On C*-algebras of Toeplitz Operators on the Harmonic Bergman Space

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Centro de Investigación y de Estudios Avanzados

del Instituto Politécnico Nacional

Unidad Zacatenco

Departamento de Matemáticas

On

C

-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

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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 algebraT(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 algebraT0 . . . 39

2.3.2 Hyperbolic case: The algebraT1 . . . 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

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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 commutativeC∗-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: theC∗-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

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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. Theory76, 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-mutativeC∗-algebra which is isomorphic to theC∗-algebra of all 2×2 matrix-valued continuous functions (fij(t)) defined on Rand such that they satisfyf12(±∞) =f21(±∞) = 0 and f11(±∞) =f22(∓∞).

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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 ofC∗-algebras generated by Toeplitz operators. This chapter mainly concerns the description of commutativeC∗-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 tripleξ = (p, E, T), whereE and T are topological spaces, andp: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 tT. LetV be an open set inT. A functionσ:V E is called alocal section of the bundleξ if p(σ(t)) =tfor all tV. IfV =T the section is called global or just section. Denote by Γ(ξ) the set of all continuous sections of the bundleξ.

Let EE ={(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 aC∗-algebra, and

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i. the functions

(x, y)7→x+y:EE E, (x, y)7→x·y:EE E, (α, x)7→αx:C×EE,

x7→x∗:EE are continuous;

ii. the subsets UV(σ, ε) ={xE:p(x)∈V and kxσ(p(x))k< ε},where V is an open subset inT,

σ is a continuous section ofξ overV,ε >0, form a basis of open sets in the space E.

Given aC-bundleξ = (p, E, T), there is the canonically definedC-algebra associated withξ. Namely, the set of all bounded continuous sections σ of ξ = (p, E, T) with componentwise operations and the following norm

kσk= sup

tT k

σ(t)k

is obviously aC∗-algebra. We shall call this algebra theC∗-algebra defined by theC∗-bundleξ = (p, E, T), and shall denote it by Γb(ξ).

Proposition 1.1.1 ([41]). 1. The functionk · k:t7→ kσ(t)k is upper semi-continuous.

2. The algebra Γb(ξ) is a Cb(T)-module, where Cb(T) is the C-algebra of all bounded continuous

functions onT.

3. If the space T is quasi-compact, then Γb(ξ) = Γ(ξ) and Cb(T) =C(T).

Recall that a space T is called quasi-completely regular if for eacht0 ∈T and each closed setY(⊂T)

which does not containt0, there exist a continuous function f :T →[0,1] such that

f(t0) = 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) : tT} 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∈ Ain the quotient algebraA(t). Let

E = G

tT

A(t)

be the disjoint union of the C∗-algebrasA(t). We define the action of the (additive) group Aon the set E: each element a ∈ A generates the mapping ga : EE 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 algebraA(t), and the collection of orbits is parameterized by points of T. The partition of E onto the disjoint orbits generates the projection

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Chapter 1. Preliminaries 3

We endow the setsE 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:t7→a(t). We denote by

e

A the set of all the above sections ˜a. For eachε >0 and each ˜aAewe introduce the set Ua, ε) ={xE :kx˜a(p(x))k< ε},

and endow the setE with the topology which prebase consists of all the setsUa, 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:ET is open.

2. The topology on T coincides with the weakest topology under which all the mappings ˜a Ae are continuous.

3. A prebase of the topology on T is given by the system of sets Va, ε) ={tT :ka˜(t)k< ε}. Then the tripleξ = (p, E, T) is a C∗-bundle.

Given aC-algebra Aand a system of its closed two-sided ideals JT ={J(t) :tT}, the C-bundle

ξ = (p, E, T) described above is called thecanonicalC-bundle defined by theC-algebraAand 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.

LetAbe aC∗-algebra,JT ={J(t) :tT}be a system of its closed two-sided ideals,ξ = (p, E, T) be

the canonicalC-bundle defined byAandJT, Γb(ξ) be theC-algebra defined by the bundleξ= (p, E, T). Then the mapping

e

π :a∈ A 7→˜aΓb(ξ)

is a morphism of the C∗-algebrasAand Γb(ξ), such that

1. kerπe=tT J(t),

2. Imeπ=Ae.

Moreover, the mapping πe :A →Aeis an isometric-isomorphism if and only if

\

tT

J(t) ={0}.

