Suppose $W\in{\cal R}(\mathbb{T})$. Then by (4.9) we have
$\|W\|_{\mu}^{2}={\bf T}(W)=\omega_{0,0}$. Therefore by (6.24)

$\|W\|_{\mu}^{2}=\int_{\mathbb{T}^{2}}\nu(x,a)\,dadx.$

Proof of Lemma 6.8. For any $f_{\varepsilon}$ satisfying (6.20)

$\Big{|}\frac{1}{2\pi}\int_{\mathbb{T}}e^{-ikx}|L_{a}*f_{\varepsilon}|^{2}\,dx\Big{|}\leq\|\operatorname{Conv}_{L_{a}}\|^{2}\leq{\bf
c}^{2},\qquad k\in\mathbb{Z}.$ (6.25)

Hence absolute values of Fourier coefficients of any function
$\nu_{\varepsilon}(a,\cdot)=|L_{a}*f_{\varepsilon}|^{2}$ do not exceed ${\bf
c}^{2}$. Existence of the weak limit (6.21) in the class of distributions with
bounded Fourier coefficients is equivalent to the existence of limits for all
Fourier coefficients:

$\xi_{m}=\lim_{\varepsilon\searrow
0}\xi_{\varepsilon,m},\qquad\xi_{\varepsilon,m}=\frac{1}{2\pi}\int_{\mathbb{T}}e^{-imx}|L_{a}*f_{\varepsilon}|^{2}\,dx,$
(6.26)

independence of these limits of $f_{\varepsilon}$, and the equation
$\xi_{m}=e^{-ima}v_{m}(a)$. The convergence (6.26) does not need to be uniform
in $m$.

We put $f_{\varepsilon}=\sum_{k}f_{\varepsilon,k}e^{ikx}$. Below for brevity
we skip the subscript $\varepsilon$ and write $f_{k}$ instead of
$f_{\varepsilon,k}$. Then $L_{a}*f_{\varepsilon}=\sum_{k}w_{k}(a)f_{k}e^{ikx}$
and

$\xi_{\varepsilon,m}=\sum_{k\in\mathbb{Z}}w_{k+m}(a)\overline{w}_{k}(a)f_{k+m}\overline{f}_{k}.$

By (6.25) $|\xi_{\varepsilon,m}|\leq{\bf c}^{2}$ for any $m\in\mathbb{Z}$.

Now we show that for any $m\in\mathbb{Z}$ convergence (6.26) takes place. We
put

$\widetilde{\xi}_{\varepsilon,m}=\sum_{k\in\mathbb{Z}}\sum_{|l^{\prime}|,|l^{\prime\prime}|\leq
B}w_{k+m}(a)\overline{w}_{k}(a)\frac{f_{k+m+l^{\prime}}\overline{f}_{k+l^{\prime\prime}}}{(2B+1)^{2}}e^{i(l^{\prime}-l^{\prime\prime})a}.$
(6.27)

Then by (2.10) and (2.5)

$\displaystyle|\xi_{\varepsilon,m}-\widetilde{\xi}_{\varepsilon,m}|$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathbb{Z}}\sum_{|l^{\prime}|,|l^{\prime\prime}|\leq
B}|w_{k+m}(a)\overline{w}_{k}(a)|\frac{|f_{k+m}\overline{f}_{k}-f_{k+m+l^{\prime}}\overline{f}_{k+l^{\prime\prime}}e^{i(l^{\prime}-l^{\prime\prime})a}|}{(2B+1)^{2}}$
(6.28) $\displaystyle\leq$
$\displaystyle\sum_{k\in\mathbb{Z}}\sum_{|l^{\prime}|,|l^{\prime\prime}|\leq
B}\\!\\!{\bf
c}^{2}\frac{|f_{k+m}-f_{k+m+l^{\prime}}e^{il^{\prime}a}|\cdot|f_{k}|+|f_{k+m+l^{\prime}}|\cdot|f_{k}-f_{k+l^{\prime\prime}}e^{il^{\prime\prime}a}|}{(2B+1)^{2}}$
$\displaystyle\leq$ $\displaystyle\frac{2{\bf c}^{2}}{2B+1}\sum_{|l|\leq
B}\|f_{\varepsilon}\|\cdot\|f_{\varepsilon}-e^{-il(x-a)}f_{\varepsilon}\|$
$\displaystyle\leq$ $\displaystyle\frac{2{\bf
c}^{2}}{2B+1}\|f_{\varepsilon}\|^{2}\sum_{|l|\leq
B}\frac{\varepsilon^{3/2}}{\sqrt{\pi}}|l|\;\leq\>2\varepsilon^{3/2}B{\bf
c}^{2}\|f_{\varepsilon}\|^{2}.$

By using in (6.27) the notation $l=l^{\prime}$, $n=k+l^{\prime}$, and
$s=l^{\prime}-l^{\prime\prime}$, we obtain:

$\widetilde{\xi}_{\varepsilon,m}=\sum_{n\in\mathbb{Z}}\sum_{|l|,|s-l|\leq
B}w_{n+m-l}(a)\overline{w}_{n-l}(a)\frac{f_{n+m}\overline{f}_{n-s}}{(2B+1)^{2}}e^{isa}.$

If we fix $n\in\mathbb{Z}$ and $s\in[-2B,2B]\cap\mathbb{Z}$ in the last sum
then $n-l\in I_{n,s}$, where

$I_{n,s}=\left\\{\begin{array}[]{cc}[n-B,n+B-s]\cap\mathbb{Z}&\mbox{ if }s\geq
0,\\\ {}[n-B-s,n+B]\cap\mathbb{Z}&\mbox{ if }s<0.\end{array}\right.$

We also put

$b_{m}=\frac{2B+1-|m|}{(2B+1)^{2}},\qquad|m|\leq 2B+1.$

Then $\sum_{|s|\leq 2B}b_{s}=1$. By (4.3)

$\widetilde{\xi}_{\varepsilon,m}=\sum_{k\in\mathbb{Z},\,|s|\leq
2B}v_{I_{k,s},m}(a)\,b_{s}f_{k+m}\overline{f}_{k-s}e^{isa}.$

Since $W$ is regular, then by Lemma 4.1 for any $m\in\mathbb{Z}$ and any
$\sigma>0$ there exists $Q_{m}=Q_{m}(\sigma)$ such that

$\\#I\geq Q_{m}\quad\mbox{implies}\quad|v_{I,m}(a)-v_{m}(a)|<\sigma.$ (6.29)

Then $\widetilde{\xi}_{\varepsilon,m}=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}$, where

