### CENTER FOR

### R

### ESEARCH AND

### ADVANCED

### STUDIES OF THE

### NATIONAL

### POLYTECHNIC

### INSTITUTE

Campus Zacatenco

Department of Mathematics

**Characterization of some model commutative**

**C*-algebras generated by Toeplitz operators**

### A DISSERTATION PRESENTED BY

:**Kevin Michael Esmeral Garcia**

### TO OBTAIN THE DEGREE OF

### DOCTOR IN SCIENCE

### IN THE SPECIALITY OF MATHEMATICS

Thesis Advisors: Dr. Nikolai Vasilevski & Dr. Egor Maximenko

**A**

**BSTRACT**

The aim of this work is to give explicit descriptions of the C*-algebras generated by Toeplitz operator whose defining symbols are invariant under dilations, rotations and translations. Therefore, we have divided this thesis in three chapters:

In Chapter2we consider Toeplitz operators acting on the weighted Bergman space A2

*λ*(Π) (Πdenotes the upper half-plane) with defining symbol invariant under dilations,

i.e., functions*φ*∈L∞(Π), such that for every h>0 the equality*φ*(z)=*φ*(hz) holds a.e.
z_{∈}Π . This class of defining symbols is calledangular, and we give a criterion for a
function to be angular:

*φ*is angular, if and only if there existsa_{∈}L_{∞}(0,*π*)such that*φ*(z)_{=}a(argz),a.e.z_{∈}Π.
The main result states that the uniform closure of the set of all Toeplitz operators
acting on the weighted Bergman space over the upper half-plane whoseL_{∞}-symbols are
angular coincides with the C*-algebra generated by the above Toeplitz operators and is
isometrically isomorphic to the C*-algebra VSO(R) of bounded functions that are very
slowly oscillating on the real line in the sense that they are uniformly continuous with
respect to the arcsinh-metric*ρ*(x,y)_{= |}arcsinhx_{−}arcsinhy_{|}on the real line.

Chapter3is devoted to the study of Toeplitz operators acting on the Fock spaceF2(C)
with defining symbol invariant under rotations, i.e., functions*ϕ*∈L∞(C), such that for
everyt_{∈}[0,2*π*) the equality*ϕ*(z)_{=}*ϕ*(e−itz) holds a.e. z_{∈}C. This class of defining symbols
is calledradial, and we give a criterion for a function to be radial:

*ϕ*is radial, if and only if there existsb_{∈}L_{∞}(R_{+})such that* _{ϕ}*(z)

_{=}b(

_{|}z

_{|}),a.e.z

_{∈}C. The principal theorem shows that the C*-algebra generated by radial Toeplitz operators withL

_{∞}-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra RO(Z

_{+}) of bounded sequences uniformly continuous with respect to the square-root-metric

*̺*(j,k)= |pj−pk|. More precisely, we prove that the sequences of eigenvalues of

radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.
Finally, in Chapter4the Toeplitz operators on the Fock space F2(Cn) are taken with
defining symbol invariant under imaginary translations, i.e., functions*ϕ*∈L∞(Cn), such
that for everyh∈Rn the equality* _{ϕ}*(z)

_{=}

*(z*

_{ϕ}_{−}ih) holds a.e. z

_{∈}Cn. This class of defining symbols is calledhorizontal, and we give a criterion for a function to be horizontal:

defining L_{∞}– symbols are L–invariant is isometrically isomorphic to the C*-algebra
T_{hor}(L

∞) generated by Toeplitz operators acting on the Fock space with horizontalL∞–
symbols. Here, a function*ψ*∈L∞(R2n) is said to beL–invariantif for everyh∈L one
gets that *ψ*(z_{−}h)_{=}*ψ*(z) a.e. z_{∈}R2n, in particular, the horizontal case corresponds to
L _{=}{0}_{×}Rn. The main result of this part states thatT_{hor}(L

∞) is isometrically isomorphic to the C*-algebra Cb,u(Rn) of bounded functions that are uniformly continuous with

respect to the usual metricd(x,y)_{= |}x_{−}y_{|}on then-dimensional real plane. More precisely,
we prove that the corresponding spectral functions form a dense subset inCb,u(Rn).

**D**

**EDICATION AND ACKNOWLEDGEMENTS**

Every work needs self efforts as well as a guidance of God and the aid of each person in your life, especially those who were very close to my heart. For that, I would like to acknowledge with gratitude, the support and love of my family, my parentsNicanorand Elvis(R.I.P), my brothersKelmar andKenny, my wife and my friends, with a special acknowledgement to theProfesor Osmin Ferrerwho have helped me in all moment.

I would like to thank Dr. Nikolai Vasilevski and Dr. Egor Maximenko for their enthusiasm, their encouragement and by showing me the way forward to achieve my objective. They were always available for my questions and they were positive and gave me generously of their time and vast knowledge. They always knew where to look for the answers to obstacles while leading me to the right source, theory and perspective.

**T**

**ABLE OF**

**C**

**ONTENTS**

**Page**

**1** **Preliminaries** **9**

1.1 Weighted Bergman space . . . 9

1.2 Fock spaces . . . 15

1.3 Toeplitz operators . . . 18

1.4 Berezin transform . . . 19

1.5 Approximation of uniformly continuous functions by convolutions . . . 21

1.6 Symplectic spaces and Lagrangian planes . . . 23

**2** **C*-algebra of angular Toeplitz operators on weighted Berman spaces**
**over the upper half–plane** **27**
2.1 Angular Toeplitz operators . . . 27

2.2 Very slowly oscillating property of the spectral functions. . . 32

2.3 Density ofΓ* _{λ}*in VSO(R) . . . 36

2.4 Strong density of Toeplitz operators in the C*-algebra of angular operators 45
**3** **C*-algebra of radial Toeplitz operators acting on Fock spaces** **47**
3.1 Radial Toeplitz operators acting on Fock spaces . . . 47

3.2 Square-root-oscillating property of the eigenvalues’ sequences . . . 51

3.3 Density ofGin RO(Z_{+}) . . . 58

3.4 Beyond the class of bounded generating symbols . . . 65

**4** **C*-algebra of horizontal Toeplitz operators acting on Fock spaces** **69**
4.1 Bargmann transform . . . 69

4.2 Horizontal Toeplitz operators . . . 72

4.3 L–invariant Toeplitz operators . . . 78

**A Appendix A** **83**

A.1 Watson’s Lemma . . . 83

A.2 Fourier transform of bounded Borel measures . . . 84

A.3 Topology on a Banach space . . . 84

A.4 Topologies onB(H) . . . 85

A.5 About multiplication operator . . . 86

A.6 Classical harmonic analysis and Wiener’s theorem . . . 89

**I**

**NTRODUCTION**

In linear algebra an infinite Toeplitz matrixT is defined by the rule:

T=

a0 a_{−}1 a_{−}2 ...
a1 a0 a−1 ...
a2 a1 a0 ...
... ... ... ...

,

where an∈C, n∈Z. In 1911Otto Toeplitzproved that the matrix T defines a bounded

operator on *ℓ*2(Z_{+}), where Z_{+}_{=}N_{∪}{0}, if and only if the numbers a_{n} are the Fourier
coefficients of a functiona∈L∞(S1), whereS1is the unit circle.

The classical Hardy space H2 can be viewed as the closed linear span in L2(S1)
of {zn_{:} _{n}

≥0}. For g∈L∞(S1), the Toeplitz operatorTgdefined byTgh=B(gh), where

B denotes the orthogonal projection from L2(S1) onto H2, is bounded and satisfies

kTgk ≤ kgk∞. The matrix ofTgwith respect to the orthonormal basis {zn: n≥0} is the

Toeplitz matrixT withanbeing the Fourier coefficients ofg. Thus, the Toeplitz operators

are a generalization of the Toeplitz matricesT.

LetX be a function space and letB be a projection ofX onto some closed subspace
Y ofX. Then the Toeplitz operator T_{g}:Y _{−→}Y with defining symbol g is given by
Tgf =B(g f). The most studied cases are whenY is either the Bergman space, the Hardy

space, or the Fock space. More recently Toeplitz operators have been also studied on many other spaces, for example on the harmonic Bergman space [38].

The Toeplitz operators have been extensively studied in several branches of mathe-matics: complex analysis, theory of normed algebras, operator theory [5, 28, 36, 53], harmonic analysis [1,17], and mathematical physics, particularly in connection with quantum mechanics [9,11], etc.

little about the properties of Toeplitz operators with general L_{∞}-symbols; though some
general results are collected, for example, in [54] the common strategy here is to study
Toeplitz operators with symbols from certain special subclasses ofL_{∞}.

The most complete results were obtained for the families of symbols that generate
commutative C*-algebras of Toeplitz operators acting on the weighted Bergman space.
They were described in a series of papers summarized in the book [50], see also [26].
These families of defining symbols lead to the following three model cases:radialsymbols,
functions on the unit disk depending only on_{|}z|,verticalsymbols, functions on the upper

half-plane depending on Imz, andangularsymbols defined on the upper half-plane and
depending only on argz. Unlike the Bergman case, on the Fock space there are only
two model cases that generate commutative C*-algebras of Toeplitz operators:radial
symbols, functions on the complex planeCdepending only on_{|}w_{|}, andhorizontal symbols,
functions on Cdepending only on Rew.

