Propriedades de regularidade de operadores de Wiener-Hopf-Hankel

2007

Ana Paula

Bran o Nolas o

Propriedades de Regularidade de Operadores de

Wiener-Hopf-Hankel

Regularity Properties of Wiener-Hopf-Hankel

2007

Ana Paula

Bran o Nolas o

Propriedades de Regularidade de Operadores de

Wiener-Hopf-Hankel

Regularity Properties of Wiener-Hopf-Hankel

Operators

teseapresentada àUniversidade de Aveiro para umprimento dosrequisitos

ne essários à obtenção do grau de Doutor em Matemáti a, realizada sob

a orientação ientí a do Doutor Luís Filipe Pinheiro de Castro, Professor

Asso iado omAgregaçãodoDepartamentodeMatemáti adaUniversidade

de Aveiro

Apoionan eiro da FCT, Bolsade

presidente Reitora da Universidade deAveiro

Doutor Vasile Stai u

ProfessorCatedráti odaUniversidadedeAveiro

Doutor Semyon Borisovi hYakubovi h

ProfessorAsso iado omAgregaçãodaFa uldadedeCiên iasdaUniversidadedoPorto

Doutor Luís Filipe Pinheiro deCastro

ProfessorAsso iado omAgregaçãodaUniversidadedeAveiro(Orientador)

Doutor Juan CarlosSan hez Rodriguez

ProfessorAuxiliardaFa uldadedeCiên iaseTe nologiadaUniversidadedoAlgarve

Doutor Frank-Olme EwalSpe k

disponibilidadee in entivoprestados aolongodestes últimos quatroanos.

À Unidade de Investigação "Matemáti a e Apli ações" e aoDepartamento

de Matemáti a da Universidade de Aveiro por todo o apoio e fa ilidades

on edidas para a realização deste trabalho e a todos os meus olegas e

amigos pelo ambiente de trabalho propor ionado e pelo apoio que sempre

manifestaram.

À Fundação para a Ciên iaeaTe nologiapelo apoio nan eiro on edido.

invertibilidade, fa torização, função quase periódi a, função semi-quase

periódi a,funçãoquaseperiódi a portroços

resumo Nesta tese estudamos as propriedades de regularidade de operadores de

Wiener-Hopf-Hankel om símbolos de Fourier perten entes às álgebrasdas

funções quase periódi as, das funções semi-quase periódi as e das funções

quase periódi as portroçose onsideramosestes operadoresa a tuarentre

espaçosdeLebesgue

L

p

(para

1

< p <

∞

). Porpropriedadesderegularidade entende-se invertibilidade lateral e bilateral, propriedade de Fredholm e

solubilidade normal.

Propomos uma teoria de fa torização para operadores de

Wiener-Hopf--Hankel om símbolos de Fourier quase periódi os, e a a tuar entre

espaços de Lebesgue

L

2

, introduzindo uma fa torização para os símbolos

de Fourier quase periódi os de tal modo que as propriedades dos fa tores

irãopermitir orrespondentesfa torizaçõesdosoperadores. Um ritériopara

a propriedade de semi-Fredholm e para a invertibilidadelateral e bilateralé

assim obtidoemtermos de determinadosíndi esdas fa torizações.

Baseado na relação delta após extensão, estabele emos um teorema do

tipo de Sarason para operadores de Wiener-Hopf-Hankel om símbolos de

Fouriersemi-quaseperiódi os,aa tuarentreespaçosdeLebesgue

L

2

. Uma

generalizaçãodoteoremadotipodeSarasonétambémobtida onsiderando

agoraosoperadoresaa tuarentreespaçosdeLebesgue

L

p

. Paraoperadores

deWiener-Hopf-Hankel omsímbolosdeFourierquaseperiódi osportroços,

a a tuar entre espaços de Lebesgue

L

2

bilateral destes operadores em termos dos valores médios e das médias

geométri as dosrepresentantesquaseperiódi osnoinnitodossímbolosde

Fourier,assim omodasdes ontinuidadesdedeterminadasfunçõesauxiliares

(no aso das funções quase periódi as por troços). Para ada aso, é

apresentada umafórmulapara o índi ede Fredholm.