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z},\,|s|\leq
2B}v_{m}(a)\,b_{s}f_{k+m}\overline{f}_{k-s}e^{isa},$ $\displaystyle\Sigma_{2}$
$\displaystyle=$ $\displaystyle\sum_{k\in\mathbb{Z},\,2B+1-|s|\geq
Q_{m}}(v_{I_{k,s},m}(a)-v_{m}(a))\,b_{s}f_{k+m}\overline{f}_{k-s}e^{isa},$
$\displaystyle\Sigma_{3}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z},\,2B+1-|s|<Q_{m}}(v_{I_{k,s},m}(a)-v_{m}(a))\,b_{s}f_{k+m}\overline{f}_{k-s}e^{isa}.$

By using Lemma 2.1, we obtain:

$\displaystyle\Big{|}\Sigma_{1}-e^{-ima}v_{m}(a)\|f_{\varepsilon}\|^{2}\Big{|}$
$\displaystyle=$ $\displaystyle\bigg{|}\sum_{|s|\leq
2B}e^{-ima}v_{m}(a)b_{s}\sum_{k\in\mathbb{Z}}\big{(}f_{k+m}\overline{f}_{k-s}e^{i(m+s)a}-|f_{k}|^{2}\big{)}\bigg{|}$
(6.30) $\displaystyle\leq$ $\displaystyle 3{\bf
c}^{2}(2B+|m|)\varepsilon\|f_{\varepsilon}\|^{2}.$

If $2B+1-|s|\geq Q_{m}$ then by using (6.29) we can estimate $\Sigma_{2}$:

$|\Sigma_{2}|\leq\sum_{k\in\mathbb{Z},\,2B+1-|s|\geq Q_{m}}\sigma
b_{s}f_{k+m}\overline{f}_{k-s}\leq\sigma\|f\|^{2}.$ (6.31)

We have:

$\displaystyle|\Sigma_{3}|$ $\displaystyle\leq$ $\displaystyle 2{\bf
c}^{2}\sum_{k\in\mathbb{Z},\,2B+1-|s|<Q_{m}}b_{s}|f_{k+m}\overline{f}_{k-s}|$
(6.32) $\displaystyle\leq$ $\displaystyle 2{\bf
c}^{2}\sum_{2B+1-|s|<Q_{m}}b_{s}\|f_{\varepsilon}\|^{2}\;\leq\;2{\bf
c}^{2}\frac{(1+Q_{m})Q_{m}}{(2B+1)^{2}}\|f_{\varepsilon}\|^{2}.$

Combining estimates (6.28), (6.30), (6.31), and (6.32), we obtain:

$\big{|}\xi_{\varepsilon,m}-e^{-ima}v_{m}(a)\|f_{\varepsilon}\|^{2}\big{|}\leq\bigg{(}(2\varepsilon^{3/2}B+6\varepsilon
B+3\varepsilon|m|){\bf c}^{2}+\sigma+2{\bf
c}^{2}\frac{(1+Q_{m})Q_{m}}{(2B+1)^{2}}\bigg{)}\|f_{\varepsilon}\|^{2}.$

This implies existence of the limits (6.26). Since the functions
$\nu_{\varepsilon}(a,\cdot)$ are nonnegative, $d\widetilde{\mu}_{W,a}$ is a
measure on $\mathbb{T}$ for any $a\in\mathbb{T}$.

###  6.4 The space ${\cal ACF}(\mathbb{T})$

###### Definition 6.2

We say that $f\in L^{\infty}(\mathbb{T})$ lies in the space ${\cal
ACF}(\mathbb{T})$ if it has absolutely converging Fourier series.

For any $f\in{\cal ACF}(\mathbb{T})$ we put

$\|f\|_{{\cal DT}}=\sum_{k\in\mathbb{Z}}|f_{k}|.$

Then for any $f\in{\cal ACF}(\mathbb{T})$ we have: $\|f\|_{{\cal
DT}}=\|\widehat{f}\|_{{\cal DT}}$ and the estimate (2.8). Assertion (14)
implies the following

###### Corollary 6.3

The space $\big{(}{\cal ACF}(\mathbb{T}),\|\cdot\|_{\cal DT}\big{)}$ is a
commutative $\mathbb{C}^{*}$-subalgebra in the $\mathbb{C}^{*}$-algebra
$\big{(}{\cal DT}(\mathbb{T}),\|\cdot\|_{\cal DT}\big{)}$.

Suppose $\nu$ satisfies (6.24) for some $W\in{\cal R}(\mathbb{T})$. Consider
the functions

$\varphi=\frac{1}{2\pi}\int_{\mathbb{T}}\nu(x,\cdot)\,dx\quad{and}\quad\psi=\frac{1}{2\pi}\int_{\mathbb{T}}\nu(\cdot,a)\,da.$
(6.33)

###### Lemma 6.9

Suppose $W\in{\cal R}(\mathbb{T})$. Then $\varphi,\psi\in{\cal
ACF}(\mathbb{T})$. Moreover,

$\|\varphi\|_{\cal DT}\leq\|W\|^{2}_{\cal DT},\quad\|\psi\|_{\cal
DT}\leq\|W\|^{2}_{\cal DT}.$

Proof. By (6.24)

$\frac{1}{2\pi}\int_{\mathbb{T}}\nu(x,a)\,dx=\sum_{n\in\mathbb{Z}}\omega_{0,n}e^{ina}.$

By (10.2) $\sum|\omega_{0,n}|\leq\|W\|_{\cal DT}^{2}$.

The case of the function $\psi$ is analogous.

###  6.5 The operator $\widehat{g}_{1}W\widehat{g}_{2}$

###### Proposition 6.1

Suppose $W\in{\cal R}(\mathbb{T})$, $g_{2}\in L^{\infty}(\mathbb{T})$, and
$g_{1},|g_{2}|^{2}\in{\cal ACF}(\mathbb{T})$. Then the operator
$\widetilde{W}=\widehat{g}_{1}W\widehat{g}_{2}$ is also regular and the
corresponding coefficients $\widetilde{\omega}_{m,n}$ satisfy

$\sum_{m,n\in\mathbb{Z}}\widetilde{\omega}_{m,n}e^{imx+ina}=|g_{1}(x)|^{2}\nu(x,a)|g_{2}(a)|^{2}.$
(6.34)

###### Corollary 6.4

Suppose $W\in{\cal R}(\mathbb{T})$, $g_{2}\in L^{\infty}(\mathbb{T})$, and
$g_{1},|g_{2}|^{2}\in{\cal ACF}(\mathbb{T})$. Then by (4.2) and (4.9)

$\|\widehat{g}_{1}W\widehat{g}_{2}\|_{\mu}^{2}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}|g_{1}(x)|^{2}\nu(x,a)|g_{2}(a)|^{2}\,dadx.$
(6.35)

Proposition 6.1 follows from Lemmas 8.1 and 8.2 while Corollary 6.4 is a
combination of Corollaries 8.1 and 8.2.