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

T

tTJ(t) = {0} we say we shall localize by points of the set T. The elements a1 and a2 of the algebra

A are called locally equivalent at the point t T (a1 ∼t a2) if and only if a1−a2 ∈J(t). The natural

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

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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 tT 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 byJtin the algebraA. Finally we introduce the system

of idealsJT ={J(t) :tT}. We have \

tT

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) :tT}. 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 measuredA(z) =dxdy, z=x+iy.As usual,L2(D)

denote the space of all measurable and square integrable functions defined inD and A2(D) its subspace

consisting of all analytic functions. This space is precisely the set of all functionsf inL2(D) such that

∂f ∂z = 0.

The orthogonal projection fromL2(D) ontoA2(D) is called the Bergman projection, denoted here by

BD. Then the Toeplitz operatorTa :A2(D)→ A2(D), with symbol ais defined by

Taf =BD(af).

Ifais a bounded function, thenTais a bounded operator onA2(D) and we havekTak ≤ kak∞andTa∗ =T¯a.

From now on, the letterK 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(C(D)) :=C{Ta : a C(D)}

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 = Ta+K is

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

IndT = 1

2π{arga(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

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Chapter 1. Preliminaries 5

z0

Figure 1.1: Elliptic pencil P(z0).

hyperbolic plane.

Each pair of geodesics, say L1 and L2, lie in a geometrically defined object, one-parameter family P of

geodesics, which is called thepencil determined byL1 and L2. 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 pencilP determined byL1 and L2 is called:

1. parabolic if L1 and L2 are parallel; in this caseP is the set of all geodesics parallel to both L1 and

L2, and cycles are called horocycles,

2. elliptic if L1 and L2 are intersecting; in this case P is the set of all geodesics passing through the

common pointL1 and L2;

3. hyperbolic if L1 and L2 are disjoint; in this case P is the set of all geodesics orthogonal to the

common orthogonal ofL1 and L2, and cycles are calledhypercycles.

Let us mention the following joint properties ofP 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 inC;

4. every cycle in C is invariant under the reflection in any geodesic inP.

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 AP 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 spaceA2(D). In particular, the classA

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

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z0

Figure 1.2: Parabolic pencilP(z0).

z0

z1

Figure 1.3: Hyperbolic pencilP(z0, z1).

in [40], each Toeplitz operatorC-algebra T(A

P) is generated by all Toeplitz operatorsTa witha∈ AP, is commutative.

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

form

Taf =BDaf,

wherea=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

γa(0)(n) = (n+ 1)

Z 1 0 a(

r)rndr, nZ+.

The algebraT(A(0)) is commutative. The isomorphic inclusion

τ0 :T(A(0))→

is generated by the following mapping of generators of the algebraT(A(0)):

τ0 :Ta7→γa(0),

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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 L2(Π) the space of all measurable and square

integrable functions defined in Π. TheBergman spaceA2(Π) is the closed subspace ofL2(Π) consisting of all analytic functions. The Bergman projection BΠ is the orthogonal projection fromL2(Π) onto A2(Π).

It has the integral form

BΠf(z) =−1

π

Z

Π

f(w)

(zw¯)2 dm(w). (1.3.1)

The anti-Bergman space is the closed subspace of L2(Π), denoted by Ae2(Π), consisting of all

mea-surable and anti-analytic functions. The orthogonal projection from L2(Π) onto Ae2(Π), called the

anti-Bergman projection, is denoted by BeΠ and its integral form is the following

e

BΠf(z) =−1

π

Z

Π

f(w)

zw)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 =BΠaf,

where a=a(θ)∈ A(0,).

