### Centro de Investigación y de Estudios Avanzados

### del Instituto Politécnico Nacional

Unidad Zacatenco

Departamento de Matemáticas

**On**

*C*

∗_{-algebras of Toeplitz Operators}

_{-algebras of Toeplitz Operators}

**on the Harmonic Bergman Space**

### Tesis que presenta

**Ma. del Carmen Lozano Arizmendi**

### para obtener el grado de

### Doctora en Ciencias

### en la Especialidad en Matemáticas

### Directora de tesis: Dra. Maribel Loaiza Leyva

**Contents**

**Introduction** **iii**

**1 Preliminaries** **1**

1.1 Local Principle . . . 1

1.1.1 Douglas-Varela local principle . . . 4

1.2 Bergman spaces on the unit disk . . . 4

1.3 Commutative Toeplitz algebras and hyperbolic geometry . . . 4

1.3.1 Commutative Toeplitz operators on the Weighted Bergman space . . . 10

1.4 The Harmonic Bergman space . . . 12

1.4.1 The Harmonic Bergman space on the unit disk . . . 12

1.4.2 The Harmonic Bergman space on the upper half-plane . . . 15

**2 On Toeplitz operators on harmonic Bergman spaces** **17**
2.1 Algebra generated by Toeplitz operators with radial symbols . . . 18

2.2 Toeplitz operators on the harmonic Bergman space on the upper half-plane . . . 21

2.2.1 Toeplitz operators with homogeneous symbols . . . 21

2.2.2 The algebra_{T}(_{A}_{∞}) . . . 25

2.2.3 Toeplitz operators with vertical symbols . . . 35

2.3 Toeplitz Algebras on the disk: parabolic and hyperbolic case . . . 38

2.3.1 Parabolic case: The algebra_{T}0 . . . 39

2.3.2 Hyperbolic case: The algebra_{T}1 . . . 40

**3 On Toeplitz operators on the weighted Harmonic Bergman space** **43**
3.1 The weighted Bergman space on the upper half-plane . . . 44

3.2 Toeplitz operators on the harmonic Bergman space on the upper half-plane . . . 45

3.2.1 Toeplitz operators with angular symbols . . . 46

3.2.2 Toeplitz operators with vertical symbols . . . 55

**Bibliography** **62**

**Introduction**

During the last decades Toeplitz operator theory and its applications in various ﬁelds, including
math-ematical physics, complex analysis, theory of normed algebras, dynamical systems, random matrix theory
and operator theory have been a quickly developing branch of mathematics. Besides, diﬀerential
oper-ators, Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and
they are a fascinating example of the fruitful interplay between topics such as operator theory, function
theory, and the theory of Banach algebras and *C*∗-algebras.

There has been growing interest in the study of Toeplitz operators on diﬀerent functions spaces. One
of the most popular setting for this research has been the Bergman space on the unit disk and on the
upper half-plane. Recent work on this area shows that there exists a close connection among the Bergman
space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. More
precisely, commutative *C*∗_{-algebras generated by Toeplitz operators on the Bergman space on the unit}
disk have been completely classiﬁed in terms of the symbols of generating Toeplitz operators.

In this thesis we study *C*∗_{-algebras generated by Toeplitz operators acting on harmonic Bergman}
spaces. This work is comprised of three parts and is organized as follows.

In Chapter 1, we begin by providing the reader with some general background material about Toeplitz
operators acting on the Bergman space on the unit disk and on the upper half-plane. We also include
an important tool to describe *C*∗-algebras, the so-called Douglas-Varela local principle. It was recently
shown, in the Bergman space on the unit disk, that a *C*∗_{-algebra generated by Toeplitz operators is}
commutative if and only if there is a pencil of hyperbolic geodesics such that the symbols of the Toeplitz
operators are constant on the cycles of this pencil. All commutative *C*∗-algebras of Toeplitz
opera-tors are classiﬁed by three types of pencils of hyperbolic geodesics on the unit disk: parabolic, elliptic
and hyperbolic. Finally, we introduce the main object treated in this thesis: the harmonic Bergman space.

In Chapter 2, a natural question appears: any class of symbols that generates a commutative *C*∗_{}
-algebra of Toeplitz operators on the Bergman space also generates a commutative*C*∗-algebra of Toeplitz
operators on the harmonic Bergman space? To answer this question we study Toeplitz operators with
radial symbols and we prove that the *C*∗_{-algebra generated by these operators is commutative. In fact,}
it is isomorphic to the algebra of all slowly oscillating sequences. To study Toeplitz operators whose
symbols are invariant under parabolic or hyperbolic Möbius transformations we pass from the unit disk
onto the upper half-plane. As in the Bergman space setting two model cases appear here: the*C*∗-algebra
generated Toeplitz operators whose symbols depend only on the imaginary part of the variable and the
*C*∗-algebra generated by Toeplitz operators whose symbols are bounded measurable homogeneous
func-tions of order zero. The ﬁrst algebra is commutative and the second one is not. This fact implies that in
the harmonic Bergman space setting on the unit disk, the Calkin algebra generated by Toeplitz operators

whose symbols are invariant under parabolic transformations is commutative and that the Calkin algebra
of the *C*∗-algebra generated by Toeplitz operators whose symbols are invariant under hyperbolic
trans-formations is not. As a result of the description of the last algebra we ﬁnd several essential diﬀerences
between Toeplitz operators acting on the Bergman space and Toeplitz operators acting on the harmonic
Bergman space. For example, we prove that the essential spectrum of a Toeplitz operator with piecewise
constant symbol depends on the angles of discontinuities and the index of a Fredholm Toeplitz operator
with such kind of symbol is always equal to zero. The results of this chapter were previously published
as: Loaiza M., Lozano C.,*On* *C*∗*-algebras of Toeplitz operators on the harmonic Bergman space.* Integr.
Equ. Oper. Theory**76**, 2013, 105-130.

Finally, in Chapter 3 we study Toeplitz operators on the weighted harmonic Bergman space on the
upper half-plane. Two classes of symbols are considered here: symbols that depend only on the vertical
variable and symbols that depend only on the angular variable. For the ﬁrst case, we prove that Toeplitz
operators with such kind of symbols generate a commutative *C*∗-algebra in every weighted harmonic
Bergman space. This algebra is isomorphic to the algebra of all very slowly oscillating functions. On the
other hand, Toeplitz operators whose symbols depend only on the angular variable generate a non
com-mutative*C*∗-algebra which is isomorphic to the*C*∗-algebra of all 2_{×}2 matrix-valued continuous functions
(*fij*(*t*)) deﬁned on Rand such that they satisfy*f*12(±∞) =*f*21(±∞) = 0 and *f*11(±∞) =*f*22(∓∞).

**Chapter 1**

**Preliminaries**

The Bergman space is deﬁned as the space of analytic functions that are square integrable with respect to the area measure. The commuting problem of Toeplitz operators acting on it, has been studied by several authors in the last decades, for example [8], [9]. This problem is stated as follows: If two Toeplitz operators commute, what can we say about their symbols?

Another interesting problem in operator theory is the description of*C*∗-algebras generated by Toeplitz
operators. This chapter mainly concerns the description of commutative*C*∗-algebras generated by Toeplitz
operators acting on Bergman spaces. In the ﬁrst section, we introduce the Douglas-Varela local principle:
an important tool that we shall use to describe these algebras. In Section 1.2 we deﬁne the Bergman space
on the unit disk and on the upper half-plane. In Section 1.3 we give some background about commutative
*C*∗_{-algebras generated by Toeplitz operators in Bergman spaces. They have been completely described}
and are close related to geometric properties of the domain. Motivated by the results concerning to
Toeplitz operators obtained for the Bergman space, in Section 1.4, we introduce the harmonic Bergman
space and we study Toeplitz operators acting on it. The harmonic Bergman space is represented in terms
of the Bergman and the anti-Bergman spaces. This fact implies that every Toeplitz operator is represented
as a 2_{×}2 matrix operator.

**1.1**

**Local Principle**

There are several techniques of localization in operator theory. In the pioneer work [31] I. Simonenko
introduced the notion of locally equivalent operators and developed a localization theory. This section
is devoted to the Douglas-Varela local principle, which gives the global description of the algebra under
study in terms of continuous sections of a certain canonically *C*∗-bundle (see, for example [41]).