In each one of the above models (for the horizontal model is proved in this Ph.D. thesis), the corresponding Toeplitz operator admit an explicit diagonalization, i.e. there exists an isometric isomorphism that transforms all Toeplitz operators of the selected type to the multiplication operators by some specific functions (we call them spectral functions, in the radial case they are just the sequences of eigenvalues). Of course, such a diagonalization immediately reveals all the main properties of the corresponding Toeplitz operators [23–25,28].

Then the next natural problem emerges: give an explicit and independent description
of the class of spectral functions (and of the algebra generated by them) for each one of
the above cases. First step in this direction was made by Suárez [46,47]. He proved that
the sequences of eigenvalues of Toeplitz operators acting on the Bergman space with
boundedradialsymbols form a dense subset in the*ℓ*_{∞}-closure of the class d1 of bounded
sequences (*σ*k)∞_{k}_{=}_{1}satisfying

sup

k

(k_{+}1)_{|}*σ*k+1−*σ*k| < +∞.

beginning of the 20th century in connection with Tauberian theory. It is worth mentioning
that the above description shows the room that radial Toeplitz operators occupy amongst
all bounded radial operators (the set of which is isomorphic to*ℓ*_{∞}(N)).

Herrera Yañez, Hutník, Maximenko and Vasilevski [30–32] continue this program
and give a description of the commutative algebra generated by Toeplitz operators acting
on the Bergman space with boundedverticalsymbols. The result states that their spectral
functions form a dense subset in the C*-algebra VSO(R_{+}) ofvery slowly oscillating
func-tionsonR_{+}, i.e. the bounded functionsR_{+}_{→}Cthat are uniformly continuous with respect
to the metric_{|}ln(x)_{−}ln(y)_{|}. This, in particular, means that the C*-algebra generated by
Toeplitz operators with bounded vertical symbols is isometrically isomorphic to VSO(R

+). This thesis is devoted to the study of remaining model cases. That is, we give an explicit description of the commutative C*-algebras generated by Toeplitz operator acting on the weighted Bergman space whose defining symbols are angular, and of the commutative C*-algebras generated by Toeplitz operator acting on the Fock space whose defining symbols are radial and horizontal. With this we complete thus the intrinsic description of the commutative C*-algebras generated by Toeplitz operators with bounded symbols for each one of the model classes that appear for the weighted Bergman space and the Fock space.

The work is divided into three chapters. In Chapter2we are interested in the Toeplitz operator with angular symbols acting on the weighted Bergman spaceA2

*λ*(Π) over the

upper half-planeΠ, which consists of all analytics functions inL_{2}(Π,d* _{ν}_{λ}*), where

d*νλ*(z)=(*λ*+1)(2y)*λ*d ydx, z=x+i y, *λ*∈(−1,+∞).

A function g_{∈}L_{∞}(Π) is said to behomogeneous of order zeroorangularif for every
h_{>}0 the equality g(hz)_{=} g(z) holds for a.e. z_{∈}Π, or, equivalently, if there exists a
functionainL_{∞}(0,*π*) such thatg(z)_{=}a(argz) for a.e. zinΠ. We denote byA

∞this class
of functions, and introduce the set T* _{λ}*(A

∞) of all Toeplitz operators acting onA*λ*2(Π) with

defining symbols inA ∞.

As was shown in [25], the uniraty operatorR*λ* : A* _{λ}*2(Π)→L2(R), where

(1) (R*λϕ*)(x)=p 1

2*λ*+1_{(}_{λ}_{+}_{1)}_{c}_{λ}_{(x)}

Z

Π

(z)−ix− ³

*λ*+2

2

´

*ϕ*(z)d*µλ*(z), x∈R,

with

(2) c*λ*(x)=

Z*π*

0 e

−2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}_{,}

_{x}

diagonalizes each Toeplitz operator T_{g} with angular symbol g(z)_{=}a(argz); that is
R* _{λ}*T

_{g}R∗

*λ*=*γ*a,*λ*I, where the spectral function*γ*a,*λ*:R→Cis given by

(3) *γ*a,*λ*(x)=

1
c*λ*(x)

Z*π*

0 a(*θ*)e

−2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}_{,}

_{x}

∈R.
In particular, this implies that the algebra generated by T* _{λ}*(A

∞) is isometrically isomor-phic to the C*-algebra generated by

(4) Γ_{λ}_{=}©_{γ}_{a}_{,}* _{λ}*: a

_{∈}L

∞(0,*π*)

ª_{.}

Chapter2describes explicitly this C*-algebra. We denote by VSO(R) the C*-algebra of very slowly oscillatingfunctions on the real line [18], which consists of all bounded functions that are uniformly continuous with respect to thearcsinh-metric

(5) *ρ*(x,y)= |arcsinhx−arcsinhy|, x,y∈R.

The main result here (Theorem2.4) states that the uniform closure ofΓ* _{λ}* coincides with
the C*-algebra VSO(R). As a consequence, the C*-algebra T

*(A*

_{λ}∞) generated by the
set T*λ*(A_{∞}) coincides just with the closure of this set of its initial generators, and is

isometrically isomorphic to VSO(R). Note that the result does not depend on a value of
the weight parameter*λ*> −1.

As a by-product of the main result, we show that the closure of the set T*λ*(A∞) in the
strong operator topology coincides with the C*-algebra of all angular operators.

With this work we finish the explicit descriptions of the above mentioned commutative C*-algebras of Toeplitz operators on the unit disk and upper half-plane. In all three cases the spectral functions oscillate at infinity with the logarithmic speed. The C*-algebras VSO(R

+) and VSO(R) corresponding to the vertical and angular cases, respectively, are isometrically isomorphic (via the change of variables v7→sinh(ln(v))), and both of them

are isometrically isomorphic to the C*-algebraCb,u(R) consisting of all bounded functions

onR that are uniformly continuous with respect to the usual metric (via the changes
of variablesv_{7→}ln(v) andv_{7→}arcsinh(v), respectively). The sequences from VSO(N) are
nothing but the restrictions toNof the functions from VSO(R_{+}).

Note that the proof in the vertical case was the simplest one because the corresponding
spectral functions*γ*v_{a}_{,}* _{λ}*admit representations in terms of the Mellin convolutions, and

the result about density was obtained just by using a convenient Dirac sequence. Unfor-tunately, in the angular case this simple approach does not work.

The key idea of the proof presented in this work is to approximate functions from
VSO(R) by* _{γ}*v

ofC0(R) functions by appropriate*γ*a,*λ*; the latter problem was solved using the duality

and the analyticity arguments (Theorem2.3).

This chapter is organized as follows: Section2.1contains criteria of angular operator
and of angular Toeplitz operator. In Section2.2we introduce formally the class of
func-tions VSO(R) and is showed that the functions of the classΓ* _{λ}*are Lipschitz continuous
with respect the arcsinh metric

*ρ*. That isΓ

_{λ}_{(}VSO(R). In Section 2.3 we prove the density of Γ

*in VSO(R), and we finish this chapter showing in Section 2.4 that the closure of {Tg: g∈A*

_{λ}_{∞}}in the strong operator topology coincides withB(A

*2(Π)).*

_{λ}Chapter 3 focuses on the study of the C*-algebra generated by radial Toeplitz
operators acting on Fock spaces. It is well known [53] that the normalized
monomi-als en(z)=zn/pn!, n∈Z+, form an orthonormal basis ofF2(C), and that the Toeplitz
operators with bounded radial symbols are diagonal with respect to this basis [28].
Namely, ifa_{∈}L_{∞}(R_{+}) and* _{ϕ}*(z)

_{=}a(

_{|}z

_{|}) a.e.z

_{∈}C, thenT

*e*

_{ϕ}_{n}

_{=}

_{γ}_{a}(n)e

_{n}, where

(6) *γ*a(n)=p1

n!

Z

R_{+}

a(pr)e−r_{r}n_{dr,} _{n}_{∈}Z
+.

From this diagonalization we have that the C*-algebra T_{rad} generated by Toeplitz
operators with radial L_{∞}-symbols is isometrically isomorphic to the C*-algebra G
generated by the setGof all sequences of the eigenvalues:

(7) G_{=}©*γ*a:a∈L_{∞}(R_{+})ª.

The main result of this part states that the uniform closure ofGcoincides with the

C*-algebra RO(Z

+) consisting of all bounded sequences*σ*:Z+−→Cthat are uniformly
continuous with respect to thesquare-root-metric

*̺*(m,n)=¯¯pm−pn ¯¯, m,n∈Z
+.

As a consequence, we obtain an explicit description of the C*-algebraG generated byG:

G_{=}RO(Z_{+}).