We associate with any $W\in{\cal R}(\mathbb{T})$ the measure $\mu_{W}$ on
${\cal X}\times{\cal X}$:

$d\mu_{W}=\nu(x,a)\,\frac{dxda}{4\pi^{2}}.$

##  7 A bistochastic operator generated by $\mu_{W}$

###### Lemma 7.1

Suppose $W$ satisfies (at least) one of the following conditions:

(1) $W=U_{F}$, where $F\in\operatorname{Aut}({\cal X},\mu)$,

(2) $W$ is an operator on ${\cal H}=L^{2}({\cal X})$, ${\cal
X}=\mathbb{Z}_{J}$,

(3) $W\in{\cal R}(\mathbb{T})$.

Then the corresponding measure $\mu_{W}$ determines a bounded operator

${\cal W}:L^{1}({\cal X},\mu)\to L^{1}({\cal X},\mu),\qquad f\mapsto{\cal
W}f=\int_{\cal X}\nu(\cdot,a)f(a)\,d\mu(a).$

This operator satisfies the estimate

$\|{\cal W}\|_{L^{1}\to L^{1}}\leq\|W\|_{\cal DT}^{2}$ (7.1)

and moreover, has the following properties

(a). ${\cal W}$ is nonnegative: ${\cal W}f\geq 0$ whenever $0\leq f\in
L^{1}({\cal X},\mu)$.

If $W$ is unitary999In case (1) this condition automatically holds. then two
more statements hold.

(b). ${\cal W}{\bf 1}_{\cal X}={\bf 1}_{\cal X}$.

(c). $\int_{\cal X}{\cal W}f(x)\,d\mu(x)=\int_{\cal X}f(a)\,d\mu(a)$ for any
$f\in L^{1}({\cal X},\mu)$.

Conditions (a)–(c) mean that ${\cal W}$ is a bistochastic (doubly stochastic)
operator.

Proof of Lemma 7.1. (1) In this case
$d\mu_{W}(x,a)=\delta(x,F(a))\,d\mu(x)d\mu(a)$ (see (6.9)). Then by
(6.5)–(6.7) for any $f\in L^{1}({\cal X})$

${\cal W}f(x)=\int_{\cal X}\delta(x,F(a))f(a)\,d\mu(a)=\int_{\cal
X}\delta(F^{-1}(x),a)f(a)\,d\mu(a)=f\circ F^{-1}(x).$ (7.2)

Hence, $\|{\cal W}f\|_{1}=\|f\|_{1}$. Therefore ${\cal W}$ is an isometry and
condition (c) holds. Conditions (a) and (b) also follow from (7.2).

(2) If ${\cal X}=\mathbb{Z}_{J}$ then by (6.13) we have:

$f(a)=\sum_{j\in\mathbb{Z}}f_{j}\eta^{ja},\quad{\cal
W}f(x)=\sum_{m,n\in\mathbb{Z}}\omega_{m,-n}f_{n}\eta^{mx}.$

Hence, by (6.17)

$\|{\cal
W}f\|_{1}\leq\sum_{m,n\in\mathbb{Z}_{J}}|\omega_{m,-n}f_{n}|\leq\|W\|_{\cal
DT}^{2}\|f\|_{1}.$

Condition (a) follows from Lemma 6.6, while Conditions (b) and (c) from Lemma
6.7.

(3) In this case for any $f\in L^{1}(\mathbb{T})$

$\|{\cal
W}f\|_{1}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}}dx\int_{\mathbb{T}}|\nu(x,a)f(a)|\,da=\frac{1}{2\pi}\int_{\mathbb{T}}\varphi(a)|f(a)|\,da,$

where $\varphi$ is determined by (6.33). By Lemma 6.9
$\|\varphi\|_{\infty}\leq\|\varphi\|_{\cal DT}\leq\|W\|_{\cal DT}^{2}$. This
implies (7.1).

To prove (a), we fix $0\leq f\in L^{1}(\mathbb{T})$ and take any nonnegative
continuous function $g:\mathbb{T}\to\mathbb{R}$. Then
$\frac{1}{2\pi}\int_{\mathbb{T}}g(x){\cal W}f(x)\,dx$ is the result of the
action of the measure $\nu$ on the non-negative function $g(x)f(a)$. This
result is nonnegative. Hence, ${\cal W}f\geq 0$.

Now suppose $W$ is unitary. Then by (6.35) for any $0\not\equiv g\in{\cal
ACF}(\mathbb{T})$

$\frac{1}{2\pi}\int_{\mathbb{T}}|g(x)|^{2}{\cal W}{\bf
1}_{\mathbb{T}}(x)\,dx=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}|g(x)|^{2}\nu(x,a)\,dxda=\|\widehat{g}W\|_{\mu}^{2}=\|\widehat{g}\|_{\mu}^{2}$

(the last equation follows from Corollary 4.1). The equation

$\|\widehat{g}\|_{\mu}^{2}=\|g\|^{2}=\frac{1}{2\pi}\int_{\mathbb{T}}|g(x)|^{2}\,dx$

implies (b).

It is sufficient to check condition (c) only for $f\geq 0$. First, consider
the case when $0\leq f\in L^{\infty}(\mathbb{T})$. Consider $g\geq 0$ such
that $f=g^{2}$. Then by Corollary 6.4

$\frac{1}{2\pi}\int_{\mathbb{T}}{\cal
W}f(x)\,dx=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}\nu(x,a)|g(a)|^{2}\,dxda=\|W\widehat{g}\|_{\mu}^{2}=\|\widehat{g}\|_{\mu}^{2}=\frac{1}{2\pi}\int_{\mathbb{T}}f(a)\,da.$

If $0\leq f\in L^{1}(\mathbb{T})$ is unbounded, by using a cut off, for any
$\varepsilon>0$ we have: $f=f_{1}+f_{2}$, $0\leq f_{1}\in
L^{\infty}(\mathbb{T})$, $\|f_{2}\|_{1}<\varepsilon$. Then by (7.1)

$\|{\cal W}f-{\cal W}f_{1}\|_{1}<\|W\|_{\cal DT}^{2}\varepsilon.$

This implies

$\bigg{|}\frac{1}{2\pi}\int_{\mathbb{T}}{\cal
W}f(x)\,dx-\frac{1}{2\pi}\int_{\mathbb{T}}{\cal
W}f_{1}(x)\,dx\bigg{|}\leq\|W\|_{\cal DT}^{2}\varepsilon.$

The estimate

$\frac{1}{2\pi}\int_{\mathbb{T}}{\cal
W}f_{1}(x)\,dx=\frac{1}{2\pi}\int_{\mathbb{T}}f_{1}(x)\,dx=\frac{1}{2\pi}\int_{\mathbb{T}}f(x)\,dx+\Delta,\qquad|\Delta|\leq\varepsilon$

finishes the proof.