Theorem 1.3.3. [40] Leta=a(θ)∈ A(0,). Then the Toeplitz operatorTa acting onA2(Π)is unitarily

equivalent to the multiplication operator γ(0,∞)I acting on L2(R). The functionγ(0,∞)(λ) is given by

γ(0,∞)(λ) = 2λ 1e−2λπ

Z π 0 a(θ)e

−2λθdθ, λ

∈R.

The algebraT(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,):Ta7→γa(0,∞)(λ),

where a=a(θ)∈ A(0,).

We consider theC-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 operatorsTa:A2(Π)→ A2(Π) of the form

Taf =BΠaf,

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 L2(R+). The functionγa(∞)(x) is given by

γa(∞)(x) =

Z

R+

a

y 2x

eydy, xR+,

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Return again to the unit diskD. With each pencil of hyperbolic geodesicsP we associate the classAP

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(z0) a elliptic pencil. The functions a = a(z) from A(z0) = AP(z0) are constant on

each Bergman cycleS(z0, r), r >0. The C∗-algebra T(A(z0)) is commutative and isomorphically

embedded into the algebra of sequences .

Passing to the general case of an arbitrary pointz0 ∈D, introduce the Möbius transformation

αz0(z) =

z0−z

1zz¯

of D onto it self, which maps to the point z0 to the point 0. Note that this mapping is

self-inverse: αz01(z) = αz0(z). The biholomorphic invariance of the hyperbolic metric implies that

P(0) = αz0(P(z0)) and A(0) = {a(αz0) : a(z) ∈ A(z0)}. We Define the unitary operator Uz0 =

U−1 z0 :L

2(D)L2(D) by

(Uz0ϕ) (z) =αz0(z)ϕ[αz0(z)].

It is easy to see that for each a(z)∈ A(z0) we have

Ta(z)=Uz0Ta(αz0(z))Uz0.

Thus, by Theorem 1.3.2 we have

Theorem 1.3.5. The C-algebra T(A(z0)) is commutative and isomorphically embedded into the

algebraℓ. This embedding

νz0 :T(A(z0))→

is generated by the mapping

νz0 :T17→γ z0 a ,

where a(z)∈ A(z0), and the sequence γaz0 is given by the formula

γz0

a (n) = (n+ 1)

Z 1 0 a

q

αz0(r)

rndr, nZ+.

2. Consider now a parabolic pencilP =P(z0). The functions a=a(z) from A(z0) =AP(z0) now are

constant on each horocycle. Introduce the Möius transformation

αz0(z) =i

z0+z

z0−z

(1.3.3)

of the unit disk Donto the upper half-plane Π, which maps the point z0 Tto the point ∞ ∈ Π.

Then, the pencilP() =αz0(P(z0)) 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(αz01(w)) :a(z)∈ A(z0)}

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Chapter 1. Preliminaries 9

The unitary operator Vz0 :L

2(Π)D, where

(Vz0ϕ) (z) =V

1

z0 Ta(αz01(w))Vz0

for eacha(z)∈ A(z0), and whereα−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 algebraT(A(z0)) is commutative. The isomorphic imbedding

τ :T(A(z0))→Cb(R+)

is generated by the following mapping of generators of the algebra T(A(z0)) :

τz0 :Ta7→γ

(z0) a (x) =

Z

R+

a

αz01

y

2x

eydy, xR+,

where a=a(z)∈ A(z0) and αz01(w) is the mapping inverse to 1.3.3.

3. Consider finally a hyperbolic pencilP =P(z1, z2). The functionsa=a(z) fromA(z1, z2) =AP(z1,z2)

now are constant on each hypercycle, i.e. on each Euclidean arc connecting the points z1 and z2.

Introduce the following Möbius transformation of the unit disk Donto the upper half-plane Π:

αz1,z2(z) =

rz

2

z1

z1−z

zz2

, (1.3.4)

where the value of the square root is selected to be in the upper half-plane. This transformation maps the points z1 and z2 on T to the points 0 and ∞ on Π, respectively. Then, the pencil

P(0,) =αz1,z2(P(z1, z2)) 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(αz11,z2(w)) :a(z)∈ A(z1, z2)},

where αz11,z2 is the mapping inverse to (1.3.4), consists of homogeneous of order zero functions. Introduce the unitary operatorWz1,z2 :L2(Π)→L2(D) as follows

(Wz1,z2ϕ) (z) =

rz

2

z1

z2−z1

(zz2)2

ϕ

rz

2

z1

z1−z

zz2

.