This chapter is organized as follows: In Section 3.1we have compiled some basic facts about radial Toeplitz operators in Fock space. In Section3.2we introduce the class RO(Z

+) and prove thatGis contained in RO(Z+).

The major part of Chapter3is occupied by a proof thatGisdensein RO(Z

+), see a
scheme in Figure 0.1. Given a sequence*σ*_{∈}RO(Z_{+}), we extend it to a sqrt-oscillating
function f onR_{+}(Proposition3.5). After the change of variables h(x)_{=}f(x2) we obtain a
bounded and uniformly continuous functionhonR. In Section1.5, using Dirac sequences
and Wiener’s division lemma, we show that functions from Cb,u(R) can be uniformly

approximated by convolutions k∗b, where k is a fixed L1(R)-function whose Fourier
transform does not vanish onR. This construction will be applied when k is the heat
kernel H (Section 3.3) and we examine the asymptotic behavior of the sequences of
eigenvalues *γ*a. It is shown there that after change of variablespn =x, the function

x_{7→}*γ*a(x2), for x sufficiently large, is close to the convolution of the symbolawith the

heat kernelH. In Section3.3also we show thatc0(Z_{+}) coincides with the uniform closure
of the set {*γ*a: a∈L∞(R+), limr→∞a(r)=0}. After that, gathering together all the pieces,
we obtain the main result of this chapter (Theorem 3.4 ). Finally, in Section 3.4 we
describe a class of generating symbols bigger thanL_{∞}(R), with eigenvalues’ sequences
still belonging to RO(Z

+), and construct an unbounded generating symbolasuch that

*γ*a∈*ℓ*_{∞}(Z_{+}) \ RO(Z_{+}).

σ_{∈}RO(Z

+)

f _{∈}RO(R_{+})

h_{∈}Cb,u(R) b∈ L_{∞}(R)

a_{∈}L_{∞}(R

+)

σ ℓ_{≈}∞ γa+c

ϑ_{∈}c0(Z+) c∈L∞(R+)

σ= f_{|}Z
+

h(x)_{=} f(x2_{)}

hL_{≈}∞ H_{∗}b

(H_{∗}b)(√j)_{≈}γa(j)
j > N

ϑ_{=}(σ_{−}γa)χ[0,N]

ϑℓ_{≈}∞γc

Figure 0.1: Scheme of the proof of density: the upper chain represents the approximation
of *σ*(j) for large values of j (j >N), and the lower one corresponds to the uniform

Finally Chapter 4 provides a detailed description of the Toeplitz operators with
horizontal symbols acting on the Fock space F2(Cn), where a function f _{∈} L

∞(Cn),
is called horizontal if for every h∈Rn the equality f(z)_{=} f(z_{−}ih) holds a.e. z_{∈}Cn.
Equivalently, f is a horizontal function, if there exists a∈ L∞(Rn) such that f(z)=
a(Rez),a.e.z_{∈}Cn.

It is well-known that the Bargmann transform is an isometric isomorphism from
L2(R2n) onto theF2(Cn) [53], and hence plays an important role in the description of
Toeplitz operators. Furthermore, this transformation relates the Toeplitz operators acting
onF2(Cn) with pseudo-differential operators acting onL_{2}(R2n). The Bargmann
trans-form is important in our approach, because the main idea as in the above model cases is
to get spectral functions*γ*H_{a} such that the Toeplitz operators are unitary equivalent to

the multiplication operators M* _{γ}*H

a acting onL2(R

n_{).}

First of all, we find a decomposition of the Bargmann transform B∗_{as composition}
of two unitary operators which allows us to diagonalize the Toeplitz operators with
horizontal symbols. More precisely, if*ϕ*(z)_{=}a(Rez) is a horizontal function, then the

Toeplitz operatorT*ϕ*is unitary equivalent to the multiplication operator BT*ϕ*B∗=*γ*aHI,

where the spectral function*γ*H_{a} :Rn_{→}Cis given by the formula

(8) *γ*H_{a}(x)=*π*−n/2

Z

Rn

a

µ _{y}

p

2

¶

e−(x−y)2_{d y,} _{x}_{∈}Rn.

A consequence of the latter diagonalization is that the C*-algebraT_{hor}(L

∞) of Toeplitz operators whose defining symbols are horizontal is isometrically isomorphic to the C*-algebraGH generated by

(9) GH_{=}n*γ*H_{a} : a_{∈}L_{∞}(Rn)

o

.

This result is generalized for functions invariant under translations over certain kind of subspaces ofCn. Recall thatR2nhas the standard symplectic form

*ω*0(z,w)=y·x′−y′·x,

for z_{=}(x,y) andw_{=}(x′_{,}_{y}′_{). It is well-known that a linear subspace}L of the symplectic
space (R2n,_{ω}_{0}) is a Lagrangian plane if for every pair (z,w)_{∈}L _{×}L, one has that

*ω*0(z,w)=0. The simplest examples of Lagrangian planes of (R2n,_{ω}_{0}) are both coordinates
planes:L_{x}_{=}Rn_{×}{0} andL_{y}_{=}{0}_{×}Rn.

Lagrangian planesL of (R2n,_{ω}_{0}), for brevity we call themL–invariant symbols. That
is,*ϕ*is aL–invariant functionif for everyh_{∈}L one gets that* _{ϕ}*(z

_{−}h)

_{=}

*(z) a.e.z*

_{ϕ}_{∈}Cn.

Using the symplectic rotations U(2n,R) we give a criterion for a function to beL–
invariant, and thus relate Toeplitz operators with this class of defining symbols with
Toeplitz operators having horizontal symbols. In fact, given any Lagrangian planeL we
can find a symplectic rotationB_{∈}U(2n,R) withBL _{=}{0}_{×}Rnand a unitary operatorV_{B}
such that for everyL–invariant function_{ϕ}_{∈}L

∞(R2n) there exists a horizontal function

*ψ*B∈L∞(R2n) (depending on*ϕ*andL) such that
V_{B}−1T*ϕ*VB=T*ψ*B.
Therefore, the C*-algebraT_{L}(L

∞) generated by Toeplitz operators withL–invariant
symbols is isometrically isomorphic toT_{hor}(L

∞). Thus, both C*-algebrasTL(L_{∞}) and

T_{hor}(L

∞) are isometrically isomorphic to the C*-algebraGH generated byGH, see (9). The main result of this chapter states that the uniform closure of GH coincides

with the C*-algebraCb,u(Rn) consisting of all bounded functions *σ*:Rn−→C that are

uniformly continuous with respect to theusual metriconRn. As a consequence, we obtain an explicit description ofGH:

GH_{=}C_{b}_{,}_{u}(Rn).

Notice that, unlike the angular and radial case, here we do not need to approximate
functions*σ*∈Cb,u(Rn) at the infinity by spectral function*γ*Ha. It follows from the structure

of the functions*γ*H_{a}, that they can be written directly as convolutions of functions a∗H,

whereH is then-dimensional heat kernelH(x)_{=}*π*−n/2e−x2.

This chapter is organized as follows: in Section4.1we write the Bargmann transform
as a composition of two unitary operators. In Section 4.2we introduce the horizontal
operators and horizontal Toeplitz operators, give some basic properties of them, including
a criterion of a operator to be horizontal. Also, we prove that the Toeplitz operators with
horizontal symbols are unitary equivalent to the multiplication operator M* _{γ}*H

a acting on
L2(Rn). In Section4.3we introduce theL–invariant functions, and establish a criterion
of a function to beL–invariant. We show that the C*-algebraT_{L}(L

∞) generated by the
Toeplitz operators whose defining symbols areL-invariant is isometrically isomorphic
toT_{hor}(L

∞). Finally, in Section4.4we prove the main result of this chapter. The proof is based on approximations of bounded uniformly continuous functions by convolutions used in Chapter3. That is, using Dirac sequences and Wiener’s division lemma, we show that the functions fromCb,u(Rn) can be uniformly approximated by convolutions k∗b,

### C

H

A

P

T

E

R

## 1

**P**

**RELIMINARIES**

In this chapter we collect several preliminary results, the main purpose here is to fix notation and to facilitate references later on. All the results are well known [14,20,29, 42,50,53,54].

**1.1**

**Weighted Bergman space**

LetΠbe the upper half-plane of the complex planeC:

(1.1) Π:_{=}{z_{∈}C: Imz_{>}0}.

Given theweight parameter*λ*_{∈}(_{−}1,_{+∞}), we introduce the following standard measure

on the upper half-planeΠ:

d*νλ*(z)=(*λ*+1)(2y)*λ*d y dx, z=x+i y.

The weighted Bergman space A2

*λ*(Π) consists of all analytic functions belonging to

L2(Π,d*νλ*). An important property of the Bergman space is contained in the following

lemma.

**Lemma 1.1.** Letn∈{0,1,2,...}. Given a compact setK⊂Π, there is a constantC_{=}C_{n}_{,}_{K}_{,}* _{λ}*,
depending onn, K and

*λ*∈(−1,+∞), such that

(1.2) sup

z∈K|f

(n)_{(z)}

| ≤CkfkA2

*λ*(Π),
for all f ∈A2

**Proposition 1.1.** The Bergman spaceA2

*λ*(Π)is a closed subspace of L2(Π,d*νλ*).