The triple*ξ* = (*p, E, T*), where*E* and *T* are topological spaces, and*p*:*E* _{→}*T* is a surjective map, is
called a *bundle*. The set *T* is called the *base* of the bundle, and *ξ*(*t*) =*p*−1_{(}_{t}_{) is called the} _{fiber over the}

*point* *t*_{∈}*T*. Let*V* be an open set in*T*. A function*σ*:*V* _{→}*E* is called a*local section* of the bundle*ξ* if
*p*(*σ*(*t*)) =*t*for all *t*_{∈}*V*. If*V* =*T* the section is called *global* or just section. Denote by Γ(*ξ*) the set of
all continuous sections of the bundle*ξ*.

Let *E*_{∨}*E* ={(*x, y*) ∈*E*_{×}*E* :*p*(*x*) =*p*(*y*)}*.* The bundle *ξ* = (*p, E, T*) is called a *C*∗_{-bundle if each}
ﬁber *ξ*(*t*) has a structure of a*C*∗-algebra, and

i. the functions

(*x, y*)_{7→}*x*+*y*:*E*_{∨}*E* _{→}*E,*
(*x, y*)_{7→}*x*_{·}*y*:*E*_{∨}*E* _{→}*E,*
(*α, x*)7→*αx*:C_{×}_{E}_{→}_{E,}

*x*_{7→}*x*∗:*E*_{→}*E*
are continuous;

ii. the subsets *UV*(*σ, ε*) ={*x*∈*E*:*p*(*x*)∈*V* and k*x*−*σ*(*p*(*x*))k*< ε*}*,*where *V* is an open subset in*T*,

*σ* is a continuous section of*ξ* over*V*,*ε >*0, form a basis of open sets in the space *E*.

Given a*C*∗_{-bundle}_{ξ}_{= (}_{p, E, T}_{), there is the canonically deﬁned}* _{C}*∗

_{-algebra associated with}

_{ξ}_{. Namely,}the set of all bounded continuous sections

*σ*of

*ξ*= (

*p, E, T*) with componentwise operations and the following norm

k*σ*_{k}= sup

*t*∈*T* k

*σ*(*t*)k

is obviously a*C*∗-algebra. We shall call this algebra the*C*∗-algebra deﬁned by the*C*∗-bundle*ξ* = (*p, E, T*),
and shall denote it by Γ*b*_{(}_{ξ}_{).}

**Proposition 1.1.1** ([41])**.** *1. The function*_{k · k}:*t*_{7→ k}*σ*(*t*)_{k} *is upper semi-continuous.*

*2. The algebra* Γ*b*_{(}_{ξ}_{)} _{is a}_{C}b_{(}_{T}_{)}_{-module, where}_{C}b_{(}_{T}_{)} _{is the}* _{C}*∗

_{-algebra of all bounded continuous}*functions onT.*

*3. If the space* *T* *is quasi-compact, then* Γ*b*_{(}_{ξ}_{) = Γ(}_{ξ}_{)} _{and}_{C}b_{(}_{T}_{) =}_{C}_{(}_{T}_{)}_{.}

Recall that a space *T* is called *quasi-completely regular* if for each*t*0 ∈*T* and each closed set*Y*(⊂*T*)

which does not contain*t*0, there exist a continuous function *f* :*T* →[0*,*1] such that

*f*(*t*0) = 0*,* *f*|*Y* ≡1*.*

**Theorem 1.1.2** (Stone-Weierstrass)**.** *Let* *ξ* = (*p, E, T*) *be a* *C*∗*-bundle over a compact, *
*quasi-completely regular space* *T. Let* _{A} *be a closed* *C*(*T*)*-submodule of* Γ(*ξ*)*, and let for each* *t* _{∈} *T* *the set*

A(*t*) ={*a*(*t*) :*a*_{∈ A}}*be dense in the fiber* *ξ*(*t*) =*p*−1_{(}_{t}_{)}_{. Then}_{A}_{= Γ(}_{ξ}_{)}_{.}

Let _{A} be a *C*∗-algebra and let *JT* = {*J*(*t*) : *t* ∈ *T*} be a system of its closed two-sided ideals,

parameterized by points of a set *T*. For each *t* _{∈}*T* introduce the quotient algebra _{A}(*t*) = _{A}*/J*(*t*). We
shall denote by *a*(*t*) the image of an element *a*_{∈ A}in the quotient algebra_{A}(*t*). Let

*E* = G

*t*∈*T*

A(*t*)

be the disjoint union of the *C*∗-algebras_{A}(*t*). We deﬁne the action of the (additive) group _{A}on the set
*E*: each element *a* _{∈ A} generates the mapping *ga* : *E* → *E* by the rule *ga* : *x*(*t*) 7→ (*x*+*a*)(*t*)*.* Then

the orbit of each point *x*=*x*(*t*) under the action of the group _{A} coincides with the whole algebra_{A}(*t*),
and the collection of orbits is parameterized by points of *T*. The partition of *E* onto the disjoint orbits
generates the projection

Chapter 1. Preliminaries 3

We endow the sets*E* and *T* with appropriate topologies in order for the triple *ξ* = (*p, E, T*) to be a
*C*∗_{-bundle. Each element} _{a}_{∈ A} _{generates the section ˜}_{a}_{:}_{T}_{→}_{E}_{by the rule ˜}_{a}_{:}_{t}_{7→}_{a}_{(}_{t}_{). We denote by}

e

A the set of all the above sections ˜*a*. For each*ε >*0 and each ˜*a*_{∈}_{A}ewe introduce the set
*U*(˜*a, ε*) =_{{}*x*_{∈}*E* :_{k}*x*_{−}˜*a*(*p*(*x*))_{k}*< ε*_{}}*,*

and endow the set*E* with the topology which prebase consists of all the sets*U*(˜*a, ε*). We endow the base
*T* of the bundle *ξ* = (*p, E, T*) with the orbit space topology under which the projection *p* : *E* _{→} *T* is
continuous.

**Proposition 1.1.3.** *1. The mapping* *p*:*E*_{→}*T* *is open.*

*2. The topology on* *T* *coincides with the weakest topology under which all the mappings* ˜*a* _{∈} *A*e *are*
*continuous.*

*3. A prebase of the topology on* *T* *is given by the system of sets*
*V*(˜*a, ε*) =_{{}*t*_{∈}*T* :_{k}*a*˜(*t*)_{k}*< ε*_{}}*.*
Then the triple*ξ* = (*p, E, T*) is a *C*∗-bundle.

Given a*C*∗_{-algebra} _{A}_{and a system of its closed two-sided ideals} _{J}_{T}_{=}_{{}_{J}_{(}_{t}_{) :}_{t}_{∈}_{T}_{}}_{, the} * _{C}*∗

_{-bundle}

*ξ* = (*p, E, T*) described above is called the*canonicalC*∗*-bundle defined by theC*∗*-algebra*_{A}*and the system*
*of ideals* *JT*.

The next result can be treated as a non commutative generalization of the Gelfand representation of a commutative Banach algebra.