##  8 Product with $\widehat{g}$

In this section we prove Proposition 6.1 and Corollary 6.4.

###  8.1 The operator $W\widehat{g}$

###### Lemma 8.1

Suppose $W\in{\cal R}(\mathbb{T})$, $g\in L^{\infty}(\mathbb{T})$, and
$|g|^{2}\in{\cal ACF}(\mathbb{T})$. Then $\widetilde{W}=W\widehat{g}$ is also
regular and the corresponding coefficients $\widetilde{\omega}_{m,n}$ satisfy

$\sum_{m,n\in\mathbb{Z}}\widetilde{\omega}_{m,n}e^{imx+ina}=\nu(x,a)|g(a)|^{2}.$
(8.1)

###### Corollary 8.1

Suppose $W\in{\cal R}(\mathbb{T})$ and $g\in{\cal ACF}(\mathbb{T})$. Then by
(4.2) and (4.9)

$\|W\widehat{g}\|_{\mu}^{2}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}\nu(x,a)|g(a)|^{2}\,dadx.$
(8.2)

Proof of Lemma 8.1. We denote by $\widetilde{\omega}_{I,m,n}$ the quantities
(4.1), corresponding to the operator $\widetilde{W}=W\widehat{g}$. Then

$\widetilde{\omega}_{I,m,n}=\frac{1}{\\#I}\sum_{k,s,j\in\mathbb{Z},\,l\in
I}W_{l+m,k}g_{k-j}\overline{W}_{l,s}\overline{g}_{s-j-n}=\frac{1}{\\#I}\sum_{k,q\in\mathbb{Z},\,l\in
I}W_{l+m,k}\overline{W}_{l,k-q+n}{\bf g}_{q},$

where ${\bf g}_{q}=\sum_{p\in\mathbb{Z}}g_{p}\overline{g}_{p-q}$. Note that
${\bf g}_{q}$ is the $q$-th Fourier coefficient of the function $|g|^{2}$.
Indeed,

${\bf g}_{q}=\langle
g,ge^{iqx}\rangle=\frac{1}{2\pi}\int_{\mathbb{T}}|g(x)|^{2}e^{-iqx}\,dx.$

By assumption of the lemma the series $\sum{\bf g}_{q}$ absolutely converges.
We put

$\sum_{q\in\mathbb{Z}}|{\bf g}_{q}|={\bf c}_{g}.$

For any $\sigma>0$ there exists $Q(\sigma)\in\mathbb{N}$ such that
$\sum_{|q|>Q(\sigma)}|{\bf g}_{q}|<\sigma$. Given a small $\sigma$ we have:
$\widetilde{\omega}_{I,m,n}=\Omega_{1}+\Omega_{2}$, where

$\displaystyle\Omega_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{k\in\mathbb{Z},\,l\in I,\,|q|\leq
Q(\sigma)}W_{l+m,k}\overline{W}_{l,k-q+n}{\bf g}_{q},$
$\displaystyle\Omega_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{k\in\mathbb{Z},\,l\in
I,\,|q|>Q(\sigma)}W_{l+m,k}\overline{W}_{l,k-q+n}{\bf g}_{q}.$

In the sum $\Omega_{1}$ the index $q$ contains only a finite number of values.
Hence for all sufficiently big $\\#I$ we have:

$\Omega_{1}=\sum_{|q|\leq Q(\sigma)}\omega_{m,n-q}{\bf
g}_{q}+\Delta,\qquad|\Delta|\leq\sum_{|q|\leq Q(\sigma)}\sigma|{\bf
g}_{q}|\leq\sigma{\bf c}_{g}.$

Now we estimate $\Omega_{2}$. Let $c_{k}$ be the majorating sequence for $W$
with $\sum c_{k}={\bf c}$. Then

$\displaystyle|\Omega_{2}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{\\#I}\sum_{k\in\mathbb{Z},\,l\in
I\,|q|>Q(\sigma)}c_{l+m-k}c_{l-k+q-n}|{\bf g}_{q}|$ $\displaystyle\leq$
$\displaystyle\sum_{j\in\mathbb{Z},\,|q|>Q(\sigma)}c_{m+j}c_{q-n+j}|{\bf
g}_{q}|=\sum_{|q|>Q(\sigma)}\widetilde{c}_{m+n-q}\sigma|{\bf g}_{q}|,$

where $\widetilde{c}_{s}=\sum_{j}c_{s+j}c_{j}$,
$\sum_{s}\widetilde{c}_{s}={\bf c}^{2}$. Hence

$|\Omega_{2}|\leq{\bf c}^{2}\sum_{|q|>Q(\sigma)}|{\bf g}_{q}|\leq{\bf
c}^{2}\sigma.$

By (10.2)

$\bigg{|}\sum_{|q|>Q(\sigma)}\omega_{m,n-q}{\bf g}_{q}\bigg{|}\leq{\bf
c}^{2}\sum_{|q|>Q(\sigma)}|{\bf g}_{q}|\leq{\bf c}^{2}\sigma.$

Therefore

$\Big{|}\widetilde{\omega}_{I,m,n}-\sum_{q\in\mathbb{Z}}\omega_{m,n+q}{\bf
g}_{q}\Big{|}\leq\sigma({\bf c}_{g}+2{\bf c}^{2}).$

Finally note that

$\displaystyle\nu(x,a)|g(a)|^{2}$ $\displaystyle=$
$\displaystyle\sum_{q,p\in\mathbb{Z}}g_{p}\overline{g}_{p-q}e^{iqa}\sum_{m,k\in\mathbb{Z}}\omega_{m,k}e^{imx+ika}$
$\displaystyle=$
$\displaystyle\sum_{m,n,q,p\in\mathbb{Z}}\omega_{m,n-q}g_{p}\overline{g}_{p-q}e^{imx+ina}.$

These computations imply that for all sufficiently big $\\#I$ the quantity
$\widetilde{\omega}_{I,m,n}$ differs arbitrarily small from

$\sum_{p,q\in\mathbb{Z}}\omega_{m,n+q}g_{p}\overline{g}_{p-q}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}e^{-imx-
ina}\nu(x,a)|g(a)|^{2}\,dxda.$

###  8.2 The operator $\widehat{g}W$

###### Lemma 8.2

Suppose $W\in{\cal R}(\mathbb{T})$ and $g\in{\cal ACF}(\mathbb{T})$. Then
$\widetilde{W}=\widehat{g}W$ is also regular and the corresponding
coefficients $\widetilde{\omega}_{m,n}$ satisfy