Proof. Let (fn)n∈Nbe a fundamental sequence of analytic functions fromA* _{λ}*2(Π) converging

onL2(Π,d*νλ*) to certain function f ∈L2(Π,d*νλ*). By Lemma1.1we see that (fn)n∈N

con-verges uniformly on every compact subsetK ofΠto certain analytic function g. However, by (1.2) we have for each compac subsetK ofΠthat

|f(z)−g(z)| ≤_{m}lim

→∞|f(z)−fm(z)| ≤CK,*λ*mlim→∞kf−fmkA*λ*2(Π)=0, z∈K.
Therefore, f is analytic and belongs toA2

*λ*(Π).

From Lemma 1.1 is follows as well that for any fixed point z∈Π the evaluation
functional*ψ*z(f)=f(z) is linear and bounded. Thus by the Riesz-Fréchet representation

theorem there exists a unique elementKz,*λ*∈A* _{λ}*2(Π) such that

*ψ*z= 〈·,Kz,

*λ*〉. The function

Kz,*λ*is the so-calledBergman kernelat a point z, and it is well known that is given by

the formula

(1.3) Kz,*λ*(w)=

µ _{i}

w−z

¶*λ*_{+}2

w_{∈}Π.

Observe that from the above definition it follows that the Bergman kernel
func-tion Kz,*λ*(w) is analytic inw and anti-analytic inz(analytic in z). On the other hand,

since the Bergman space A2

*λ*(Π) is a closed subspace of L2(Π,d*νλ*), there exists the

unique orthogonal projectionB*λ*fromL2(Π,d*νλ*) ontoA* _{λ}*2(Π). This projection is called

theBergman projectionand has the integral representation

(B*λ*f)(z)=i*λ*+2

Z

Π

f(w)

(z_{−}w)*λ*+2d*νλ*(w), f ∈L2(Π,d*νλ*).

**Representations of the weighted Bergman space**

The Bergman space can be characterized as the set ofL2(Π,d* _{ν}_{λ}*) functions which satisfy
the Cauchy-Riemann equation

(1.4) *∂*

*∂*zf(z)=0.

Passing to polar coordinates we have the tensor decomposition

(1.5) L2(Π,d*νλ*) :=L2

³

R+,r*λ*+1dr

´

⊗L2

µ

[0,*π*],2
*λ*

*π* (*λ*+1) sin
*λ _{θ}*

_{d}

_{θ}¶

,

and rewriting (1.4) we have that the Bergman space A2

*λ*(Π) is the set of all functions

satisfying the equation

(1.6)

µ

r *∂*

*∂*r+i
*∂*
*∂θ*

¶

1.1. WEIGHTED BERGMAN SPACE

LetU1,*λ*=p1* _{π}*(M⊗Id) be the unitary operator from

L2

³

R+,r*λ*+1dr

´

⊗L2

µ

[0,*π*],2
*λ*

*π* (*λ*+1) sin
*λ _{θ}*

_{d}

_{θ}¶

onto L2(R)_{⊗}L_{2}

³

[0,*π*],2* _{π}λ*(

*λ*

_{+}1) sin

*λθ*d

*θ*´, where M :L2¡R+,r

*λ*+1dr¢

_{−→}L

_{2}(R) is the Mellin transform defined by the rule

(1.7) ¡M*ψ*¢(x) :=p1_{2}

*π*

Z

R_{+}

r−ix+*λ*2* _{ψ}*(r)dr.

Its inverse Mellin transform M−1_{:}_{L}_{2}_{(}R)_{−→}L_{2}¡R+,r*λ*+1dr¢is given by

(1.8) ¡M−1* _{ψ}*¢

_{(r) :}

_{=}

_{p}1 2

*π*

Z

R

ri*ξ*−*λ*2−1* _{ψ}*(

*)d*

_{ξ}*.*

_{ξ}Observe that

(M_{⊗}Id)

µ

r *∂*

*∂*r+i
*∂*
*∂θ*

¶_{¡}

M−1_{⊗}_{Id}¢_{=}

µ

i*ξ*_{−}

µ

*λ*

2+1

¶¶

Id_{+}i *∂*

*∂θ*=i

µµ

*ξ*_{+}

µ

*λ*

2+1

¶

i

¶

Id_{+} *∂*

*∂θ*

¶

.

Hence, the image of the weighted Bergman spaceA2

*λ*(*π*) under the unitary operatorU1,*λ*

A_{1,}_{λ}_{=}U_{1,}* _{λ}*¡A2

*λ*(Π)

¢

it is the closed subspace ofL2(R)⊗L2

³

[0,*π*],2* _{π}λ*(

*λ*+1) sin

*λθ*d

*θ*

´

, consisting of all functions

*ψ*(*ξ*,*θ*) satisfying the equation

µµ

*ξ*_{+}

µ

*λ*

2+1

¶

i

¶

Id_{+} *∂*

*∂θ*

¶

*ψ*(*ξ*,*θ*)_{=}0.

The generalL2(R)_{⊗}L_{2}

³

[0,*π*],2* _{π}λ*(

*λ*+1) sin

*λθ*d

*θ*

´

-solution of this equation has the form

*ψ*(*ξ*,*θ*)= f(*ξ*)e

−³*ξ*+³*λ*+22

´

i´*θ*

p

2*λ*_{(}_{λ}_{+}_{1)}_{c}
*λ*(*ξ*)

, f∈L2(R),

where the function c*λ*:R−→R+is of the form

(1.9) c*λ*(x)=

Z*π*

0 e

−2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}_{,}

_{x}

∈R.

and

°

**Lemma 1.2.** Let c*λ*(x)be the function given in (1.9). The following properties hold.

(i)The functionc*λ*can be written as

(1.10) c*λ*(x)=

*π*Γ(_{λ}_{+}1)e−*π*x
2*λ*¯¯Γ¡*λ*+2

2 +ix¢¯¯

2, x∈R.

(ii)The function c*λ*is infinitely smooth, for everyp=0,1,2,...its p-th derivative is given

by the integral

(1.11) dpc*λ*(x)

dxp =(−2)p

Z*π*

0 *θ*

p_{e}_{−}2x*θ*_{sin}*λ*
*θ*d*θ*,

moreover, it has the following asymptotic behavior at+∞:

(1.12) dpc*λ*(x)
dxp ∼

(_{−}2)p_{Γ}_{(}_{λ}

+p+1)

(2x)*λ*_{+}p_{+}1 , as x→ +∞.

Proof. (i) By definition of Beta function and by [22, Eq: 3.892-1] one gets that:

c*λ*(x)=

Z*π*

0 e

−2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}=

Z*π*

0 e

i*βθ*_{sin}v−1_{θ}_{d}_{θ}_{, where} _{β}_{=}_{2ix,}_{v}_{=}_{λ}_{+}_{1,}

= *π*e

iβπ

2

2v_{−}1_{v}B

³_{v}

+*β*+1
2 ,

v_{−}*β*+1
2

´= *π*e

x*π*

2*λ*_{(}_{λ}_{+}_{1)}_{B}¡*λ*+2

2 +ix,*λ*+22−ix

¢= *π*

Γ(_{λ}_{+}2)e−*π*x
2*λ*_{(}_{λ}_{+}_{1)}¯¯_{Γ}¡*λ*_{+}2

2 +ix¢¯¯ 2

= *π*

Γ(_{λ}_{+}1)e−*π*x
2*λ*¯¯_{Γ}¡*λ*_{+}2

2 +ix¢¯¯

2, x∈R.

(ii) Sincec*λ*(x)=e−2x*π*c*λ*(−x) for eachx∈R, we give the proof of the equality for the case

x_{≥}0. Given p_{∈}{0,1,2,...} note that*θ*pe−2x*θ*sin*λθ*_{≤}*θ*psin*λθ*for each*θ*_{∈}(0,*π*) and each

x_{≥}0. Thus, by the Leibniz’s rulec*λ*is infinitely smooth and

dpc*λ*(x)

dxp =(−2)
pZ*π*

0 *θ*

p_{e}_{−}2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}_{,}

_{x}

∈R.