Let_{A}be a*C*∗-algebra,*JT* ={*J*(*t*) :*t*∈*T*}be a system of its closed two-sided ideals,*ξ* = (*p, E, T*) be

the canonical*C*∗_{-bundle deﬁned by}_{A}_{and}_{J}_{T}_{, Γ}*b*_{(}_{ξ}_{) be the}* _{C}*∗

_{-algebra deﬁned by the bundle}

_{ξ}_{= (}

_{p, E, T}_{).}Then the mapping

e

*π* :*a*_{∈ A 7→}˜*a*_{∈}Γ*b*_{(}_{ξ}_{)}

is a morphism of the *C*∗-algebras_{A}and Γ*b*_{(}_{ξ}_{), such that}

1. ker*π*e=_{∩}*t*∈*T* J(t),

2. Ime*π*=_{A}e_{.}

Moreover, the mapping *π*e :_{A →}_{A}eis an isometric_{∗}-isomorphism if and only if

\

*t*∈*T*

*J*(*t*) =_{{}0_{}}*.*

Having a *C*∗-algebra _{A} and system *JT* of its closed two-sided ideals (with or without) the property

T

*t*∈*TJ*(*t*) = {0} we say we shall *localize by points of the set* *T*. The elements *a*1 and *a*2 of the algebra

A are called *locally equivalent at the point* *t*_{∈} *T* (*a*1 ∼*t* *a*2) if and only if *a*1−*a*2 ∈*J*(*t*). The natural

projections *πt*:A → A(*t*) identify the elements of the algebraAlocally equivalent at the point*t*, and the

**1.1.1** **Douglas-Varela local principle**

Let _{A} be a *C*∗_{-algebra with identity} _{e}_{, and} _{Z} _{be a central commutative} * _{C}*∗

_{-subalgebra, containing}

*e*. Denote by *T* the compact set of maximal ideals of the algebra _{Z}; then, _{Z ∼}= *C*(*T*). For each point
*t*_{∈}*T* denote by *Jt* the maximal ideal of the algebra Z which corresponds to the point *t*, and denote by

*J*(*t*) =_{A ·}*Jt* the closed two sided ideal generated by*Jt*in the algebraA. Finally we introduce the system

of ideals*JT* ={*J*(*t*) :*t*∈*T*}. We have _{\}

*t*∈*T*

*J*(*t*) =_{{}0_{}}*.*

**Theorem 1.1.4** (Douglas-Varela local principle)**.** *The algebra* _{A} *is* ∗_{-isomorphic and isometric to the}

*algebra of all (global) continuous sections of the* *C*∗*-bundle, defined by the algebra* _{A} *and the system of*
*ideals* *JT* = {*J*(*t*) :*t*∈ *T*}*. Moreover, the* ∗*-bundle topology on* *T* *coincides with hull kernel topology of*

*the compact* *T.*

**1.2**

**Bergman spaces on the unit disk**

Denote by Dthe complex unit disk with the area measure* _{dA}*(

*) =*

_{z}*=*

_{dxdy, z}*+*

_{x}*As usual,*

_{iy.}*2(D)*

_{L}denote the space of all measurable and square integrable functions deﬁned inD _{and} _{A}2_{(}D_{) its subspace}

consisting of all analytic functions. This space is precisely the set of all functions*f* in*L*2(D) such that

*∂f*
*∂z* = 0*.*

The orthogonal projection from*L*2_{(}_{D}_{) onto}_{A}2_{(}_{D}_{) is called the Bergman projection, denoted here by}

*B*D. Then the Toeplitz operator*Ta* :A2(D)→ A2(D), with symbol *a*is deﬁned by

*Taf* =*B*D(*af*)*.*

If*a*is a bounded function, then*Ta*is a bounded operator onA2(D) and we havek*Ta*k ≤ k*a*k∞and*Ta*∗ =*T*¯*a*.

From now on, the letter_{K} shall denote the ideal of all compact operators on the space under
consid-eration.

The Toeplitz *C*∗-algebra over D is deﬁned as the unital * _{C}*∗-algebra

_{T}(

*(D)) :=*

_{C}*∗*

_{C}_{{}

*:*

_{T}_{a}

_{a}_{∈}

*(D)*

_{C}_{}}

generated by all Toeplitz operators with continuous symbols. In the paper [6] due to Coburn were obtained
that this algebra is irreducible and contains the whole ideal _{K}. In fact, it was proved that the Calkin
algebra _{T}(*C*(D))_{/}_{K} is isomorphic to the algebra * _{C}*(T), where T =

*D. The operator*

_{∂}*=*

_{T}*+*

_{T}_{a}*is*

_{K}Fredholm if and only if its symbol is invertible i.e.,*a*(*t*)_{6}= 0 onT, and

Ind*T* =_{−} 1

2*π*{arg*a*(*t*)}T*.*

**1.3**

**Commutative Toeplitz algebras and hyperbolic geometry**

This section concerns to the *C*∗-algebras generated by Toeplitz operators on the Bergman space. We
begin with some basic facts from hyperbolic geometry on the unit disk. A straight (geodesic or hyperbolic)
line is the part of an Euclidean circle orthogonal to T_{. There is a unique geodesic passing through any}

Chapter 1. Preliminaries 5

*z*0

Figure 1.1: Elliptic pencil _{P}(*z*0).

hyperbolic plane.

Each pair of geodesics, say *L*1 and *L*2, lie in a geometrically deﬁned object, one-parameter family P of

geodesics, which is called the*pencil* determined by*L*1 and *L*2. Each pencil has an associated familyC of

lines, called *cycles*, which are orthogonal trajectories to the geodesics forming the pencil.

**Definition 1.3.1.** The pencil_{P} determined by*L*1 and *L*2 is called:

1. *parabolic* if *L*1 and *L*2 are parallel; in this caseP is the set of all geodesics parallel to both *L*1 and

*L*2, and cycles are called *horocycles*,

2. *elliptic* if *L*1 and *L*2 are intersecting; in this case P is the set of all geodesics passing through the

common point*L*1 and *L*2;

3. *hyperbolic* if *L*1 and *L*2 are disjoint; in this case P is the set of all geodesics orthogonal to the

common orthogonal of*L*1 and *L*2, and cycles are called*hypercycles*.

Let us mention the following joint properties of_{P} and _{C}:

1. each point in the hyperbolic plane lies on exactly one cycle in _{C},

2. with possibly one exception (common point of geodesics in an elliptic pencil), each point in the
hyperbolic plane lies on exactly one geodesic in _{P};

3. all geodesics in _{P} are orthogonal to every cycle in_{C};

4. every cycle in _{C} is invariant under the reﬂection in any geodesic in_{P}.

It turns out that there is a deep connection between the structure of the commutative algebras of Toeplitz operators and the geometric properties of pencils of hyperbolic lines.

With each pencil of lines _{P} we associate the class _{A}_{P} of measurable functions (symbols) *a* = *a*(*z*)
such that are constant on cycles, and for which the corresponding Toeplitz operators *Ta* are bounded on

the Bergman space_{A}2_{(}_{D}_{). In particular, the class}_{A}

P contains all*L*∞(D)-functions that are constant on
cycles. For simplicity we restrict ourselves to the case of bounded symbols. That is, in what follows by

*z*0

Figure 1.2: Parabolic pencil_{P}(*z*0).

*z*0

*z*1

Figure 1.3: Hyperbolic pencil_{P}(*z*0*, z*1).

in [40], each Toeplitz operator*C*∗_{-algebra} _{T}_{(}_{A}

P) is generated by all Toeplitz operators*Ta* with*a*∈ AP,
is commutative.

Denote by _{A}(0) the *C*∗-algebra of bounded measurable symbols which depend only on*r* =√*zz*¯, and
consider the Toeplitz operator algebra _{T}(_{A}(0)) generated by all operators *Ta* : A2(D) → A2(D) of the

form

*Taf* =*B*D*af,*

where*a*=*a*(*r*)_{∈ A}(0).

**Theorem 1.3.2.** *[40] Let* *a*=*a*(*r*)_{∈ A}(0)*. Then the Toeplitz operatorTa, acting on* A2(D)*, is unitarily*

*equivalent to the multiplication operator* *γa*(0)*I, acting onℓ*+_{2}*. The sequence* *γa*(0) ={*γa*(0)(*n*)}*n*∈Z+ *is given*

*by*

*γ _{a}*(0)(

*n*) = (

*n*+ 1)

Z 1
0 *a*(

√

*r*)*rndr, n*_{∈}Z_{+}_{.}

The algebra_{T}(_{A}(0)) is commutative. The isomorphic inclusion

*τ*0 :T(A(0))→*ℓ*∞

is generated by the following mapping of generators of the algebra_{T}(_{A}(0)):

*τ*0 :*Ta*7→*γa*(0)*,*

Chapter 1. Preliminaries 7

Now, we introduce the Bergman space on the upper half-plane. Denote by Π the upper half-plane
with the area measure *dm*(*z*) =*dxdy*, *z* =*x*+*iy* and by *L*2_{(Π) the space of all measurable and square}

integrable functions deﬁned in Π. The*Bergman space*_{A}2(Π) is the closed subspace of*L*2(Π) consisting of
all analytic functions. The Bergman projection *B*Π is the orthogonal projection from*L*2(Π) onto A2(Π).