$\sum_{m,n\in\mathbb{Z}}\widetilde{\omega}_{m,n}e^{imx+ina}=|g(x)|^{2}\nu(x,a).$
(8.3)

###### Corollary 8.2

Suppose $W\in{\cal R}(\mathbb{T})$ and $g\in{\cal ACF}(\mathbb{T})$. Then by
(4.2) and (4.9)

$\|\widehat{g}W\|_{\mu}^{2}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}|g(x)|^{2}\nu(x,a)\,dxda.$
(8.4)

Proof of Proposition 8.2. We denote by $\widetilde{\omega}_{I,m,n}$ the
quantities (4.1), corresponding to the operator $\widetilde{W}=\widehat{g}W$.
Then

$\displaystyle\widetilde{\omega}_{I,m,n}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{k,s,j\in\mathbb{Z},\,l\in
I}g_{l+m-k}W_{k,j}\overline{g}_{l-s}\overline{W}_{s,j+n}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{k,s,j\in\mathbb{Z},\,l\in
I}g_{m-k}\overline{g}_{-s}W_{k+l,j}\overline{W}_{s+l,j+n}$ $\displaystyle=$
$\displaystyle\frac{1}{\\#I}\sum_{k,s,j\in\mathbb{Z},\,l\in
I+s}g_{m-k}\overline{g}_{-s}W_{k-s+l,j}\overline{W}_{l,j+n}$ $\displaystyle=$
$\displaystyle\sum_{k,s\in\mathbb{Z}}g_{m-k}\overline{g}_{-s}\omega_{I+s,k-s,n}.$

We define ${\bf c}_{g}=\sum_{j\in\mathbb{Z}}|g_{j}|$. Given $\sigma>0$ we take
$M=M(\sigma)>|m|$ such that

$\sum_{|j|\geq(M-|m|)/2}|g_{j}|<\sigma.$ (8.5)

We take $N=N(\sigma,M)$ such that for any $j$, $|j|\leq M$ and any interval
$J\subset\mathbb{Z}$, $\\#J>N$

$|\omega_{J,j,n}-\omega_{j,n}|<\sigma.$ (8.6)

Then $\widetilde{\omega}_{I,m,n}=\Sigma_{0}+\Sigma_{1}+\Sigma_{2}+\Sigma_{3}$,
where

$\displaystyle\Sigma_{0}$ $\displaystyle=$
$\displaystyle\sum_{k,s\in\mathbb{Z}}g_{m-k}\overline{g}_{-s}\omega_{k-s,n},$
$\displaystyle\Sigma_{1}$ $\displaystyle=$ $\displaystyle\sum_{|k-s|\geq
M}g_{m-k}\overline{g}_{-s}\omega_{k-s,n},$ $\displaystyle\Sigma_{2}$
$\displaystyle=$
$\displaystyle-\sum_{|k-s|<M}g_{m-k}\overline{g}_{-s}(\omega_{I+s,k-s,n}-\omega_{k-s,n}),$
$\displaystyle\Sigma_{3}$ $\displaystyle=$ $\displaystyle\sum_{|k-s|\geq
M}g_{m-k}\overline{g}_{-s}\omega_{I,k-s,n}.$

First, we transform the sum $\Sigma_{0}$:

$\displaystyle\Sigma_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}e^{-imx-
ina}\sum_{k,s\in\mathbb{Z}}g_{m-k}\overline{g}_{-s}e^{i(m-k+s)x}\omega_{k-s,n}e^{i(k-s)x+ina}\,dxda$
(8.7) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}e^{-imx-
ina}|g(x)|^{2}\nu(x,a)\,dxda.$

We estimate $\Sigma_{1}$ by using (10.1) and (8.5):

$|\Sigma_{1}|\leq{\bf c}^{2}\sum_{|k-s|\geq M}|g_{m-k}||g_{-s}|\leq 2{\bf
c}^{2}\sum_{\beta\in\mathbb{Z},|\alpha|\geq(M-|m|)/2}|g_{\alpha}||g_{\beta}|\leq
2{\bf c}^{2}{\bf c}_{g}\sigma.$ (8.8)

We have by (8.6):

$|\Sigma_{2}|\leq\sigma\sum_{k,s\in\mathbb{Z}}|g_{m-k}||g_{-s}|=\sigma{\bf
c}_{g}^{2}.$ (8.9)

The sum $\Sigma_{3}$ is estimated in the same way as $\Sigma_{1}$:

$|\Sigma_{3}|\leq{\bf c}^{2}\sum_{|k-s|\geq M}|g_{m-k}||g_{-s}|\leq 2{\bf
c}^{2}{\bf c}_{g}\sigma.$ (8.10)

Combining (8.7), (8.8), (8.9), and (8.10), we see that

$\bigg{|}\widetilde{\omega}_{I,m,n}-\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}e^{-imx-
ina}|g(x)|^{2}\nu(x,a)\,dxda\bigg{|}\leq(4{\bf c}^{2}{\bf c}_{g}+{\bf
c}_{g}^{2})\sigma.$

This implies (8.3).

##  9 Weaker assumptions on $g$

Unfortunately we do not know if equation (6.35) remains valid for $W\in{\cal
R}(\mathbb{T})$ and arbitrary $g_{1},g_{2}\in L^{\infty}(\mathbb{T})$. In this
section we present two partial results in this direction. More precisely, we
show that equations (8.2) and (8.4) hold for $g$ lying in sets which are
larger than declared in Corollaries 8.1 and 8.2, in particular, for $g$ equal
to indicators of intervals.

###  9.1 Computation of $\|W\widehat{g}\|_{\mu}$

###### Lemma 9.1

Suppose $W\in{\cal R}(\mathbb{T})$ and $g\in L^{\infty}(\mathbb{T})$. Then
equation (8.2) remains valid.

Proof. For any $\sigma>0$ there exists $g_{*}\in{\cal ACF}(\mathbb{T})$ such
that

$\|g_{*}\|_{\infty}\leq\|g\|_{\infty}\quad\mbox{and}\quad\|g-g_{*}\|<\sigma.$

Indeed, it is sufficient to take $\widetilde{g}$ equal to a finite, but
sufficiently long part of the Fourier series of $g$, so that
$\big{|}\|g\|^{2}-\|\widetilde{g}\|^{2}\big{|}<\sigma$. Then we define $g_{*}$
as a cut off of $\widetilde{g}$:

$g_{*}(x)=\left\\{\begin{array}[]{ccc}\widetilde{g}(x)&\mbox{ if
}&|\widetilde{g}(x)|\leq\|g\|_{\infty},\\\ \|g\|_{\infty}&\mbox{ if
}&\widetilde{g}(x)>\|g\|_{\infty},\\\ -\|g\|_{\infty}&\mbox{ if
}&\widetilde{g}(x)<-\|g\|_{\infty}.\\\ \end{array}\right.$

The function $g_{*}(x)$ is Lipschitz. Hence, it lies in ${\cal
ACF}(\mathbb{T})$.