On the other hand, the asymptotic behavior is easily analized using the Watson’s Lemma
(Proposition A.1). Writing*θ*psin*λθ* as *θλ*+p¡sin* _{θ}θ*¢

*λ*, where ¡sin

*¢*

_{θ}θ*λ*is infinitely smooth

near 0, we obtain:

Z*π*

0 *θ*

p_{e}_{−}2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}∼

Γ(_{λ}_{+}p_{+}1)

(2x)*λ*+p+1 , asx→ +∞,

thus the proof is completed. _{}

**Example 1.1.** Letz_{∈}ΠandK_{z}_{,}* _{λ}* the Bergman kernel. Consider

Kz,*λ*(w)=

µ _{i}

rei*α*_{−}_{z}

¶*λ*_{+}2

1.1. WEIGHTED BERGMAN SPACE

by [22, Eq: 3.194.3] we have that

(U1,*λ*Kz,*λ*)(x,*α*)=

i*λ*+2

*π*p2

Z

R_{+}

r−ix+*λ*2

(rei*α*_{−}_{z)}*λ*_{+}2dr=

i*λ*+2

*π*(_{−}z)*λ*_{+}2p_{2}

Z

R_{+}

ru−1
(r*β*+1)vdr

= i

*λ*_{+}2
*π*(−z)*λ*+2p_{2}*β*

−uB(u,v_{−}u),

where u_{= −}ix_{+}*λ*+2

2 ,*β*= −e
iα

z = e

i(*α*−*π*)

z ,v=*λ*+2 andBis the Beta function. Hence

(U_{1,}* _{λ}*Kz,

*λ*)(x,

*α*)=

i*λ*_{+}2

*π*(−z)*λ*+2p_{2}

·_{e}i(*α*_{−}*π*)

z

¸ix_{−}³*λ*+2
2

´

B

µ

*λ*+2

2 −ix,

*λ*+2

2 +ix

¶

= i

*λ*+2

*π*(−z)*λ*+2p_{2}

·_{e}i(*α*−*π*)

z

¸ix_{−}³*λ*_{+}2
2

´_{¯}

¯Γ¡*λ*+2
2 +ix¢¯¯

2

Γ(_{λ}_{+}2)

=

(−i)*λ*+2(ei*π*)

(*λ*+2)

2 e−x*α*−i−

³_{λ}

+2

2

´

(z)ix_{+}*λ*+2
2 p2

e

x*π*¯_{¯}_{Γ}¡*λ*+2
2 +ix¢¯¯

2

*π*Γ(_{λ}_{+}2)

.

Now, since (ei*π*_{)}(*λ*+_{2}2)

=i*λ*+2, by (1.10) item (i) we have for each*α*_{∈}(0,*π*) that

(1.13) (U_{1,}* _{λ}*Kz,

*λ*)(x,

*α*)=

e−x*α*−i
³_{λ}

+2 2

´

*α*_{(z)}_{−}ix_{−}³*λ*+2
2

´

2*λ*+12(_{λ}_{+}1)c* _{λ}*(x)

, x_{∈}R_{+}.

**Lemma 1.3.** The unitary operatorU1,*λ*is an isometric isomorphism of the spaceL2

¡

Π,d* _{µ}_{λ}*¢
onto L2(R)

_{⊗}L

_{2}¡[0,

*],2*

_{π}*λ*(

_{λ}_{+}1) sin

*λ*d

_{θ}*¢ under which the Bergman space A2*

_{θ}*λ*(Π) is

mapped onto

(1.14) A_{1,}_{λ}_{=}

f(*ξ*)e−(
³

*ξ*_{+}³*λ*_{2}_{+}1´i´*θ*

p

2*λ*_{(}_{λ}_{+}_{1)}_{c}
*λ*(*ξ*)

, f _{∈}L2(R)

.

Let R0,*λ*:L2(R)−→A1,*λ*⊂L2(R)⊗L2¡[0,*π*],2*λ*(*λ*+1) sin*λθ*d*θ*¢be the isometric
em-bedding given by

(1.15) ¡R0,*λ*f

¢

(*ξ*,*θ*)= f(*ξ*)e

−³*ξ*+³*λ*+22

´

i´*θ*

p

2*λ*_{(}_{λ}_{+}_{1)}_{c}
*λ*(*ξ*)

.

The adjoint operatorR∗

0,*λ*:A1,*λ*⊂L2(R)⊗L2

¡

[0,*π*],2*λ*(*λ*+1) sin*λθ*d*θ*¢−→L2(R) has the
form

(1.16) ³R∗
0,*λψ*

´

(*ξ*)_{=}

s

2*λ*_{(}_{λ}_{+}_{1)}

c*λ*(*ξ*)

Z*π*

0 *ψ*(*ξ*,*θ*)e

−³*ξ*−³*λ*+22

´

Note that

R∗

0R0=IL2(R):L2(R)−→L2(R);

R_{0}R∗

0 =B1,*λ*:L2(R)⊗L2

³

[0,*π*],2*λ*(*λ*_{+}1) sin*λθ*d*θ*´_{−→}A_{1,}_{λ}

whereB_{1,}_{λ}_{=}U_{1,}* _{λ}*B

_{Π}

_{,}

*U−1*

_{λ}1,*λ*is the orthogonal projection from

L2(R)⊗L2

³

[0,*π*],2*λ*(*λ*+1) sin*λθ*d*θ*

´

ontoA_{1,}* _{λ}*.

Now the operatorR_{λ}_{=}R∗

0,*λ*U1,*λ*maps the spaceL2(Π,d*µλ*) ontoL2(R), and its

restric-tion

R*λ*|A2

*λ*(Π)=R
∗

0,*λ*U1,*λ*:A

2

*λ*(Π)−→L2(R)

is an isometric isomorphism. The adjoint operator

R∗

*λ*=U1,∗*λ*R0,*λ*:L2(R)−→A

2

*λ*(Π)⊂L2(Π,d*µλ*)

is an isometric isomorphism fromL2(R) onto the Bergman spaceA2

*λ*(Π).

**Remark 1.1.**

R* _{λ}*R∗

*λ*=IdL2(R):L2(R)−→L2(R);

R∗

*λ*R*λ*=B*λ*:L2

¡

Π,d* _{µλ}*¢

_{−→}A2

*λ*(Π);

**Theorem 1.2.** The isometric isomorphismR∗

*λ*=R0,∗*λ*U1,*λ*:A

2

*λ*(Π)−→L2(R)is given by

(1.17) ¡R∗

*λ*f

¢

(z)_{=}

Z

R

zi*ξ*−
³_{λ}

+2 2

´ _{f}_{(}

*ξ*)

p

2*λ*_{+}1_{(}_{λ}_{+}_{1)}_{c}
*λ*(*ξ*)

d*ξ*.

**Corollary 1.1.** The inverse isomorphismR_{λ}_{=}R∗_{0,}* _{λ}*U

_{1,}

*:A2*

_{λ}*λ*(Π)−→L2(R)is given by

(1.18) (R*λϕ*)(x)=p 1

2*λ*+1_{(}_{λ}_{+}_{1)}_{c}_{λ}_{(x)}

Z

Π

(z)−ix− ³

*λ*+2
2

´

*ϕ*(z)d*µλ*(z), x∈R,

where the Lebesgue measure*µλ*is given by

(1.19) d*µλ*(z)=

2*λ*
*π*r

*λ*+1_{sin}*λ _{θ}*

_{drd}

_{θ}_{,}

_{z}

1.2. FOCK SPACES

**Example 1.2.** Let z∈ΠandK_{z}_{,}* _{λ}*be the Bergman kernel ofA2

*λ*(Π). SinceR*λ*=R0,∗*λ*U1,*λ*,

by (1.13) and (1.16) we get

(R* _{λ}*Kz,

*λ*)(x)=(R0,∗

*λ*U1,

*λ*Kz,

*λ*)(x)=

s

2*λ*_{(}_{λ}_{+}_{1)}

c*λ*(x)

Z*π*

0 (U1,*λ*Kz,*λ*)(x,*θ*)e

−³x−³*λ*+22

´

i´*θ*_{sin}*λ*
*θ*d*θ*

=

s

2*λ*_{(}_{λ}_{+}_{1)}

c*λ*(x)

Z*π*

0

e−x*θ*−i
³

*λ*+2

2

´

*θ*_{(z)}−ix−³*λ*+22

´

2*λ*_{+}1_{2}_{(}_{λ}

+1)c*λ*(x)

e− ³

x−³*λ*+22

´

i´*θ*_{sin}*λ*
*θ*d*θ*

= (z)

−ix−³*λ*+22

´

c3/2* _{λ}* (x)p2

*λ*+1

_{(}

_{λ}_{+}

_{1)}

Z*π*

0 e

−2x*θ*_{sin}*λ _{θ}*

_{d}

_{θ}_{,}

_{x}

∈R.

Therefore

(1.20) (R*λ*Kz,*λ*)(x)=

(z)−ix− ³

*λ*+2

2

´

p

2*λ*+1_{(}_{λ}_{+}_{1)}_{c}_{λ}_{(x)}, x∈
R.

**Example 1.3.** By (1.18) we have for each h>0 that

¡

R*λ*Dh,*λϕ*

¢

(x)_{=}_{p} 1

2*λ*+1_{(}_{λ}_{+}_{1)}_{π}_{c}_{λ}_{(x)}

Z

Π

(z)−ix−³*λ*+22

´

h*λ*+22* _{ϕ}*(hz)d

_{µ}*λ*(z)

= h

ix

p

2*λ*_{+}1_{(}_{λ}_{+}_{1)}_{π}_{c}

*λ*(x)

Z

Π

(z)−ix−³*λ*+22

´

*ϕ*(z)d*µλ*(z)

=hix(R*λϕ*)(x)=(MEihR*λϕ*)(x).
This clearly forces

(1.21) R* _{λ}*D

_{h}

_{,}

*R∗*

_{λ}*λ*=MEh, ∀h∈R+.