It has the integral form

*B*Π*f*(*z*) =−1

*π*

Z

Π

*f*(*w*)

(*z*_{−}*w*¯)2 *dm*(*w*)*.* (1.3.1)

The *anti-Bergman space* is the closed subspace of *L*2(Π), denoted by _{A}e2_{(Π), consisting of all }

mea-surable and anti-analytic functions. The orthogonal projection from *L*2(Π) onto _{A}e2_{(Π), called the }

anti-Bergman projection, is denoted by *B*eΠ and its integral form is the following

e

*B*Π*f*(*z*) =−1

*π*

Z

Π

*f*(*w*)

(¯*z*_{−}*w*)2 *dm*(*w*)*.* (1.3.2)

Let _{A}(0*,*_{∞}) be the algebra of all bounded measurable and homogeneous (of order zero) functions
deﬁned in Π. Also we consider the Toeplitz operator algebra _{T}(_{A}(0*,*_{∞})) which is generated by all the
operators *Ta*:A2(Π)→ A2(Π) of the form

*Taf* =*B*Π*af,*

where *a*=*a*(*θ*)_{∈ A}(0*,*_{∞}).

**Theorem 1.3.3.** *[40] Leta*=*a*(*θ*)_{∈ A}(0*,*_{∞})*. Then the Toeplitz operatorTa* *acting on*A2(Π)*is unitarily*

*equivalent to the multiplication operator* *γ*(0*,*∞)_{I}_{acting on}* _{L}*2

_{(}

_{R}

_{)}

*(0*

_{. The function}_{γ}*,*∞)

_{(}

_{λ}_{)}

_{is given by}*γ*(0*,*∞)(*λ*) = 2*λ*
1_{−}*e*−2*λπ*

Z *π*
0 *a*(*θ*)*e*

−2*λθ _{dθ,}*

_{λ}∈R_{.}

The algebra_{T}(_{A}(0*,*_{∞})) is commutative. The isomorphic inclusion
*τ*(0*,*∞) :T(A(0*,*∞))→*Cb*(R)

is generated by the following mapping of generators of the algebra _{T}(_{A}(0*,*_{∞})) :
*τ*_{(0}_{,}_{∞}_{)}:*Ta*7→*γa*(0*,*∞)(*λ*)*,*

where *a*=*a*(*θ*)_{∈ A}(0*,*_{∞})*.*

We consider the*C*∗_{-algebra} _{A}_{(}_{∞}_{) of all bounded measurable symbols that depend only on the }
imag-inary part *y* of a variable *w* = *x*+*iy* (vertical symbols), and consider the Toeplitz operator algebra

T(_{A}(_{∞})) generated by all operators*Ta*:A2(Π)→ A2(Π) of the form

*Taf* =*B*Π*af,*

where *a*=*a*(*y*)_{∈ A}(_{∞}).

**Theorem 1.3.4.** *[40] Let* *a*=*a*(*y*)∈ A(∞)*. Then the Toeplitz operator* *Ta* *acting on* A2(Π) *is unitarily*

*equivalent to the multiplication operator* *γa*(∞)*I, acting on* *L*2(R+)*. The functionγa*(∞)(*x*) *is given by*

*γ _{a}*(∞)(

*x*) =

Z

R_{+}

*a*

*y*
2*x*

*e*−*ydy,* *x*_{∈}R_{+}_{,}

Return again to the unit diskD. With each pencil of hyperbolic geodesics_{P} we associate the class_{A}_{P}

of measurable functions *a*=*a*(*z*) that are constant on cycles, and the corresponding Toeplitz operators
*Ta* are bounded on the Bergman space A2(D). We shall show that each Toeplitz operator *C*∗-algebra

generated by all operators *Ta* with *a* ∈ AP is commutative. These are the only commutative Toeplitz
operator algebras on the unit disk. We have the following results.

1. Let _{P} = _{P}(*z*0) a elliptic pencil. The functions *a* = *a*(*z*) from A(*z*0) = A_{P}(*z*0) are constant on

each Bergman cycle*S*(*z*0*, r*), *r >*0. The *C*∗-algebra T(A(*z*0)) is commutative and isomorphically

embedded into the algebra of sequences *ℓ*_{∞}.

Passing to the general case of an arbitrary point*z*0 ∈D, introduce the Möbius transformation

*αz*0(*z*) =

*z*0−*z*

1_{−}*zz*¯

of D onto it self, which maps to the point _{z}_{0} to the point 0. Note that this mapping is

self-inverse: *α*−_{z}_{0}1(*z*) = *αz*0(*z*). The biholomorphic invariance of the hyperbolic metric implies that

P(0) = *αz*0(P(*z*0)) and A(0) = {*a*(*αz*0) : *a*(*z*) ∈ A(*z*0)}. We Deﬁne the unitary operator *Uz*0 =

*U*−1
*z*0 :*L*

2_{(}_{D}_{)}_{→}* _{L}*2

_{(}

_{D}

_{) by}

(*Uz*0*ϕ*) (*z*) =*α*′*z*0(*z*)*ϕ*[*αz*0(*z*)]*.*

It is easy to see that for each *a*(*z*)_{∈ A}(*z*0) we have

*Ta*(*z*)=*Uz*0*Ta*(*αz*_{0}(*z*))*Uz*0*.*

Thus, by Theorem 1.3.2 we have

**Theorem 1.3.5.** *The* *C*∗_{-algebra}_{T}_{(}_{A}_{(}_{z}_{0}_{))} _{is commutative and isomorphically embedded into the}

*algebraℓ*_{∞}*. This embedding*

*νz*0 :T(A(*z*0))→*ℓ*∞

*is generated by the mapping*

*νz*0 :*T*17→*γ*
*z*0
*a* *,*

*where* *a*(*z*)_{∈ A}(*z*0)*, and the sequence* *γaz*0 *is given by the formula*

*γz*0

*a* (*n*) = (*n*+ 1)

Z 1
0 *a*

q

*αz*0(*r*)

*rndr, n*_{∈}Z_{+}_{.}

2. Consider now a parabolic pencil_{P} =_{P}(*z*0). The functions *a*=*a*(*z*) from A(*z*0) =A_{P}(*z*0) now are

constant on each horocycle. Introduce the Möius transformation

*αz*0(*z*) =*i*

*z*0+*z*

*z*0−*z*

(1.3.3)

of the unit disk Donto the upper half-plane Π, which maps the point _{z}_{0} _{∈}Tto the point _{∞ ∈} Π.

Then, the pencil_{P}(_{∞}) =*αz*0(P(*z*0)) on the upper half-plane Π consists of all semi-lines which are

parallel in Euclidean sense to positive semi-axis _{{}*w* = 0 +*iv* : *v* _{∈} R_{+}_{}}, the set of all horocycles

coincides with the set of all Euclidean straight lines parallel to the real axis being the boundary

R_{=}_{∂}_{Π of the upper half-plane, and the set}

A(_{∞}) =_{{}*a*(*α*−_{z}_{0}1(*w*)) :*a*(*z*)_{∈ A}(*z*0)}

Chapter 1. Preliminaries 9

The unitary operator *Vz*0 :*L*

2_{(Π)}_{→}_{D}_{, where}

(*Vz*0*ϕ*) (*z*) =*V*−

1

*z*0 *Ta*(*α*−*z*_{0}1(*w*))*Vz*0

for each*a*(*z*)_{∈ A}(*z*0), and where*α*−1(*w*) is the mapping inverse to (1.3.3). Thus as a direct corollary

of Theorem 1.3.3

**Theorem 1.3.6.** *The algebra*_{T}(_{A}(*z*0)) *is commutative. The isomorphic imbedding*

*τ* :_{T}(_{A}(*z*0))→*Cb*(R+)

*is generated by the following mapping of generators of the algebra* _{T}(_{A}(*z*0)) :

*τz*0 :*Ta*7→*γ*

(*z*0)
*a* (*x*) =

Z

R_{+}

*a*

*α*−_{z}_{0}1

_{y}

2*x*

*e*−*ydy,* *x*_{∈}R_{+}_{,}

*where* *a*=*a*(*z*)_{∈ A}(*z*0) *and* *α*−*z*01(*w*) *is the mapping inverse to 1.3.3.*

3. Consider ﬁnally a hyperbolic pencil_{P} =_{P}(*z*1*, z*2). The functions*a*=*a*(*z*) fromA(*z*1*, z*2) =AP(*z*1*,z*2)

now are constant on each hypercycle, i.e. on each Euclidean arc connecting the points *z*1 and *z*2.