By using the triangle inequality, (1.5) and Assertion (7), we have:

$\Big{|}\|W\widehat{g}\|_{\mu}-\|W\widehat{g}_{*}\|_{\mu}\Big{|}\leq\|W(\widehat{g}-\widehat{g}_{*})\|_{\mu}\leq\|W\|\,\|(\widehat{g}-\widehat{g}_{*})\|_{\mu}\leq\sigma\|W\|.$
(9.1)

On the other hand, by (8.2)

$\displaystyle\\!\\!\\!\\!\Big{|}\|W\widehat{g}_{*}\|_{\mu}^{2}-\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}\nu(x,a)|g(a)|^{2}\,dadx\Big{|}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\Big{|}\int_{\mathbb{T}}\varphi(a)(|g_{*}(a)|^{2}-|g(a)|^{2})\,da\Big{|},$
$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{T}}\nu(x,\cdot)\,dx.$

By using the estimate $\|\varphi\|_{\infty}\leq\|\varphi\|_{\cal
DT}\leq\|W\|_{\cal DT}^{2}$ (Lemma 6.9), we obtain:

$\Big{|}\|W\widehat{g}_{*}\|_{\mu}^{2}-\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}\nu(x,a)|g(a)|^{2}\,dadx\Big{|}\leq\|W\|_{\cal
DT}^{2}\sigma$ (9.2)

Hence, Lemma 9.1 is a combination of (9.1) and (9.2).

###  9.2 Computation of $\|\widehat{g}W\|_{\mu}$

###### Definition 9.1

We say that $g\in L^{\infty}(\mathbb{T})$ is almost ${\cal ACF}(\mathbb{T})$
(the notation is $g\in{\cal AACF}(\mathbb{T})$) if there exists a continuous
in $\varepsilon\in(-\varepsilon_{0},\varepsilon_{0})$ family of functions
$g_{\varepsilon}\in L^{\infty}(\mathbb{T})$ such that the following conditions
hold:

A1. $g_{0}=g$,

A2. if $\varepsilon\neq 0$ then $g_{\varepsilon}\in{\cal ACF}(\mathbb{T})$,

A3. $|g_{\varepsilon}-g|\leq|g_{\varepsilon}-g_{-\varepsilon}|$ if
$0\leq\varepsilon<\varepsilon_{0}$,

A4. $\lim_{\varepsilon\to 0}\|g_{\varepsilon}-g_{-\varepsilon}\|=0$.

A typical example of an ${\cal AACF}(\mathbb{T})$ function is the indicator of
an interval (see Lemma 9.3 below).

###### Lemma 9.2

Suppose $W\in{\cal R}(\mathbb{T})$ and $g\in{\cal AACF}(\mathbb{T})$. Then
equation (8.4) remains valid.

Proof. Let $g_{\varepsilon}$ be the family from Definition 9.1. By A3 and A1
for any $\varepsilon\in(0,\varepsilon_{0})$ and any $X\in{\cal B}$

$0\leq\|(\widehat{g}_{\varepsilon}-\widehat{g})W\pi_{X}\|^{2}\leq\|(\widehat{g}_{\varepsilon}-\widehat{g}_{-\varepsilon})W\pi_{X}\|^{2}.$

This inequality implies

$0\leq\|(\widehat{g}_{\varepsilon}-\widehat{g})W\|_{\mu}^{2}\leq\|(\widehat{g}_{\varepsilon}-\widehat{g}_{-\varepsilon})W\|_{\mu}^{2},\qquad\varepsilon\in(0,\varepsilon_{0}).$

By A2 and (8.4)
$\|(\widehat{g}_{\varepsilon}-\widehat{g})W\|_{\mu}^{2}\leq\Delta_{\varepsilon}$,

$\Delta_{\varepsilon}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}|g_{\varepsilon}(x)-g_{-\varepsilon}(x)|^{2}\nu(x,a)\,dxda\leq\|g_{\varepsilon}-g_{-\varepsilon}\|^{2}\|\psi\|_{\infty},\quad\psi=\frac{1}{2\pi}\int_{\mathbb{T}}\nu(\cdot,a)\,da$

(see also (6.33)). Note that $\|\psi\|_{\infty}\leq\|\psi\|_{\cal DT}$. Hence
by A4 and Lemma 6.9 $\Delta_{\varepsilon}\to 0$ as $\varepsilon\to 0$.

By triangle inequality

$\Big{|}\|\widehat{g}_{\varepsilon}W\|_{\mu}^{2}-\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}|g(x)|^{2}\nu(x,a)\,dadx\Big{|}=\big{|}\|\widehat{g}W\|_{\mu}^{2}-\|\widehat{g}_{\varepsilon}W\|_{\mu}^{2}\leq\Delta_{\varepsilon}.$

Hence equation (8.4) follows.

###  9.3 Indicators of intervals

###### Lemma 9.3

For any interval $I\subset\mathbb{T}$ the function ${\bf 1}_{I}$ lies in
${\cal AACF}$.

Proof. We put $g_{0}={\bf 1}_{I}$. For $\varepsilon\in(0,1)$ we define

$\displaystyle g_{\varepsilon}(x)$ $\displaystyle=$
$\displaystyle\max\Big{\\{}0,1-\frac{1}{\varepsilon}\operatorname{dist}(x,I)\Big{\\}},$
$\displaystyle g_{-\varepsilon}(x)$ $\displaystyle=$ $\displaystyle
1-\max\Big{\\{}0,1+\frac{1}{\varepsilon}\operatorname{dist}(x,\mathbb{T}\setminus
I)\Big{\\}}.$

Conditions A1–A4 from Definition 9.1 also obviously hold. Hence ${\bf
1}_{I}\in{\cal AACF}$.

##  10 Coefficients $\omega_{m,n}$

In this section $W\in{\cal DT}(\mathbb{T})$, $\\{c_{k}\\}_{k\in\mathbb{Z}}$ is
its majorating sequence, and ${\bf c}=\sum c_{k}=\|W\|_{\cal DT}$.

###### Lemma 10.1

For any $W\in{\cal R}(\mathbb{T})$ and any $m,n\in\mathbb{Z}$

$\omega_{m,n}=\overline{\omega}_{-m,-n}.$

Proof. This equation follows from the identity
$\omega_{I,m,n}=\overline{\omega}_{I+m,-m,-n}$ for any interval
$I\subset\mathbb{Z}$ .