Then the Toeplitz operator algebra T(A(z1, z2)) is unitary equivalent to the algebra

T(A(0,)) =Wz11,z2T(A(z1, z2))Wz1,z2.

Thus by Theorem 1.3.4 we have

Theorem 1.3.7. The algebraT(A(z1, z2)) is commutative. The isomorphic embedding

τ(z1,z2):A(z1, z2)→Cb(R)

is generated by following mapping of generators of the algebra T(A(z1, z2)),

τ(z1,z2) :Ta7→γa(z1,z2)(λ) =

2λ 1e−2λπ

Z π 0

z11,z2(eiθ)e−2λθdθ, λR,

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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 (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(z), (1.3.5)

which is normalized to a be a probability measure in D. Let L2(D, dνα) be the Hilbert space with the

scalar product

hf, giα =

Z

D

f(z)g(z)dνα(z).

The weighted Bergman space A2α(D), where α(1,), is defined as the subspace ofL2(D, dνα) which

consists of all analytic functions in D. This is a closed subspace of L2(D, dνα). And the orthogonal

Bergman projection

D fromL2(D, dνα) onto A2α(D) is given by

(BDαf)(z) =

Z

D

f(w)

(1zw)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 A2α(D).

Given a radial function a=a(r)∈L(0,1), consider the Toeplitz operator,

a :A2α(D)→ A2α(D),

Taαf =BDαaf.

Theorem 1.3.9. [40] For anya=a(r)L[0,1), the Toeplitz operatorTα

a acting onA2α(D)is unitarily

equivalent to the multiplication operatorγa,αI acting on 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(1r)αdr, nZ +.

Corollary 1.3.10. The C-algebra Tα generated by all Toeplitz operators Taα with symbols a = a(r) ∈

L[0,1)is commutative and is isometrically imbedded in . 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

(z) = 1 π

dx dy (2y)2 =

1 2πi

dz dz¯ (2 Imz)2,

where z = x+iy. For α (1,) let L2(Π, dµα) be the space consisting of all measurable functions

satisfying

kfk2=

Z

Π|

f(z)|2 α(z)

1/2

<, where

dµα(z) = (α+ 1)(2 Imz)α

1

2πidzdz¯ = (α+ 1)(2 Imz)

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Chapter 1. Preliminaries 11

Denote by ,·iα the inner product in L2(Π, dµα) given by

hf, giα =

Z

Π

f(z)g(z)dµα(z).

Theweighted Bergman spaceA2α(Π) on the upper half-plane is the closed subspace ofL2(Π, dµα) consisting

of all analytic functions. Ifα = 0 thenA2

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

A2(Π).

The orthogonal projection fromL2(Π, dµα) ontoA2α(Π) is denoted byBΠα and is given by the integral

formula

Πf(z) = (α+ 1)

Z

Πf(w)

ww zw

α+2

=+2

Z

Π

f(w)

(zw)α+2 dµα(w). (1.3.6)

The function

(z, w) =

+2

(zw)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 measuredAr given by

dAr(z) = (2r+ 1)K(z, z)−rdx dy,

wherer >12 andK(z, w) is the Bergman reproducing kernel given byK(z, w) = −1

π(zw)2. We note that

dAr and dµα generate equivalent norms whenα= 2r. In fact,

dAr(z) = (2r+ 1)πr(2 Imz)2rdxdy

= (2r+ 1)πr−1(2 Imz)2rdxdy

π =πα2−1(α+ 1)(2 Imz)αdxdy

π =πα2−1α(z).

For any functionaL(Π) the Toeplitz operator

a, with symbola, acting onA2α(Π) is the operator

defined by

Taα(f) =BΠαaf, (1.3.8) for all f ∈ A2α(Π).