Introduce the following Möbius transformation of the unit disk D_{onto the upper half-plane Π:}

*αz*1*,z*2(*z*) =

r_{z}

2

*z*1

*z*1−*z*

*z*_{−}*z*2

*,* (1.3.4)

where the value of the square root is selected to be in the upper half-plane. This transformation
maps the points *z*1 and *z*2 on T to the points 0 and ∞ on Π, respectively. Then, the pencil

P(0*,*_{∞}) =*αz*1*,z*2(P(*z*1*, z*2)) on the upper half-plane Π consists of all Euclidean semi-circles centered

at the origin, the set of all hypercycles with the set of all Euclidean rays outgoing from the origin, and the set

A(0*,*_{∞}) =_{{}*a*(*α*−_{z}_{1}1_{,z}_{2}(*w*)) :*a*(*z*)_{∈ A}(*z*1*, z*2)}*,*

where *α*−_{z}_{1}1_{,z}_{2} is the mapping inverse to (1.3.4), consists of homogeneous of order zero functions.
Introduce the unitary operator*Wz*1*,z*2 :*L*2(Π)→*L*2(D) as follows

(*Wz*1*,z*2*ϕ*) (*z*) =

r_{z}

2

*z*1

*z*2−*z*1

(*z*_{−}*z*2)2

*ϕ*

r_{z}

2

*z*1

*z*1−*z*

*z*_{−}*z*2

*.*

Then the Toeplitz operator algebra _{T}(_{A}(*z*1*, z*2)) is unitary equivalent to the algebra

T(_{A}(0*,*_{∞})) =*W _{z}*−

_{1}1

_{,z}_{2}

_{T}(

_{A}(

*z*1

*, z*2))

*Wz*1

*,z*2

*.*

Thus by Theorem 1.3.4 we have

**Theorem 1.3.7.** *The algebra*_{T}(_{A}(*z*1*, z*2)) *is commutative. The isomorphic embedding*

*τ*(*z*1*,z*2):A(*z*1*, z*2)→*Cb*(R)

*is generated by following mapping of generators of the algebra* _{T}(_{A}(*z*1*, z*2))*,*

*τ*_{(}_{z}_{1}_{,z}_{2}_{)} :*Ta*7→*γa*(*z*1*,z*2)(*λ*) =

2*λ*
1_{−}*e*−2*λπ*

Z *π*
0

*aα*−_{z}_{1}1_{,z}_{2}(*eiθ*)*e*−2*λθdθ, λ*_{∈}R_{,}

**1.3.1** **Commutative Toeplitz operators on the Weighted Bergman space**

In this part we recall some facts concerning weighted Bergman spaces (for more details see [40]).

Let *dν*(*z*) = *dxdy, z* = *x*+*iy*, the standard Lebesgue area measure on the unit disk. For each
*α*_{∈}(_{−}1*,*_{∞}), we introduce the measure

*dνα*(*z*) =

*λ*+ 1

*π* (1− |*z*|

2_{)}*λ _{dν}*

_{(}

_{z}_{)}

_{,}_{(1.3.5)}

which is normalized to a be a probability measure in D_{. Let} * _{L}*2

_{(}D

_{, dν}_{α}_{) be the Hilbert space with the}

scalar product

h*f, g*_{i}*α* =

Z

D

*f*(*z*)*g*(*z*)*dνα*(*z*)*.*

The *weighted Bergman space* _{A}2* _{α}*(D

_{), where}

_{α}_{∈}

_{(}

_{−}

_{1}

_{,}_{∞}

_{), is deﬁned as the subspace of}

*2*

_{L}_{(}D

_{, dν}_{α}_{) which}

consists of all analytic functions in D. This is a closed subspace of * _{L}*2(D

*). And the orthogonal*

_{, dν}_{α}Bergman projection*Bα*

D from*L*2(D*, dνα*) onto A2*α*(D) is given by

(*B*D*αf*)(*z*) =

Z

D

*f*(*w*)

(1_{−}*zw*)2+*αdνα*(*w*)*.*

**Theorem 1.3.8.** *[40] Given a pencil* _{P} *of geodesics, consider the set of* *L*_{∞}*-symbols which are *
*con-stant on corresponding cycles. Then the* *C*∗_{-algebra generated by Toeplitz operators with such symbols is}

*commutative on each weighted Bergman space* _{A}2* _{α}*(D)

_{.}Given a radial function *a*=*a*(*r*)∈*L*_{∞}(0*,*1), consider the Toeplitz operator,*Tα*

*a* :A2*α*(D)→ A2*α*(D),

*T _{a}αf* =

*B*D

*αaf.*

**Theorem 1.3.9.** *[40] For anya*=*a*(*r*)_{∈}*L*_{∞}[0*,*1)*, the Toeplitz operatorTα*

*a* *acting on*A2*α*(D)*is unitarily*

*equivalent to the multiplication operatorγa,αI* *acting on* *ℓ*2*. The sequenceγa,α*={*γa,α*(*n*)}*n*∈Z_{+} *is*

*γa,α*(*n*) =

1

*B*(*n*+ 1*, α*+ 1)

Z 1 0

*a*(√*r*)*rn*(1_{−}*r*)*α _{dr,}*

_{n}_{∈}

_{Z}+

*.*

**Corollary 1.3.10.** *The* *C*∗*-algebra* _{T}*α* *generated by all Toeplitz operators* *Taα* *with symbols* *a* = *a*(*r*) ∈

*L*_{∞}[0*,*1)*is commutative and is isometrically imbedded in* *ℓ*_{∞}*. The isometric imbeddingτα* *is generated by*

*the mapping*

*τα*:*Taα* 7→*γa,α.*

Now, we consider the upper half-plane Π and the normalized area measure

*dµ*(*z*) = 1
*π*

*dx dy*
(2*y*)2 =

1
2*πi*

*dz dz*¯
(2 Im*z*)2*,*

where *z* = *x*+*iy*. For *α* _{∈} (_{−}1*,*_{∞}) let *L*2(Π*, dµα*) be the space consisting of all measurable functions

satisfying

k*f*_{k}2*,α*=

Z

Π|

*f*(*z*)_{|}2 _{dµ}*α*(*z*)

1*/*2

*<*_{∞}*,*
where

*dµα*(*z*) = (*α*+ 1)(2 Im*z*)*α*

1

2*πidzdz*¯ = (*α*+ 1)(2 Im*z*)

Chapter 1. Preliminaries 11

Denote by _{h·}*,*_{·i}*α* the inner product in *L*2(Π*, dµα*) given by

h*f, g*_{i}*α* =

Z

Π

*f*(*z*)*g*(*z*)*dµα*(*z*)*.*

The*weighted Bergman space*_{A}2* _{α}*(Π) on the upper half-plane is the closed subspace of

*L*2(Π

*, dµα*) consisting

of all analytic functions. If*α* = 0 then_{A}2

*α*(Π) is the (unweighted) Bergman space on the upper half-plane

A2(Π).