We have the estimates

$\begin{array}[]{ccc}\displaystyle\sum_{n\in\mathbb{Z}}|\omega_{I,m,n}|&\leq&\displaystyle\frac{1}{\\#I}\sum_{j,n\in\mathbb{Z},\,l\in
I}c_{l+m-j}c_{l-j-n}={\bf c}^{2},\\\
\displaystyle\sum_{m\in\mathbb{Z}}|\omega_{I,m,n}|&\leq&\displaystyle\frac{1}{\\#I}\sum_{j,n\in\mathbb{Z},\,l\in
I}c_{l+m-j}c_{l-j-n}={\bf c}^{2}.\end{array}$ (10.1)

These estimates imply the following

###### Corollary 10.1

If $W$ is regular then (compare with (6.17))

$\sum_{n\in\mathbb{Z}}|\omega_{m,n}|\leq{\bf
c}^{2},\quad\sum_{n\in\mathbb{Z}}|\omega_{n,m}|\leq{\bf c}^{2}\quad\mbox{ for
any $m\in\mathbb{Z}$}.$ (10.2)

We put

$P_{M}(m)=\\{n\in\mathbb{Z}:|n+m|\geq 2M\\}.$

###### Lemma 10.2

For any $m\in\mathbb{Z}$

$\sum_{n\in P_{M}(m)}|\omega_{m,n}|\leq 2{\bf c}\sum_{|k|\geq M}c_{k}.$ (10.3)

Proof. For any integer interval $I$

$\displaystyle\sum_{n\in P_{M}(m)}|\omega_{I,m,n}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{\\#I}\sum_{j\in\mathbb{Z},\,n\in P_{M}(m),\,l\in
I}c_{l-j+m}c_{l-j-n}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z},\,n\in
P_{M}(m)}c_{m+n-k}c_{k}\;=\;\sum_{k\in\mathbb{Z},\,s\in P_{M}(0)}c_{s-k}c_{k}$
$\displaystyle\leq$ $\displaystyle\sum_{k\in\mathbb{Z},\,|m|\geq
M}c_{m}c_{k}+\sum_{m\in\mathbb{Z},\,|k|\geq M}c_{m}c_{k}\;=\;2{\bf
c}\sum_{|k|\geq M}c_{k}.$

In the limit $\\#I\to\infty$ we obtain (10.3).

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# Comment on “Universal and Non-Universal Correction Terms of Bose Gases in
Dilute Region: A Quantum Monte Carlo Study” [J. Phys. Soc. Jpn. 91, 024001
(2022)]

Adam Rançon  Univ. Lille, CNRS UMR 8523 - PhLAM - Laboratoire des Lasers
Atomes et Molécules, F-59000 Lille, France

Recently, Masaki-Kato et al. have investigated the thermodynamics of a
hardcore Bose gas on the cubic lattice in the dilute limit Masaki-Kato et al.
(2022). In this regime, the system is expected to be very-well described as a
gas of bosons in the continuum with effective mass $m=\hbar^{2}/(2ta^{2})$,
with $t$ the hopping amplitude on the lattice and $a$ the lattice spacing, and
a contact interaction characterised by its s-wave scattering length $a_{s}$
Rançon (2014). By comparing their numerical results obtained by quantum Monte-
Carlo simulations to the results of Bogoliubov theory, they were able to
estimate numerically the ratio $a_{s}/a=0.316(2)$, as well as the constant
$c_{3}$ describing the non-universal next order correction in the expansion in
$na_{s}^{3}$, with $n$ the density.

The goals of this comment are two-fold. Firstly, we show that the s-wave
scattering length of hardcore bosons on the cubic lattice can be computed
analytically and reads Rançon (2014)

$a_{s}/a=\frac{8\sqrt{6}\pi^{2}}{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right)}\simeq
0.31487023\ldots$ (1)

where $\Gamma(z)$ is the Gamma function. This results can be obtained by
computing the T-matrix of the Bose-Hubbard model in the limit of infinite
interactions (implementing effectively the hardcore constraint). It is in
agreement with the estimate obtained by Masaki-Kato et al. Secondly, we show
that the results obtained in Masaki-Kato et al. (2022) agree with first order
spin-wave calculations that take the hardcore constraint into account exactly,
without referring to the Bose-Hubbard model with finite interaction.

## 1) Analytic calculation of the s-wave scattering length-

The s-wave scattering length is obtained from the low-energy limit $E\to 0$ of
the T-matrix $T(E)$, which is obtained by resumming the ladder diagrams
describing the multiple interactions between two bosons Rançon and Dupuis
(2011); Castin (2007). This generalizes directly to the case of bosons on a
lattice with on-site interaction $U$, for which Rançon and Dupuis (2011)

$T(0)=\left(\frac{1}{U}+\int_{\bf q}\frac{1}{2\epsilon_{\bf q}}\right)^{-1},$
(2)

where $\epsilon_{\bf q}=2t(3-\cos(q_{x})-\cos(q_{y})-\cos(q_{z}))$ is the
dispersion on the cubic lattice (such that $\epsilon_{\bf q}\simeq t{\bf
q}^{2}$ at low momenta, from which we read the effective mass) and $\int_{\bf
q}\equiv\int_{BZ}\frac{d^{3}q}{(2\pi)^{3}}$ with $BZ$ the first Brillouin zone
$[-\pi,\pi[^{3}$. Here and thereafter, we use the lattice spacing $a$ as units
of length and $\hbar=1$.

The scattering length is defined by

$\frac{4\pi a_{s}}{m}=8\pi ta_{s}=T(0),$ (3)

and in the limit $U\to\infty$ this leads to

$a_{s}=\frac{1}{\int_{\bf q}\frac{4\pi t}{\epsilon_{\bf q}}}.$ (4)

Evaluating the integral numerically, one recovers the result quoted in the
note added in proof of Masaki-Kato et al. (2022), based on Castin (2007).
However, it so happen that the integral

$I_{3}=\int_{\bf q}\frac{1}{3-\cos(q_{x})-\cos(q_{y})-\cos(q_{z})},$ (5)

known as one of Watson’s triple integrals Borwein et al. (2009), can be
performed analytically and gives the rather amazing result

$I_{3}=\frac{\sqrt{6}}{96\pi^{3}}\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right),$
(6)

which allows us to obtain Eq. (1). A similar analysis shows that on the square
lattice, one finds the two-dimensional scattering length of hardcore bosons is
$a_{s}/a=e^{-C}/(2\sqrt{2})\simeq 0.198506\ldots$ with $C$ the Euler constant
Rançon and Dupuis (2011); Rançon (2014).