**1.2**

**Fock spaces**

In this section we recall some elementary results about Fock spaces. First, we consider the entire functions on the n-dimensional complex plane and some basic properties of them. For more detail see for example [41,53]

Let Cn be the n-dimensional complex plane. The addition, scalar multiplication and conjugation are defined onCncomponentwise. We will use the following standard notation: z=x+i y=(z1,z2,...,zn)∈Cn;

z_{·}w_{=} Xn

k_{=}1

zkwk; z2=z·z= n

X

k_{=}1

z2_{k}; _{|}z_{|}2_{=}z_{·}z_{=}Xn

k_{=}1|
zk|2.

Consider the following partial differential operators

(1.22) *∂*

*∂*zj =

1 2

µ

*∂*
*∂*xj−i

*∂*
*∂*yj

¶

; *∂*

*∂*zj =

1 2

µ

*∂*
*∂*xj+i

*∂*
*∂*yj

**Definition 1.1.** LetO_{⊂}Cn be a open set. A function f:O_{→}Cis calledholomorphic(on
O) if f _{∈}C1(O) and satisfies the system of partial differential equations

(1.23) *∂*f

*∂*zj

(z)_{=}0, for1_{≤} j_{≤}nandz_{∈}Cn.

We denote byH(O)the space of holomorphic functionsonO.

The following proposition is a generalization of Theorem 2.2 in [45] and was proved
by Miroslav Engliš, [16, Proposition 1] forO_{(}Cn, and by Folland [21, Proposition 1.69]
forO_{=}Cn.

**Proposition 1.2.** Let O be a domain in Cn, O _{=}{z_{∈}Cn: z_{∈}O}, and F be an analytic
function onO_{×}O such that

F(z,z)_{=}0, z∈O.
ThenF vanishes identically onO_{×}O.

A function f is said to beentirewhen it is analytic on the whole complex planeCn.
i.e., f _{∈}H(Cn). Forn_{=}1, it is well known that a function f is entire if and only if f has
a power series expansion

f(z)_{=} X∞

n_{=}0
anzn

with infinite radius of convergence. Also, it is well known that every bounded entire function is a constant function (Liouville’s Theorem) which is a consequence of the most fundamental result in complex analysis, namely, the identity theorem.

**Proposition 1.3**(identity theorem). Suppose that f is an entire function. If there is a
pointz_{∈}Csuch that f(n)(z)_{=}0for alln_{=}0,1,2,..., then f_{≡}0onC.

Amulti-index*α*=(*α*1,...,*α*n) is an element ofZ_{+}n. Next, we fix the standard

multi-index notation:

z*α*

=z*α*_{1}1z*α*_{2}2···z*α*nn; z*α*=z1*α*1z2*α*2···zn*α*n;

µ

*∂*
*∂*z

¶*α*

=

µ

*∂*
*∂*z1

¶*α*1
···

µ

*∂*
*∂*zn

¶*α*n
;

µ

*∂*
*∂*z

¶*α*

=

µ

*∂*
*∂*z1

¶*α*1
···

µ

*∂*
*∂*zn

¶*α*n
;

*α*!=*α*1!···*α*n!; |*α*| =
n

X

j=1

*α*j;

*∂α*f _{=} *∂*

|*α*|

*∂*z*α*_{1}1*∂*z*α*_{2}2···*∂*z*α*nn

; *∂α*f _{=} *∂*

|*α*|

1.2. FOCK SPACES

The lack of bounded entire functions is one of the key differences between the theory
of Fock spaces and the more classical theories of Hardy and Bergman spaces. Next, we
introduce the Fock spaces and give some their basic properties. For any *ς*>0 let us

denote by gn,*ς* the normed Gaussian measure onCnwith respect to the density

(1.24) dg_{n}_{,}* _{ς}*(z)

_{=}³

*ς*

*π*

´n

e−*ς*|z|2_{d}_{µ}_{n}_{(z),}

where*µ*nis the usual Lebesgue measure onL2(R2n). Observe thatdgn,*ς*is a probability

measure, in effect

gn,*ς*(Cn)=

Z

Cn

dgn,*ς*(z)=

³_{ς}

*π*

´n n_{Y}

j_{=}1

Z2*π*

0

Z_{+∞}

0 e
−*ς*r2_{j}_{r}

jdrjd*θ*j=(2*ς*)n
n

Y

j_{=}1

Z_{+∞}

0 e
−*ς*r2_{j}_{r}

jdrj=1.

**Definition 1.2.** TheFock space(also known as the Segal–Bargmann space, see [2,8,19,
44])F2

*ς*(Cn) consists of all entire functions that are square integrable onCnwith respect

to the Gaussian measure (1.24).

Let {e* _{α}*}

*α*

_{∈}Zn

+ be the system of functions inF

2

*ς*(Cn) given by the rule

(1.25) e*α*(z)=

s

*ς*|*α*|

*α*! z
*α*_{,}

*α*∈Zn
+.

By the integral formula of the Gamma function one can easily check thatz*α*_{,}_{z}*β*®

=*δα*,*β*.

Since every function f ∈F2

*ς*(Cn) has an expansion in Taylor series

f(z)_{=} X

*α*∈Zn
+

*∂α*f(0)
*α*! z

*α*_{,} _{z}

∈Cn,

it is easy to see that span©e*α*:*α*∈Zn_{+}

ª

is a dense subset of F2

*ς*(Cn). Thus the system

{e*α*}*α*∈Zn

+ form an orthonormal basis forF

2

*ς*(Cn).

**Proposition 1.4.** The Fock spaceF2

*ς*(Cn)is a closed subspace of L2(Cn,dgn,*ς*).

For any fixed z_{∈}Cn, the evaluation functional_{ψ}_{z}(f)_{=} f(z) is linear and bounded.
Thus, by the Riesz representation theorem there exists a unique elementkz,*ς*∈F*ς*2(Cn)

such that*ψ*z= 〈·,kz,*α*〉; that is

f(z)_{=}

Z

Cn

f(w)kz,*ς*(w)dgn,*ς*(w).

The functionkz,*ς*is given by the formula

Since the Fock spaceF2

*ς*(Cn) is a closed subspace ofL2(Cn,dgn,*ς*), there exists the

unique orthogonal projection Pn,*ς* from L2(Cn,dgn,*ς*) onto F*ς*2(Cn). This projection is

called theBargmann projectionand has the integral representation

(1.27) (P_{n}_{,}* _{ς}*f)(z)

_{=}Z

Cn

f(w)e*ς*wz_{dg}

n,*ς*(w), f∈L2(Cn,dgn,*ς*).

**1.3**

**Toeplitz operators**

Next, we introduce the Toeplitz operators acting on either the Bergman spaces or the Fock spaces and give some their basic properties.

LetHL_{2}(O) be the closed subspace ofL_{2}(O) consisting of holomorphic functions on
the non-empty open setO_{⊆}C, with reproducing kernelK and letP:L_{2}(O)_{→}HL_{2}(O)
be the orthogonal projection from L2(O) ontoHL_{2}(O) given in terms of the reproducing
kernel K as

(P f)(z)_{=}

Z

O

f(w)K(z,w)dmu(w), f ∈L2(O).

Now, suppose that*φ*_{∈}L_{∞}(O), define a linear operatorT* _{φ}*:HL

_{2}(O)

_{−→}HL

_{2}(O) by

T*φ*f =P(*φ*f), f ∈HL2(O).

This is called theToeplitz operator, with defining symbol*φ*, acting onHL_{2}(O). So, the
Toeplitz operator is one of the form “multiply then project”, that is, multiply by*φ*and

then project back into the holomorphic subspace. The following proposition summarizes some the most important properties of Toeplitz operators.

**Proposition 1.5.** [50, Theorem 2.81] Let*α*,*β*∈C, and f,g_{∈}L

∞(O), then
(a)the operatorTf is bounded onHL2(O)andkTfk ≤ kfk_{∞},

(b)T*α*f+*β*g=*α*Tf +*β*Tg,

(c)T∗

f =Tf.

If*φ*is an unbounded function, then we can still define the Toeplitz operatorT*φ* in

the same way, except thatT*φ* may be undounded.

**Example 1.4.** (i). If HL_{2}(O)_{=}A2

*λ*(Π), then the reproducing kernelK is given in (1.3)

and P is the Bergman projection fromL2(Π,d*νλ*) ontoA* _{λ}*2(Π). Thus the Toeplitz operator

T*ϕ*with defining symbol*ϕ*∈L∞(Π) acting onA*λ*2(Π) has the integral form

(T* _{ϕ}*f)(z)

_{=}i

*λ*

_{+}2Z

Π

*ϕ*(w)f(w)

1.4. BEREZIN TRANSFORM

(ii). IfHL_{2}(O)_{=}F2

*ς*(Cn), then the reproducing kernelK is given in (1.26) andP is the

Bargmann projection from L2(Cn,dgn,*ς*) ontoF*ς*2(Cn). Thus, the Toeplitz operator T*φ*

with defining symbol*φ*∈L∞(Cn) acting onF*ς*2(Cn) has the following integral form

(T*φ*f)(z)=

Z

Cn

*φ*(w)f(w)e*ς*wzdgn,*ς*(w), z∈Cn.