The orthogonal projection from*L*2(Π*, dµα*) ontoA2*α*(Π) is denoted by*B*Π*α* and is given by the integral

formula

*Bα*_{Π}*f*(*z*) = (*α*+ 1)

Z

Π*f*(*w*)

*w*_{−}*w*
*z*_{−}*w*

*α*+2

*dµ*
=*iα*+2

Z

Π

*f*(*w*)

(*z*_{−}*w*)*α*+2 *dµα*(*w*)*.* (1.3.6)

The function

*Kα*(*z, w*) =

*iα*+2

(*z*_{−}*w*)*α*+2 (1.3.7)

is the weighted Bergman reproducing kernel (for details see for example [36] and [40]).
Some works (for example [31]) use the measure*dAr* given by

*dAr*(*z*) = (2*r*+ 1)*K*(*z, z*)−*rdx dy,*

where*r >*_{−}1_{2} and*K*(*z, w*) is the Bergman reproducing kernel given by*K*(*z, w*) = −1

*π*(*z*−*w*)2. We note that

*dAr* and *dµα* generate equivalent norms when*α*= 2*r*. In fact,

*dAr*(*z*) = (2*r*+ 1)*πr*(2 Im*z*)2*rdxdy*

= (2*r*+ 1)*πr*−1(2 Im*z*)2*rdxdy*

*π*
=*πα*2−1(*α*+ 1)(2 Im*z*)*αdxdy*

*π*
=*πα*2−1*dµ _{α}*(

*z*)

*.*

For any function*a*_{∈}*L*_{∞}(Π) the Toeplitz operator*Tα*

*a*, with symbol*a*, acting onA2*α*(Π) is the operator

deﬁned by

*T _{a}α*(

*f*) =

*B*

_{Π}

*αaf,*(1.3.8) for all

*f*

_{∈ A}2

*(Π)*

_{α}*.*

Given a vertical symbol*a*=*a*(*y*)_{∈}*L*_{∞}(R_{+}). Consider the Toeplitz operator_{T}α

*a* :A2*α*(Π)→ A2*α*(Π),

*T _{a}αf* =

*B*

_{Π}

*αaf.*

**Theorem 1.3.11.** *[40] For any* *a* = *a*(*y*) _{∈} *L*_{∞}(R_{+})_{, the Toeplitz operator}_{T}_{a}α_{acting on}_{A}2* _{α}*(Π)

_{is}*unitarily equivalent to the multiplication operator* *γa,αI, acting on* *L*2(R+)*. The function* *γa,α*(*x*) *is as*

*follows*

*γa,α*(*x*) =

*xα*+1
Γ(*α*+ 1)

Z _{∞}

0

*a*(*t/*2)*tαe*−*xtdt*

= *a*
Γ(*α*+ 1)

Z _{∞}

0 *a*

_{t}

2*x*

**Corollary 1.3.12.** *The* *C*∗*-algebra* _{T}*α* *generated by all Toeplitz operators* *Taα* *with symbols* *a* = *a*(*y*) ∈

*L*_{∞}(R_{+})* _{is commutative and is isometrically imbedded in}_{C}_{b}*(R

_{+})

_{. The isometric imbedding}_{τ}_{α}_{is generated}*by the mapping*

*τα*:*Taα* 7→*γa,α.*

Let now *a*=*a*(*θ*) a function that depends only on the angular variable.

**Theorem 1.3.13.** *[40] Givena*=*a*(*θ*)_{∈}*L*_{∞}(0*, π*)*, the Toeplitz operatorT _{a}α*

*acting on*

_{A}2

*(Π)*

_{α}*is unitarily*

*equivalent to the multiplication operatorγa,αI, acting on*

*L*2(R)

*. The functionγa,α*(

*ζ*)

*is*

*γa,α*(*ζ*) =

Z *π*
0

*e*−2*ζθ*sin*α _{θdθ}*−1Z

*π*0

*a*(*θ*)*e*−2*ζθ*sin*α _{θdθ, ζ}*

∈R_{.}

**Corollary 1.3.14.** *The* *C*∗_{-algebra}_{T}_{α}_{generated by all Toeplitz operators}_{T}α

*a* *with symbols* *a* = *a*(*θ*) ∈

*L*_{∞}(0*, π*)*is commutative and is isometrically imbedded inCb*(R)*. The isometric imbeddingτα* *is generated*

*by the mapping*

*τα*:*Taα* 7→*γa,α.*

**1.4**

**The Harmonic Bergman space**

**1.4.1** **The Harmonic Bergman space on the unit disk**

Recall that an *anti-analytic function* is deﬁned as the conjugate of an analytic function. So, the
space of all anti-analytic functions in *L*2(D) is called the anti-Bergman space, denoted by _{A}e2(D) and

characterized by the equation

*∂f*
*∂z* = 0*.*

This is a closed subspace of *L*2_{(}_{D}_{) and the anti-Bergman projection} _{B}_{e}

D is the orthogonal projection

from*L*2(D_{) onto} _{A}e2_{(}D_{). The Toeplitz operator}* _{T}*e

_{a}_{:}

_{A}e2

_{(}D

_{)}

_{→}

_{A}e2

_{(}D

_{), with symbol}

_{a}_{, is deﬁned by}

e

*Tag*=*B*eD(*ag*)*.*

There are interesting relations between analytic functions and harmonic functions. For example, if
*f* = *u*+*iv* is an analytic function, its real part *u*(*x, y*) and its imaginary part, *v*(*x, y*), satisfy Laplace
equation

∆*u*= *∂*2*u*
*∂x*2 +

*∂*2*u*
*∂y*2 = 0

and thus both are real harmonic functions. It follows from the Cauchy-Riemann equations that every
ana-lytic function is harmonic. On the other hand, a pair of functions (*u, v*) that satisﬁes the Cauchy-Riemann
equations is said to be a *conjugate pair*, and *v* is called the *harmonic conjugate* of *u*. The objective of
this section is to study complex-valued harmonic functions that are square integrable in the unit disk.

The *harmonic Bergman space* *b*2(D) is the closed subspace of * _{L}*2(D) which consists of all

complex-valued functions *f*(*z*) =*u*(*z*) +*iv*(*z*), with*u* and *v* real-valued functions, such that
∆*f* = ∆*u*+*i*∆*v*= 0*.*

Chapter 1. Preliminaries 13

analytic and anti-analytic function. The representation is unique up to a constant value.

For a function *f* _{∈} *b*2(D), the derivative *∂f*

*∂z* is an analytic function. Let *f*1 be an analytic function

such that *∂f*1
*∂z* =

*∂f*

*∂z* and let*f*2 =*f*−*f*1. Then

*f* =*f*1+*f*2*.*

It is known that the function *f*1 belongs to the space A2(D), and *f*2, to the spaceAe2(D). Thus, the

harmonic Bergman space is the (non-direct) sum of the Bergman and the anti-Bergman space.

We show some important results about harmonic functions, for details see [1].

**Proposition 1.4.1.** *Given a compact set* *K*_{⊂}D_{, there exists a constant}_{C}_{K}_{, depending on}_{K}_{such that}

|*u*(*z*)_{| ≤}*CK*k*u*k2*,*

*for all* *u*_{∈}*b*2_{(}_{D}_{)}_{.}

**Proposition 1.4.2.** *Let* (*un*) *be a sequence of harmonic functions in* D *that converges uniformly on*

*compact subsets of* D _{to a function}_{u. Then,}_{u}* _{is a harmonic function in}* D

_{.}Now, we shall prove that the harmonic Bergman space is a closed subspace of*L*2(D).

**Theorem 1.4.3.** *The harmonic Bergman spaceb*2(D) * _{is a closed subspace of}_{L}*2(D)

_{.}*Proof.* Let*K* be a compact subset ofD_{and (}_{u}_{j}_{) a Cauchy sequence in}* _{L}*2

_{(}D

_{). Then, by Proposition 1.4.1}

there exists a constant *CK* such that

|*uj*(*z*)−*uk*(*z*)| ≤*CK*k*uj*−*uk*k2

for all *z*_{∈}*K* and all *j, k*. Since (*uj*) is Cauchy sequence in *b*2(D) the inequality above implies that (*uj*)

is a Cauchy sequence in *C*(*K*). Hence (*uj*) converges uniformly on *K*. It follows from Proposition 1.4.2

(*uj*) converges uniformly to a function*u* that is harmonic on D. Therefore, the sequence (*uj*) converges

in*b*2(D).

For*z*_{∈}D, the map_{u}_{7→}* _{u}*(

*) is a linear functional on*

_{z}*2(D). From Proposition 1.4.1 it follows that the*

_{b}point evaluation is continuous on *b*2_{(}_{D}_{). From the Riesz Representation Theorem there exists a unique}

function*K*(*z,*_{·}), named the reproducing kernel for*b*2_{(}_{D}_{), that has the following reproducing formula}

*u*(*z*) =_{h}*u, K*(*z,*_{·})_{i}*,*

for every *u*_{∈}*b*2(D_{). And the explicit formula of} _{K}_{(}_{z,}_{·}_{) is given by (see for example [1])}

*K*(*z, w*) = 1
*π*

1
(1_{−}*wz*)2 +

1
*π*

1
(1_{−}*zw*)2 −

1

*π.* (1.4.1)

The reproducing kernel*K*(*z,*_{·}) has the following properties:

• *K*(*z,*_{·}) is a real valued function.