## 2) Spin-wave calculation of the ground-state energy of hardcore bosons-

In the dilute limit, the thermodynamics of hardcore bosons can be recovered in
at least two ways. Mapping hardcore bosons on the quantum spin-$1/2$ XY model
in a transverse field, a perturbation theory in $1/S$ to lowest order (spin-
wave or semi-classical approximation) corresponds to a Bogoliubov theory for
magnons in the dilute limit. Careful comparison, between the $1/S$ expansion
and quantum Monte-Carlo on the square lattice have shown that the spin-wave
approximation is in very good agreement with the numerics already at this
order Coletta et al. (2012). Another possibility is to use a (quantum) Weiss
theory for hardcore bosons, that treat the on-site contraint exactly and the
hopping at mean-field. This can then be improved by including loop
corrections, that can be organized within the framework of the (lattice)
functional renormalization group Rançon (2014). Both methods are equivalent,
and the ground-state energy per unit-volume in the grand-canonical ensemble at
lowest order (equivalent to Bogoliubov theory for dilute bosons) reads in
d-dimensions Rançon (2014)

$E_{GS}=-\frac{dt}{2}\left(1+\frac{\mu}{2dt}\right)^{2}+\frac{1}{2}\int_{\bf
q}\left(E_{\bf q}-\epsilon_{\bf q}\right),$ (7)

where $E_{\bf q}=\sqrt{\epsilon_{\bf
q}\left(\left(\frac{\mu}{2dt}\right)^{2}\epsilon_{\bf
q}+2dt\left(1-\left(\frac{\mu}{2dt}\right)^{2}\right)\right)}$ is the
Bogoliubov dispersion of the spin-waves. The first term corresponds to the
mean-field approximation, while the second term is the $1/S$ correction (with
$S=1/2$) coming from the fluctuations of the Bogoliubov excitations. This
result is valid for $-2dt\leq\mu\leq 2dt$ and works very well compared to
Monte-Carlo in this whole range in $d=2$ Coletta et al. (2012). Since
fluctuations are expected to be weaker in higher dimensions, this
approximation is expected to be very good in dimension three.

In three dimensions, the dilute limit corresponds to $\mu=-6t+\delta\mu$, with
$\delta\mu/t\ll 1$. Using Eq. (4) and rewriting the energy as

$E_{GS}=-\frac{\delta\mu^{2}}{16\pi ta_{s}}+\frac{1}{2}\int_{\bf
q}\left(E_{\bf q}-\epsilon_{\bf
q}-\frac{\delta\mu^{2}}{12t}+\frac{\delta\mu^{2}}{2\epsilon_{\bf q}}\right),$
(8)

a careful analysis shows that

$\frac{1}{2}\int_{\bf q}\left(E_{\bf q}-\epsilon_{\bf
q}-\frac{\delta\mu^{2}}{12t}+\frac{\delta\mu^{2}}{2\epsilon_{\bf
q}}\right)=\frac{8t}{15\pi^{2}}\left(\frac{\delta\mu}{t}\right)^{5/2}+\mathcal{O}\left(\delta\mu^{3}\right),$
(9)

and we recover the ground-state energy of a dilute Bose gas in the grand-
canonical ensemble, including the Lee-Huang-Yang correction

$E_{GS}=-\frac{m\delta\mu^{2}}{8\pi
a_{s}}\left(1-\frac{64}{15\pi}\sqrt{ma_{s}^{2}\delta\mu}+C_{\rm cor.}\right).$
(10)

Note that the correction $C_{\rm cor.}$, of order $ma_{s}^{2}\delta\mu$, is
non-universal and comes in part from lattice effects, and we have estimated it
numerically to be $C_{\rm cor.}=\alpha
ma_{s}^{2}\delta\mu+o(ma_{s}^{2}\delta\mu)$, with a conservative estimate
$\alpha\simeq 3.43(1)$. From this, we recover the energy per particles in the
canonical ensemble,

$E/N=\frac{2\pi
a_{s}n}{m}\left(1+\frac{128}{15\pi}\sqrt{na_{s}^{3}}+\frac{4(256-9\pi^{2}\alpha)}{9\pi}na_{s}^{3}+\mathcal{O}(na_{s}^{3})\right).$
(11)

Note that correction term of order $na_{s}^{3}$ explicitly written here is not
the full contribution, as we expect the next order in the loop (or $1/S$)
expansion to also contribute at this order (in addition to a universal term in
$na_{s}^{3}\log(na_{s}^{3})$). Masaki-Kato et al. have estimated the full
contribution at order $na_{s}^{3}$ to be $c_{3}=130(40)$, and with
$\frac{4(256-9\pi^{2}\alpha)}{9\pi}\simeq-6.9(1)$ we conclude that the first
order in $1/S$ contributes for a few percent to $c_{3}$ in the opposite
direction than the next order.

In summary, we have shown that the results of Masaki-Kato et al. are in very
good agreement with the analytical results from the spin-wave approximation
describing a dilute gas hardcore bosons. The latter includes the Lee-Huang-
Yang correction as well as some of the non-universal corrections at the next
order in $na_{s}^{3}$. We have also shown that the scattering-length $a_{s}$,
Eq. (1), naturally appears in the ground-state energy. This comment leaves
open some interesting questions. On the simulation side, it would be
interesting to see if using $a_{s}$ as an input, and not as a fitting
parameter, would give a better estimate of $c_{3}$. It would also be
interesting to compare the numerical results to the spin-wave result Eq. (7)
away from the dilute limit (i.e. in the range $|\mu|\leq 6t$), to see if it
compares as favourably as in two-dimensions. On the theory side, it would be
interesting to compute the next correction in $1/S^{2}$ to the energy (as has
been done in two dimensions in Coletta et al. (2012)) to have a theoretical
estimate of $c_{3}$ and see how the universal correction in
$na_{s}^{3}\log(na_{s}^{3})$ is recovered.

## References

  * Masaki-Kato et al. (2022) A. Masaki-Kato, Y. Motoyama, and N. Kawashima, J. Phys. Soc. Jpn. 91, 024001 (2022).
  * Rançon (2014) A. Rançon, Phys. Rev. B 89, 214418 (2014).
  * Rançon (2014) A. Rançon, arXiv:1404.3425 (2014).
  * Rançon and Dupuis (2011) A. Rançon and N. Dupuis, Phys. Rev. B 84, 174513 (2011).
  * Castin (2007) Y. Castin, Proceedings of the International School of Physics “Enrico Fermi” 164, 289–349 (2007).
  * Borwein et al. (2009) J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, _Lattice Sums Then and Now_ (Cambridge University Press, 2009).
  * Coletta et al. (2012) T. Coletta, N. Laflorencie, and F. Mila, Phys. Rev. B 85, 104421 (2012).