Given a vertical symbola=a(y)L(R+). Consider the Toeplitz operatorTα

a :A2α(Π)→ A2α(Π),

Taαf =BΠαaf.

Theorem 1.3.11. [40] For any a = a(y) L(R+), the Toeplitz operator Taα acting on A2α(Π) is

unitarily equivalent to the multiplication operator γa,αI, acting on L2(R+). The function γa,α(x) is as

follows

γa,α(x) =

+1 Γ(α+ 1)

Z

0

a(t/2)tαextdt

= a Γ(α+ 1)

Z

0 a

t

2x

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Corollary 1.3.12. The C-algebra Tα generated by all Toeplitz operators Taα with symbols a = a(y) ∈

L(R+)is commutative and is isometrically imbedded inCb(R+). The isometric imbeddingταis 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 operatorTaα acting onA2α(Π)is unitarily equivalent to the multiplication operatorγa,αI, acting on L2(R). The functionγa,α(ζ) is

γa,α(ζ) =

Z π 0

e−2ζθsinαθdθ−1Z π 0

a(θ)e−2ζθsinαθdθ, ζ

∈R.

Corollary 1.3.14. The C-algebra Tα generated by all Toeplitz operators Tα

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 L2(D) is called the anti-Bergman space, denoted by Ae2(D) and

characterized by the equation

∂f ∂z = 0.

This is a closed subspace of L2(D) and the anti-Bergman projection Be

D is the orthogonal projection

fromL2(D) onto Ae2(D). The Toeplitz operatorTea:Ae2(D)Ae2(D), with symbola, is defined by

e

Tag=BeD(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= 2u ∂x2 +

2u ∂y2 = 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 b2(D) is the closed subspace of L2(D) which consists of all

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

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Chapter 1. Preliminaries 13

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

For a function f b2(D), the derivative ∂f

∂z is an analytic function. Let f1 be an analytic function

such that ∂f1 ∂z =

∂f

∂z and letf2 =ff1. Then

f =f1+f2.

It is known that the function f1 belongs to the space A2(D), and f2, 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 KD, there exists a constantCK, depending on K such that

|u(z)| ≤CKkuk2,

for all ub2(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 ofL2(D).

Theorem 1.4.3. The harmonic Bergman spaceb2(D) is a closed subspace ofL2(D).

Proof. LetK be a compact subset ofDand (uj) a Cauchy sequence inL2(D). Then, by Proposition 1.4.1

there exists a constant CK such that

|uj(z)−uk(z)| ≤CKkujukk2

for all zK and all j, k. Since (uj) is Cauchy sequence in b2(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 functionu that is harmonic on D. Therefore, the sequence (uj) converges

inb2(D).

ForzD, the mapu7→u(z) is a linear functional onb2(D). From Proposition 1.4.1 it follows that the

point evaluation is continuous on b2(D). From the Riesz Representation Theorem there exists a unique

functionK(z,·), named the reproducing kernel forb2(D), that has the following reproducing formula

u(z) =hu, K(z,·)i,

for every ub2(D). And the explicit formula of K(z,·) is given by (see for example [1])

K(z, w) = 1 π

1 (1wz)2 +

1 π

1 (1zw)2 −

1

π. (1.4.1)

The reproducing kernelK(z,·) has the following properties:

K(z,·) is a real valued function.

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Sinceb2(D) is a closed subspace ofL2(D), there exists a unique orthogonal projection fromL2(D) onto

b2(D). We denote it byQ and using (1.4.1) we have that

Qf =BDf +BeDf

1 π

Z

D

f(w)dA(w).

Leaving the compact perturbation aside, Q=BD+BeD. Like in the case of Toeplitz operators acting

on the Bergman space, a Toeplitz operatorTba acting on the harmonic Bergman space with symbol a, is

given by the formula

b

Taf =Q(af), fb2(D). (1.4.2)

When aL(D) the Toeplitz operator Tbais always bounded and kTak ≤ kak.

Each function ub2(D) is written in the form

u= [BDuBDu(0)] + [(IBD)u+BDu(0)].