Since*b*2(D) is a closed subspace of* _{L}*2(D), there exists a unique orthogonal projection from

*2(D) onto*

_{L}*b*2_{(}_{D}_{). We denote it by}_{Q}_{and using (1.4.1) we have that}

*Qf* =*B*D*f* +*B*eD*f*−

1
*π*

Z

D

*f*(*w*)*dA*(*w*)*.*

Leaving the compact perturbation aside, *Q*=*B*D+*B*eD. Like in the case of Toeplitz operators acting

on the Bergman space, a Toeplitz operator* _{T}*b

_{a}_{acting on the harmonic Bergman space with symbol}

_{a}_{, is}

given by the formula

b

*Taf* =*Q*(*af*)*,* *f* ∈*b*2(D)*.* (1.4.2)

When *a*_{∈}*L*_{∞}(D_{) the Toeplitz operator} * _{T}*b

_{a}_{is always bounded and}

_{k}

_{T}_{a}_{k ≤ k}

_{a}_{k}

_{∞}

_{.}

Each function *u*_{∈}*b*2(D_{) is written in the form}

*u*= [*B*D*u*−*B*D*u*(0)] + [(*I* −*B*D)*u*+*B*D*u*(0)]*.*

Note that the term *B*D*u*−*B*D*u*(0) belongs to*z*A2(D) and (*I*−*B*D)*u*+*B*D*u*(0) is in Ae2(D). It is easy

to prove that the last two spaces are mutually orthogonal. Thus, the harmonic Bergman space on the unit disk is represented as the direct sum

*b*2(D_{) =}_{z}_{A}2_{(}D_{)}_{⊕}_{A}e2_{(}D_{)}_{.}

Let *U*:_{A}e2_{(}_{D}_{)}_{→A}2_{(}_{D}_{) be the unitary operator given by}

(*U f*)(*z*) =*f*(*z*)*.* (1.4.3)
We consider the unitary operator * _{U}*e

_{:}

*2*

_{b}_{(}

_{D}

_{)=}

_{z}_{A}2

_{(}

_{D}

_{)}

_{⊕}

_{A}

_{e}2

_{(}

_{D}

_{)}

_{→}

_{z}_{A}2

_{(}

_{D}

_{)}

_{⊕A}2

_{(}

_{D}

_{) given in matrix form by}

e

*U* = *I* 0

0 *U*

!

*.* (1.4.4)

The following theorem diﬀers from the original in [11] just by the constant factor 1

*π*. This constant

appears here because we are using the area measure of the unit disk instead of the normalized area measure which was used in the original theorem.

**Theorem 1.4.4** ([11])**.** *On the spacez*_{A}2(D)_{⊕ A}2(D)

e

*UT*b*aU*e∗ = *Ta*−
1

*π*(1⊗*a*¯) Γ*a*−1*π*(1⊗*a*∗)

Γˆ*a* *T*ˆ*a*

!

(1.4.5)

*where* ˆ*a*(*z*) =*a*(¯*z*)*, a*∗(*z*) =*a*(¯*z*)*,*(1_{⊗}*a*∗)(*h*) =_{h}*h, a*∗_{i}1 *and*

Γ*af* =*B*D(*aU f*) (1.4.6)

*is the small Hankel operator.*

Let_{T}(*C*(D)) be the* _{C}*∗-algebra generated by all Toeplitz operators with continuous symbols acting on

the harmonic Bergman space. In next theorem we identify each element of the Calkin algebraT(*C*(D_{))}_{/}_{K}

Chapter 1. Preliminaries 15

**Theorem 1.4.5** ([11])**.** *The sequence*

0_{−→ K}_{−−→ T}*i* (*C*(D_{))}_{−−→}*π* _{C}_{(}T_{)}_{−→}_{0}

*is a short exact sequence; that is, the quotient algebra*_{T}(*C*(D))_{/}_{K} _{is}_{∗}_{-isometrically isomorphic to}* _{C}*(T)

_{,}*where* *π* *is the symbol map which maps* *Tφ*+*K* *toφ*|T*.*

We conclude this section with a result about the Fredholm index of a Fredholm operator in_{T}(*C*(D)).

**Theorem 1.4.6** ([11])**.** *Let* *A*_{∈ T}(*C*(D))_{. If}_{A}* _{is Fredholm, then}*Ind

*= 0*

_{A}

_{.}**1.4.2** **The Harmonic Bergman space on the upper half-plane**

Recall now some known facts about Toeplitz operators on the upper half-plane. The harmonic
Bergman space of *L*2_{(Π), denoted by} * _{b}*2

_{(Π), is the closed subspace of}

*2*

_{L}_{(Π) consisting of all }

complex-valued harmonic functions. It is well known that *b*2(Π) = _{A}2_{(Π) +}_{A}_{e}2_{(Π). Since} _{A}2_{(Π) and} _{A}_{e}2_{(Π) are}

mutually orthogonal spaces, the representation for*b*2(Π) given above is actually a direct sum (see [43] for
details). Then, if we denote by*Q*the orthogonal projection from*L*2_{(Π) onto}* _{b}*2

_{(Π), we have the equation}

*Q*=*B*Π+*B*eΠ*.* (1.4.7)

Let*J* denote the (non linear) operator deﬁned in *b*2(Π) by the formula
*Jf* = ¯*f .*

From equations (1.3.1) and (1.3.2) it is easy to prove that

e

*B*Π=*JB*Π*J.* (1.4.8)

For a bounded function*a* _{∈}*L*_{∞}(Π) the Toeplitz operator * _{T}*b

_{a}_{:}

*2*

_{b}_{(Π)}

_{→}

*2*

_{b}_{(Π) and the small Hankel}

operator*Ha*:A2(Π)→Ae2(Π) are deﬁned by the equations

b

*Ta*=*QaI*= (*B*Π+*B*eΠ)*aI, Haf* =*B*eΠ(*af*)*.*

In 2009, Choe and Nam made a decomposition of a Toeplitz operator acting on the harmonic Bergman space on the upper half-plane in terms of Toeplitz operators acting on the Bergman and the anti-Bergman space and of operators between these two spaces. Explicitly

**Theorem 1.4.7** ([5])**.** *For a bounded symbol* *a* *the operator* *T*b*a* *has the following matrix representation*

*based on the decomposition* *b*2_{(Π) =}_{A}2_{(Π)}_{⊕}_{A}_{e}2_{(Π)}

b

*Ta*= _{H}Ta*JH*¯*aJ*
*a* *JTa*∗*J*

!

**Chapter 2**

**On Toeplitz operators on harmonic**

**Bergman spaces**

Commutative*C*∗-algebras generated by Toeplitz operators acting on the Bergman spaces on the unit
disk have been recently an important object of study. In [14] Grudsdy, Quiroga and Vasilevski
charac-terized all commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk.
Every class of symbols that produce a commutative*C*∗-algebra of Toeplitz operators arises as the set of
functions invariant under the action of a maximal abelian group of Möbius transformations of the unit
disk. There are three classes of such symbols: elliptic, which is realized by radial symbols on the unit
disk, parabolic, which is realized by vertical symbols, and hyperbolic, which is realized by symbols that
depend only on the angular variable on the upper half-plane.

In the context of algebras of Toeplitz operators on the harmonic Bergman space two important re-sults were obtained in the work of Guo and Zheng (see [11]). First, they found that the Calkin algebra of the Toeplitz algebra generated by Toeplitz operators with continuous symbols is isomorphic to the algebra of continuous functions deﬁned in the boundary of the disk. On the other hand, they also found that the Fredholm index of every Fredholm Toeplitz operator with continuous symbol is zero. This is one of the ﬁrst results that show diﬀerences between the harmonic Bergman space and the Bergman space.

Motivated by these recent research about Toeplitz operators on the Bergman space and on the
har-monic Bergman space, in this chapter, we introduce and study Toeplitz operators acting on the harhar-monic
Bergman space with symbols do not have to be continuous. In fact, this chapter is based on the three
model cases that generate commutative *C*∗-algebras of Toeplitz operators in the Bergman space. We
prove that, in the harmonic Bergman space, Toeplitz operators with radial symbols generate a
commu-tative *C*∗_{-algebra which is isomorphic to the algebra of all slowly oscillating sequences. We also prove}
that the Calkin algebra of the *C*∗-algebra generated by Toeplitz operators whose symbols are invariant
under parabolic transformations is commutative. The most interesting case is the *C*∗-algebra generated
by Toeplitz operators whose symbols are invariant under hyperbolic transformations. We prove that the
Calkin algebra of the*C*∗_{-algebra generated by Toeplitz operators with this type of symbols is isomorphic}
and isometric to the *C*∗-algebra of all 2_{×}2 matrix-valued continuous functions (*fij*(*t*)) deﬁned on Rand

such that they satisfy *f*12(±∞) =*f*21(±∞) = 0.