The next theorem shows that there is a one-to-one-correspondence between the Toeplitz operators and their bounded defining symbols.

**Proposition 1.6.** Let*φ*_{∈}L_{∞}(O). ThenT* _{φ}*is zero if and only if

_{φ}_{≡}0almost everywhere inO.

The corresponding result for Toeplitz operators acting on Bergman spaces over the unit disk is well known (see for example [50, Theorem 2.8.2]). To extend it to the upper half-plane case, we introduce the Caley transform

Ψ:Π_{−→}D, w_{7→} w−i
w_{+}i,
the corresponding unitary operatorU:A2

*λ*(D)−→A*λ*2(Π) given by the rule
*ϕ*7→(*ϕ*◦Ψ)Ψ′

and observe thatU∗_{T}

*ϕ*U=T*ϕ*_{◦}Ψ−1.

On the other hand, the corresponding result (Proposition1.6) for Toeplitz operators
acting on Fock spaces was proved by Berger and Coburn in [9, Theorem 4]. There are
other classes of functions such that Proposition 1.6is true. For example, Folland [20,
P. 140] extended Proposition1.6for Toeplitz operatorsTaacting on Fock spaceF*ς*2(Cn)

whose defining symbolsabelong to the class of unbounded functions which satisfy the inequality

(1.28) _{|}a(z)_{| ≤}const e*δ*|z|2_{, for some}_{δ}_{<}_{1.}

**1.4**

**Berezin transform**

**Definition 1.3.** HL_{2}(O) be the closed subspace ofL_{2}(O,d* _{µ}*) consisting of holomorphic
functions on the non-empty open setO

_{⊆}C(orCn), with reproducing kernel K. LetSbe a bounded linear operator onHL

_{2}(O), theBerezin transformofSis defined by

(1.29) S(z)e _{=}〈SKz,Kz〉

〈Kz,Kz〉 , z∈

O.

In particular, the Berezin transform of a function*φ*_{∈}L_{∞}(O) (denoted by* _{φ}*e) is definded
as the Berezin transform of the Toeplitz operatorT

*φ*. Likewise,

*φ*e=Tf

*φ*.

For each bounded operator, the Berezin transform is a bounded real-analytic function
on a domain ofCn. Indeed, observe that ifS_{∈}B(HL2(O)), thenSe_{∈}L

∞(O), and by the Cauchy-Schwarz inequality it satisfies the relation

(1.30) _{k}Sek∞≤ kSk.

**Example 1.5.** The Berezin transform of an operator V ∈B(A2

*λ*(Π)) is the function

e

V:Π_{→}Cdefined by

(1.31) Ve(z) :_{=}(2Imz)*λ*+2_{i}*λ*+2Z

Π

(V K_{z}_{,}* _{λ}*)(w)

(w_{−}z)*λ*+2 d*νλ*(w), z∈Π,

where Kz,*λ* is the Bergman kernel (1.3). Thus forV =T*ϕ*with*ϕ*∈L∞(Π) one gets for
every z_{∈}Πthat

e

*ϕ*(z)_{=}(2Imz)*λ*+2

Z

Π

*ϕ*(w)

|z−w|2(*λ*+2)d*νλ*(w).

**Example 1.6.** The Berezin transform of a bounded operatorSon the Fock spaceF2

*ς*(Cn)

is the functionSe:Cn_{→}Cdefined by

(1.32) S(z)e _{=}e−*ς*|z|2Z

Cn

(Sk_{z})(w)e*ς*wz_{dg}

n,*ς*(w), z∈Cn.

Thus, forS_{=}T*φ* with*φ*∈L∞(Cn) one has for every z∈Cn that

e

*φ*(z)=e−*ς*|z|2

Z

Cn

*φ*(w)e*ς*(zw+zw)dgn,*ς*(w)=

Z

Cn

*φ*(w)e−*ς*(z−w)·(z−w)d*µ*n(w)

=

Z

Cn

*φ*(w)e−*ς*|z−w|2d*µ*n(w).

1.5. APPROXIMATION OF UNIFORMLY CONTINUOUS FUNCTIONS BY CONVOLUTIONS

Proof. IfS=0, thenS(w)e =0 for all w∈O trivially. Conversely, suppose thatS(w)e _{=}0
for allw∈O. DefineF:O_{×}O_{→}CbyF(z,w)_{= 〈}SK_{w},K_{z}_{〉}, z,w_{∈}O. The functionF has
an integral representation as

F(z,w)_{=}

Z

O

(SKw)(*ζ*)Kz(*ζ*)d*µ*(*ζ*)=

Z

O

(SKw)(*ζ*)K*ζ*(z)d*µ*(*ζ*), z,w∈O.

From this observe that F is analytic in the first variable, that is, for fixed w_{∈}O the
functionz7→F(z,w) is analytic onO. Furthermore, using that

F(z,w)_{=}Kz,S∗Kz®= 〈S∗Kz,Kw〉,

we see that F is analytic in the second variable. It follows from the fact for an analytic
function gonO the functionw_{7→}g(w) is analityc onO. On the other hand,F(z,z)_{=}0 for
all z_{∈}O, thus by Lemma1.2one gets that F(z,w)_{=}0 onO_{×}O. Equivalently, (SK_{w})(z)_{=}

〈SKw,Kz〉 =0 for all z,w∈O, and hence SKw=0 for w∈O. Now, given an arbitrary

f _{∈}H(O) andw_{∈}O we have

(S∗_{f}_{)(w)}_{=}_{S}∗_{f}_{,}_{K}_{w}®_{= 〈}_{f}_{,}_{SK}_{w}_{〉 =}_{0.}

ThusS∗_{=}_{0, and therefore}_{S}_{=}_{0.} _{}

**1.5**

**Approximation of uniformly continuous**

**functions by convolutions**

In this section we recall a technique that permits us to approximate bounded uniformly continuous functions by convolutions with a fixed kernel satisfying Wiener’s condition. We could not find Proposition1.9in the literature, but it is based on well-known ideas and can be considered as a variation of the Wiener’s Tauberian theorem. The constructions of this section can be generalized to abelian locally compact groups.

Let f:Rn_{→}C. Recall that the functionΩ_{f} : [0,_{+∞}]_{−→}[0,_{+∞}] defined by

(1.33) Ω_{f}(* _{δ}*) :

_{=}sup©

_{|}f(x)

_{−}f(y)

_{|}: x,y

_{∈}R, d(x,y)

_{≤}

*ª.*

_{δ}is the modulus of continuity of f with respect to the usual metric d on R (or simply
modulus of continuity). Hence, if f:Rn_{−→}Cis a bounded function, then f is said to be
uniformly continuouswhenever lim

*δ*→0

The convolution of two complex-valued functions onRn is itself a complex-valued function onRndefined by:

(1.34) (f_{∗}g)(x)_{=}

Z

Rn

f(y)g(x_{−}y)d y, x∈Rn.

It is well known that the convolution of f and gexists if f _{∈}L1(Rn) and g_{∈}L_{p}(Rn) where
1_{≤}p_{≤ +∞}. In this case f_{∗}g_{∈}Lp(Rn) and

kf∗gkp≤ kfk1kgkp.

**Proposition 1.8.** L1(Rn)∗L∞(Rn)⊂Cb,u(Rn).

Proof. Let f _{∈}L1(Rn) and g_{∈}L_{∞}(Rn). Then forh_{∈}Rnone obtains

|(f∗g)(x+h)−(f∗g)(x)| ≤

Z

Rn|

f(x_{−}h_{−}t)_{−}f(x_{−}t)_{| |}g(t)_{|}dt

≤ kgk∞

Z

Rn|

f(x_{−}h_{−}t)_{−}f(x_{−}t)_{|}dt

y_{=}x_{−}t

= kgk∞

Z

Rn|

f(y_{−}h)_{−}f(y)_{|}d y

= kgk∞kLhf−fk1, x∈Rn.

Here*τ*hdenotes the translation operator byh,(see (A.18)) . Thus, since for each f∈L1(Rn)

the mapping h_{7→}*τ*hf fromRn intoL1(Rn) is uniformly continuous (CorollaryA.1), we

have that f_{∗}gbelongs toCb,u(Rn).

In Proposition1.8we can change by an equality. i.e.,L1(Rn)_{∗}L_{∞}(Rn)_{=}C_{b}_{,}_{u}(Rn) (see
[33, p. 283, 32-45]). Therefore, any function in {k∗f: f ∈L∞(Rn)} belongs toCb,u(Rn).

**Proposition 1.9.** Ifk_{∈}L1(Rn)andk(t)ˆ _{6=}0for each t_{∈}Rn, then{k_{∗}f: f_{∈}L_{∞}(Rn)}is a
dense subset of Cb,u(Rn).