Note that the term BDuBDu(0) belongs tozA2(D) and (IBD)u+BDu(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

b2(D) =zA2(D)Ae2(D).

Let U:Ae2(D)→A2(D) be the unitary operator given by

(U f)(z) =f(z). (1.4.3) We consider the unitary operator Ue:b2(D)=zA2(D)Ae2(D)zA2(D)⊕A2(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 spacezA2(D)⊕ A2(D)

e

UTbaUe∗ = Ta− 1

π(1⊗a¯) Γa−1π(1⊗a∗)

Γˆa Tˆa

!

(1.4.5)

where ˆa(z) =az), a∗(z) =az),(1a∗)(h) =hh, ai1 and

Γaf =BD(aU f) (1.4.6)

is the small Hankel operator.

LetT(C(D)) be theC∗-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

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Chapter 1. Preliminaries 15

Theorem 1.4.5 ([11]). The sequence

0−→ K−−→ Ti (C(D))−−→π C(T)−→0

is a short exact sequence; that is, the quotient algebraT(C(D))/K is-isometrically isomorphic to C(T),

where π is the symbol map which maps +K toφ|T.

We conclude this section with a result about the Fredholm index of a Fredholm operator inT(C(D)).

Theorem 1.4.6 ([11]). Let A∈ T(C(D)). If A is Fredholm, thenIndA= 0.

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 L2(Π), denoted by b2(Π), is the closed subspace of L2(Π) consisting of all

complex-valued harmonic functions. It is well known that b2(Π) = A2(Π) +Ae2(Π). Since A2(Π) and Ae2(Π) are

mutually orthogonal spaces, the representation forb2(Π) given above is actually a direct sum (see [43] for details). Then, if we denote byQthe orthogonal projection fromL2(Π) ontob2(Π), we have the equation

Q=BΠ+B. (1.4.7)

LetJ denote the (non linear) operator defined in b2(Π) by the formula Jf = ¯f .

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

e

BΠ=JBΠJ. (1.4.8)

For a bounded functiona L(Π) the Toeplitz operator Tba :b2(Π) b2(Π) and the small Hankel

operatorHa:A2(Π)→Ae2(Π) are defined by the equations

b

Ta=QaI= (BΠ+BeΠ)aI, Haf =BeΠ(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 Tba has the following matrix representation

based on the decomposition b2(Π) =A2(Π)Ae2(Π)

b

Ta= HTa JH¯aJ a JTaJ

!

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(23)

Chapter 2

On Toeplitz operators on harmonic

Bergman spaces

CommutativeC∗-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 commutativeC∗-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 theC-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 f12(±∞) =f21(±∞) = 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.

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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+2 =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 ˆa(z) =az) =a(z)

and a∗(z) =a(z) = ¯a(z).Thus

e

UTbaUe∗ = Ta− 1

π(1⊗¯a) Γa−1π(1⊗¯a)

Γa Ta

!

, (2.1.1)

whereUe 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

L2(D, dA) =L2([0,1], rdr)L2

T,dt

it

. Consider the unitary operator

U1 =I⊗ F :L2([0,1], rdr)⊗L2

T,dt

it

L2([0,1], rdr)2,

where the Fourier transformF :L2T,dt

it

2 is given by

F :f 7→cn=

1

2π

Z

T

f(t)tndt it. Thus the image of the Bergman spaceA2

1=U1(A2(D)) coincides with the subspace ofL2([0,1], rdr)⊗2

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

cn(r) =

( p

2(n+ 1)cnrn, if n∈Z+,

0, if nZ,

whereZ+=N∪ {0},Z=Z\Z+.

For eachnZ+ consider the unitary operator

un:L2([0,1], rdr)→L2([0,1], rdr)

given by the rule

un(f)(r) =

1

n+ 1rn

n+1f(rn+11 ).

Finally, define the unitary operator

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Chapter 2. On Toeplitz operators on harmonic Bergman spaces 19

by the following formula

U2 :{cn(r)}n∈Z7→

n

(u|n|cn)(r)

o

n∈Z.