The results of this chapter were previously published as: Loaiza M., Lozano C., *On* *C*∗_{-algebras of}

*Toeplitz operators on the harmonic Bergman space.* Integr. Equ. Oper. Theory **76**, 2013, 105-130.

**2.1**

**Algebra generated by Toeplitz operators with radial symbols**

It is well known that Toeplitz operators with radial symbols acting on the Bergman space on the unit
disk generate a commutative *C*∗-algebra; see [19]. This algebra is described in terms of multiplication
operators acting on the Hilbert space of all square summable sequences*ℓ*+_{2} =*ℓ*2(Z+); see [40].

The ﬁrst question emerging here is the following: is the *C*∗-algebra generated by Toeplitz operators
with radial symbols and acting on the harmonic Bergman space, commutative? Can we ﬁnd the explicit
form of the spectrum of a Toeplitz operator with radial symbol? The aim of this section is to give the
aﬃrmative answer to the above questions.

Let *a*(*z*) = *a*(_{|}*z*_{|}) _{∈} *L*_{∞}(D) be a radial function. In Theorem 1.4.4 we have that ˆ* _{a}*(

*) =*

_{z}*(¯*

_{a}*) =*

_{z}*(*

_{a}*)*

_{z}and *a*∗(*z*) =*a*(*z*) = ¯*a*(*z*)*.*Thus

e

*UT*b*aU*e∗ = *Ta*−
1

*π*(1⊗¯*a*) Γ*a*−1*π*(1⊗¯*a*)

Γ*a* *Ta*

!

*,* (2.1.1)

where*U*e and Γ are the operators given by (1.4.4) and (1.4.6), respectively.

Now, we recall some results and notation from [40]. Passing to polar coordinates

*L*2(D_{, dA}_{) =}* _{L}*2

_{([0}

_{,}_{1]}

_{, rdr}_{)}

_{⊗}

*2*

_{L}

T_{,}dt

*it*

*.*
Consider the unitary operator

*U*1 =*I*⊗ F :*L*2([0*,*1]*, rdr*)⊗*L*2

T_{,}dt

*it*

→*L*2([0*,*1]*, rdr*)_{⊗}*ℓ*2*,*

where the Fourier transform_{F} :*L*2T_{,}dt

*it*

→*ℓ*2 is given by

F :*f* _{7→}*cn*=

1

√

2*π*

Z

T

*f*(*t*)*t*−*ndt*
*it.*
Thus the image of the Bergman space_{A}2

1=*U*1(A2(D)) coincides with the subspace of*L*2([0*,*1]*, rdr*)⊗*ℓ*2

which consists of all sequences _{{}*cn*(*r*)}*n*∈Z such that

*cn*(*r*) =

( p

2(*n*+ 1)*cnrn,* if *n*∈Z+*,*

0*,* if *n*_{∈}Z_{−}_{,}

whereZ_{+}_{=}N_{∪ {}_{0}_{}}_{,}Z_{−}_{=}Z_{\}Z_{+}_{.}

For each*n*_{∈}Z_{+} consider the unitary operator

*un*:*L*2([0*,*1]*, rdr*)→*L*2([0*,*1]*, rdr*)

given by the rule

*un*(*f*)(*r*) =

1

√

*n*+ 1*r*
− *n*

*n*+1* _{f}*(

*+11 )*

_{r}n

_{.}Finally, deﬁne the unitary operator

Chapter 2. On Toeplitz operators on harmonic Bergman spaces 19

by the following formula

*U*2 :{*cn*(*r*)}*n*∈Z7→

n

(*u*_{|}_{n}_{|}*cn*)(*r*)

o

*n*∈Z*.*

Then the space _{A}2

2 :=*U*2(A21) coincides with the space of all sequences {*dn*(*r*)}*n*∈Z, where

*dn*=*un*

q

2(*n*+ 1)*cnrn*

=√2*cn,*

for*n*_{∈}Z_{+}, and * _{d}_{n}*(

*)*

_{r}_{≡}0, for

_{n}_{∈}Z

_{−}.

Let*ℓ*0(*r*) =√2. We have *ℓ*0(*r*) ∈*L*2([0*,*1)*, rdr*) and k*ℓ*0(*r*)k= 1. Denote by *L*0 the one-dimensional

subspace of *L*2_{([0}_{,}_{1)}_{, rdr}_{) generated by} _{ℓ}

0(*r*), then the one-dimensional projection *P*0 of *L*2([0*,*1)*, rdr*)

onto *L*0 has the form

(*P*0*f*)(*r*) =h*f, ℓ*0i ·*ℓ*0=

√

2Z 1

0

*f*(*r*)√2*r dr.*

Let *ℓ*−_{2} = *ℓ*2(Z−). Then *ℓ*2 = *ℓ*+2 ⊕*ℓ*−2 and denote by *p*+(*p*−) the orthogonal projection of *ℓ*2 onto

*ℓ*+_{2}(*ℓ*−_{2}). Introduce the sequences*χ*_{±}=_{{}*χ*_{±}(*n*)_{}}*n*∈Z∈*ℓ*_{∞}, where*χ*_{±}(*n*) = 1 for*n*∈Z_{±} and*χ*_{±}(*n*) = 0 for

*n*_{∈}Z_{∓}. Then* _{p}*± =

_{χ}_{±}

*.*

_{I}Now_{A}2

2 =*L*0⊗*ℓ*+2, and the orthogonal projection *B*2 of*ℓ*2(*L*2([0*,*1)*, rdr*)) =*L*2([0*,*1)*, rdr*)⊗*ℓ*2 onto

A22 has the form*B*2 =*P*0⊗*p*+.

**Theorem 2.1.1.** *The unitary operator* *U*2*U*1 *gives an isometric isomorphism of the space* *L*2(D) *onto*

*L*2([0*,*1)*, rdr*)_{⊗}*ℓ*2 *under which* A2(D) *is mapped onto* *L*0 ⊗*ℓ*+2 *and* *B*D *is unitarily equivalent to the*

*projection* *P*0⊗*p*+*.*

Consider the operator*R*:*L*2_{(}_{D}_{)}_{→}* _{ℓ}*+

2, deﬁned by

*R*:*f*(*z*)_{7→}

( p

2(*n*+ 1)

√

2*π*

Z

D

*f*(*z*)¯*zndA*(*z*)

)

*n*∈Z_{+}

*.*

The restriction *R*_{|}_{A}2_{(}D) to the space A2(D) is an isometric isomorphism. The adjoint operator *R*∗ :

*ℓ*+_{2} _{→ A}2(D) is given by

{*cn*}*n*∈Z_{+} 7→

1

√

2*π*

X

*n*∈Z_{+}

q

2(*n*+ 1)*cnzn.*

It is trivial to prove that *R*∗*R* =*B*D.

The following theorem is due to K. Korenblum and K. Zhu [19], see also [40].

**Theorem 2.1.2.** *Let* *a*(*r*) *be a measurable function on the segment* [0*,*1]*. Then the Toeplitz operator* *Ta*

*acting on* _{A}2(D) _{is unitarily equivalent to the multiplication operator}_{γ}_{a}_{I}_{acting on}* _{ℓ}*+

2*, where*

*γa*(*n*) = (*n*+ 1)

Z 1 0

*a*(√*r*)*rndr, n*_{∈}Z_{+}* _{.}* (2.1.2)

Denote by*SO*1(Z+) the set of all *ℓ*∞-sequences that slowly oscillate in the sense of Schmidt (see [37])
i.e.

*SO*1(Z+) =

*x*∈*ℓ*∞: lim*k*+1

*n*+1→1

|*xk*−*xn*|= 0

*.*

Alternatively, *SO*1(Z+) consists of all bounded functionsZ+ → C uniformly continuous with respect to

the logarithmic metric