Proof. By Proposition1.8every function in {k∗f: f ∈L∞(R)} belongs to Cb,u(R). Next,
the density is proved by means of Wiener’s Division Lemma and LemmaA.1as follows:
Let (hn)n_{∈}N be a Dirac sequence such that the functions ˆhnhave compact supports. For

example, (hn)n∈N can be defined by (A.22). Since bk(t)6=0 for each t∈R, by Wiener’s

Division Lemma (TheoremA.4) for everyn_{∈}Nthere existsq_{n}_{∈}L_{1}(R) such thath_{n}_{=}
k_{∗}qn. Now, given*ψ*∈Cb,u(R), we construct a sequence (wn)n∈Nby the rulew_{n}=q_{n}∗*ψ*.

Thenwn∈L∞ and the sequence (k∗wn)n∈N takes values in the set {k∗f: f ∈L_{∞}(R)}.

Finally, applying the identities

k∗wn=k∗qn∗*ψ*=hn∗*ψ*

1.6. SYMPLECTIC SPACES AND LAGRANGIAN PLANES

**1.6**

**Symplectic spaces and Lagrangian planes**

This section contains a brief summary of basic concepts of the theory of the symplectic group and related topics.

**Definition 1.4**(symplectic group). A real 2n_{×}2nmatrixBis said to be symplectic if
it satisfies the conditions:

(1.35) BTJB_{=}BJBT_{=}J,

where J is the “ standard symplectic matrix” given by

(1.36) J=

Ã

0 I_{n}

−In 0

!

.

Here 0 and Inare then×nzero and identity matrices. The set of all symplectic matrices

is denoted by Sp(2n,R).

A symplectic matrix is invertible and has determinant 1. In fact, if B_{∈}Sp(2n,R) ,
thenB−1_{∈}_{Sp(2n,}R). It is well-known that Sp(2n,R) form a group, and is calledthe (real)
symplectic group.

Recall that askew-symmetric bilinear form*ω*is a bilinear form such that

*ω*(z,w)_{= −}*ω*(w,z)

for allz,w_{∈}R2n. Notice that if* _{ω}*is a skew-symmetric bilinear form, then all vectors z
are isotropic. i.e., for every z

_{∈}R2none gets that

*ω*(z,z)_{=}0.

**Definition 1.5.** A bilinear form onR2nis called asymplectic formif it is a non-degenerate
skew-symmetric bilinear form.

The special skew-symmetric bilinear form*ω*0onR2ndefined by

(1.37) *ω*0(z,w)=y·x′−y′·x,

for z=(x,y) andw=(x′,y′) is symplectic; it is calledthe standard symplectic formofR2n.
The standard symplectic form *ω*0can be re-written in a convenient way using the
symplectic standard matrix J given in (1.36):

Let*ψ*:R2n_{→}R2n be a linear mapping. The condition

*ω*0(*ψ*(z),*ψ*(w))=*ω*0(z,w),
is equivalent toBT_{JB}

=J, whereBis the matrix of*ψ*in the canonical basis ofR2n, that
is,B_{∈}Sp(2n,R). We can thus redefine the symplectic group by saying that it is the group
of all linear automorphism ofR2nwhich preserve the standard symplectic form_{ω}_{0}.

**Definition 1.6** (Lagrangian plane). A n-dimensional linear subspace L of R2n is
said to be aLagrangian planeof the symplectic space (R2n,_{ω}_{0}) if_{ω}_{0}(z,w)_{=}0, for every
z,w∈L. The set of all Lagrangian planes in (R2n,_{ω}_{0}) is denoted by Lag(2n,R).

**Example 1.7.** The simplest examples of Lagrangian planes of (R2n,_{ω}_{0}) are both
coordi-nates planes:L_{x}_{=}Rn_{×}{0} andL_{y}_{=}{0}_{×}Rn, and so is the diagonal∆_{=}{(x,x): x_{∈}Rn}.

LetM(n,C) be the algebra of complex matrices of dimensionn. Denote by U(n,C) the
unitary subgroup ofM(n,C) consisting of all complex matrices U of n_{×}nsuch that

U∗_{U}_{=}_{UU}∗_{=}_{I}

n,

whereU∗_{=}_{U}T_{. Define the mapping}_{ι}_{:}M(n,C)_{→}M(n,C) by the rule

(1.39) *ι*(C)_{=}

Ã

A _{−}B

B A

!

,

where the matrixC_{∈}M(n,C) is the formC_{=}A_{+}iBwith A andBreal matrices. It is
easy to see from the definition of*ι*that it is an injective mapping.

**Lemma 1.4.** The mapping*ι*given in(1.39)satisfies the following properties:

• For everyU,V,_{∈}M(n,C)and* _{α}*,

_{β}_{∈}Cthe mapping

*is lineal. That is,*

_{ι}*(*

_{ι}*U*

_{α}_{+}

*V)*

_{β}_{=}

*α ι*(U)_{+}*β ι*(V).

• For everyU,V,_{∈}M(n,C)the mapping* _{ι}*is multiplicative. That is,

*(UV)*

_{ι}_{=}

*(U)*

_{ι}*(V).*

_{ι}• For everyU_{∈}M(n,C)one gets that* _{ι}*(U∗)

_{=}

*(U)T. Define the set U(2n,R) of real matricesn*

_{ι}_{×}nby the rule

U(2n,R)_{=}* _{ι}*(U(n,C)).
(1.40)

1.6. SYMPLECTIC SPACES AND LAGRANGIAN PLANES

Recall that ifC=A+iB∈U(n,C), thenCC∗_{=}I_{n}. Equivalently

In=(A+iB)(AT−iBT)=A AT+BBT+i(BAT−ABT).

That is, A AT+BBT=In andBAT=ABT. Thus by (1.39) one can see the symplectic

rotations U(2n,R) as:

(1.41) U(2n,R)_{=}

(Ã

A −B

B A

!

: A AT+BBT=In and ABT=BAT, A,B∈M(n,R)

)

**Proposition 1.10.**

U(2n,R)_{=}Sp(2n,R)_{∩}O(2n,R).

Proof. Let U_{∈}U(2n,R). Then by (1.41) there are A,B_{∈}M(n,R) with A AT_{+}BBT_{=}I_{n}
and ABT_{=}BAT such that

U_{=}

Ã

A −B

B A

!

.

Therefore

UUT=

Ã

A _{−}B

B A

!Ã

AT BT

−BT AT

!

=

Ã

A AT_{+}BBT ABT_{−}BAT
BAT_{−}ABT A AT_{+}BBT

!

=

Ã

I_{n} 0
0 I_{n}

!

=I2n.

Analogously,UTU=I2n. This implies thatU∈O(2n,R). On the other hand,

U JUT=

Ã

B A

−A B

!Ã

AT BT

−BT AT

!

=

Ã

BAT_{−}ABT A AT_{+}BBT

−(A AT+BBT) BAT−ABT

!

=J.

In the same way is showed thatUTJU=J. ThusU∈Sp(2n,R) and hence

U(2n,R)_{⊂}Sp(2n,R)_{∩}O(2n,R).

Now, ifV∈Sp(2n,R)_{∩}O(2n,R), thenV JVT_{=}J andV VT_{=}VTV_{=}I_{2}_{n}. We thus get that
V J=JV, from this equality it is easy to see that the matrixV belongs to U(2n,R). _{}
By (1.35) and Proposition1.10a 2n_{×}2nmatrixBbelongs to U(2n,R) if it commutes
with the standard symplectic matrix J, that is:

(1.42) BJ=JB.

### C

H

A

P

T

E

R

## 2

**C*-**

**ALGEBRA OF ANGULAR**

**T**

**OEPLITZ OPERATORS**

**ON WEIGHTED**

**B**

**ERMAN SPACES OVER THE UPPER**

**HALF**

**–**

**PLANE**

In this chapter we show that the uniform closure of the set of all Toeplitz operators
acting on the weighted Bergman space over the upper half-plane whoseL_{∞}-symbols are
angular coincides with the C*-algebra generated by the above Toeplitz operators and is
isometrically isomorphic to the C*-algebra VSO(R) of bounded functions that are very
slowly oscillating on the real line in the sense that they are uniformly continuous with
respect to the arcsinh-metric*ρ*(x,y)_{= |}arcsinhx_{−}arcsinhy_{|}on the real line.

**2.1**

**Angular Toeplitz operators**

In this section we characterize the angular Toeplitz operators acting on weighted Bergman spaces. The characterization is based on the notion of angular operator. So, we will first introduce the angular operators and study their basic properties, including a simple criterion for an operator to be angular.

LetB(A2

*λ*(Π)) be the algebra of all linear bounded operators acting on the Bergman

spaceA2

*λ*(Π). Givenh∈R+, letD*λ*,h∈B(A

2

*λ*(Π)) be thedilationoperator defined by

(2.1) D*λ*,hf(z)=h

*λ*+2
2 f(hz).

Next, we introduce the angular operators acting on the Bargman spaceA2