Then the space A2

2 :=U2(A21) coincides with the space of all sequences {dn(r)}n∈Z, where

dn=un

q

2(n+ 1)cnrn

=√2cn,

fornZ+, and dn(r)0, fornZ.

Let0(r) =√2. We have 0(r) ∈L2([0,1), rdr) and k0(r)k= 1. Denote by L0 the one-dimensional

subspace of L2([0,1), rdr) generated by

0(r), then the one-dimensional projection P0 of L2([0,1), rdr)

onto L0 has the form

(P0f)(r) =hf, ℓ0i ·0=

2Z 1

0

f(r)√2r dr.

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

+2(2). Introduce the sequencesχ±={χ±(n)}n∈Z∈, whereχ±(n) = 1 forn∈Z± andχ±(n) = 0 for

nZ. Thenp± =χ±I.

NowA2

2 =L0⊗+2, and the orthogonal projection B2 of2(L2([0,1), rdr)) =L2([0,1), rdr)⊗2 onto

A22 has the formB2 =P0⊗p+.

Theorem 2.1.1. The unitary operator U2U1 gives an isometric isomorphism of the space L2(D) onto

L2([0,1), rdr)2 under which A2(D) is mapped onto L0 ⊗+2 and BD is unitarily equivalent to the

projection P0⊗p+.

Consider the operatorR:L2(D)+

2, defined by

R:f(z)7→

( p

2(n+ 1)

2π

Z

D

f(zzndA(z)

)

n∈Z+

.

The restriction R|A2(D) to the space A2(D) is an isometric isomorphism. The adjoint operator R∗ :

+2 → A2(D) is given by

{cn}n∈Z+ 7→

1

2π

X

n∈Z+

q

2(n+ 1)cnzn.

It is trivial to prove that RR =BD.

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 A2(D) is unitarily equivalent to the multiplication operator γaI acting on +

2, where

γa(n) = (n+ 1)

Z 1 0

a(√r)rndr, nZ+. (2.1.2)

Denote bySO1(Z+) the set of all ∞-sequences that slowly oscillate in the sense of Schmidt (see [37]) i.e.

SO1(Z+) =

 

x∞: limk+1

n+1→1

|xkxn|= 0

  .

Alternatively, SO1(Z+) consists of all bounded functionsZ+ → C uniformly continuous with respect to

the logarithmic metric

Figure

Figure 1.1: Elliptic pencil P(z0).
Figure 1 1 Elliptic pencil P z0 . View in document p.11
Figure 1.2: Parabolic pencil P(z0).
Figure 1 2 Parabolic pencil P z0 . View in document p.12
Figure 2.1:t �8sech2 tπ4
Figure 2 1 t 8sech2 t 4. View in document p.33
Figure 2.2:t �2sech2 tπ2
Figure 2 2 t 2sech2 t 2. View in document p.33
Figure 2.3: 14
Figure 2 3 14. View in document p.34
Figure 2.5: Function t � sech2 tπ2
Figure 2 5 Function t sech2 t 2. View in document p.36
Figure 2.4: − 132
Figure 2 4 132. View in document p.36
Figure 2.6: t2 csch2(tπ).
Figure 2 6 t2 csch2 t . View in document p.37
Figure 2.7: 14
Figure 2 7 14. View in document p.38
Figure 2.8: Functions x1 and x2 at θ0 = π/2
Figure 2 8 Functions x1 and x2 at 0 2. View in document p.39
Figure 2.9: Functions x1 and x2 at θ0 = π/3
Figure 2 9 Functions x1 and x2 at 0 3. View in document p.40
Figure 2.10: Derivative of the function x1
Figure 2 10 Derivative of the function x1. View in document p.40
Figure 2.11: φz0(P(z0)) on the upper half-plane.
Figure 2 11 z0 P z0 on the upper half plane . View in document p.46
Figure 2.12: φz0,z1(P(z0, z1)) on the upper half-plane.
Figure 2 12 z0 z1 P z0 z1 on the upper half plane . View in document p.47
Figure 3.1: Function x1(λ).
Figure 3 1 Function x1 . View in document p.60

Referencias

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