Finalmente, de volta aos operadores de Wiener-Hopf-Hankel om símbolos

de Fouriernasubálgebradasfunçõesquaseperiódi as

AP W

,aa tuarentre espaços de Lebesgue

L

2

, onsideramos o aso mais geralde operadores de

Wiener-Hopf-Hankel om símbolos matri iais de Fourier

AP W

. Para estes operadores, obtemos um ritério para a invertibilidade e a propriedade de

semi-Fredholmbaseadonahipótesede umespe í o onjunto de Hausdor

pie ewise almostperiodi fun tion

abstra t In this thesis we study the regularity properties of Wiener-Hopf-Hankel

operatorswithFouriersymbolsbelongingtothealgebrasofalmostperiodi ,

semi-almostperiodi andpie ewisealmostperiodi fun tionsandwe onsider

these operators a ting between

L

p

Lebesgue spa es (for

1

< p <

∞

). By regularity properties one means one-sided and both-sided invertibility,

Fredholm property and normal solvability.

We propose a fa torization theory for Wiener-Hopf-Hankel operators with

almostperiodi Fouriersymbols,anda tingbetween

L

2

Lebesguespa es,by

introdu inga fa torization on ept forthe almostperiodi Fouriersymbols

su h that the properties of the fa tors will allow orresponding operator

fa torizations. A riterionforthesemi-Fredholmproperty and forone-sided

and both-sidedinvertibilityis thereforeobtainedupon ertainindi es ofthe

fa torizations.

Based on the delta relation after extension, we establish a Sarason's

type theorem for Wiener-Hopf-Hankel operators with semi-almost periodi

Fourier symbols and a ting between

L

2

Lebesgue spa es. We also derive

a generalization of the Sarason's type theorem, the so- alled

Dudu hava--Saginashvili's type theorem, when we onsider the same kind of operators

a tingnowbetween

L

p

Lebesguespa es. ForWiener-Hopf-Hankeloperators

withpie ewisealmostperiodi Fouriersymbols,a tingbetween

L

2

Lebesgue

one-sided invertibility of these operators, based on the mean motions and

geometri meanvaluesof the almostperiodi representativesof theFourier

symbolsatminusandplusinnity,aswellasonthedis ontinuitiesof ertain

auxiliaryfun tions (inthe ase of pie ewise almostperiodi fun tions). For

ea h ase, formulaeforthe Fredholmindex of theoperators areprovided.

Finally, we return to Wiener-Hopf-Hankel operators with Fourier symbols

in the subalgebra of almost periodi fun tions

AP W

, a ting between

L

2

Lebesgue spa es, and we onsider the more general ase of

Wiener-Hopf-Hankeloperatorswithmatrix

AP W

Fouriersymbols. Fortheseoperatorswe a hieveaninvertibilityandsemi-Fredholm riterionbasedontheassumption

List of Symbols v

Introdu tion ix

1 Convolution Type Operators 1

1.1 Some denitions,notations, and histori al notes . . . 2

1.2 Regularity properties . . . 8

1.3 Relationsbetween onvolution typeoperators . . . 11

1.3.1 Relationsbetween bounded linearoperators . . . 11

1.3.2 Fa torizationof Wiener-Hopf plus Hankeloperators . . . 13

1.3.3 Relationbetween Wiener-Hopf-Hankeloperators and Wiener-Hopf operators . . . 18

1.3.4 Relationbetween Wiener-Hopf-Hankeloperators and Toeplitz-Hankeloperators . . . 23

1.4 Ne essary onditions forthesemi-Fredholmproperty . . . 27

2 Fourier Symbols 33 2.1 Almost periodi fun tions . . . 34

2.2 Semi-almostperiodi fun tions. . . 41

2.3 Pie ewise almost periodi fun tions . . . 45

3 Wiener-Hopf plus Hankel Operators with Almost Periodi Symbols 53 3.1

AP

fa torizations . . . 56

3.3 Invertibility of Wiener-Hopf plus Hankel operators with

AP

symbols. . . . 73

4 Wiener-Hopf-Hankel Operators with Semi-Almost Periodi Symbols 77 4.1 A Sarason's typetheorem . . . 78

4.1.1 Motivation. . . 78

4.1.2 First approa h . . . 79

4.1.3 A stronger version of the Sarason's type theorem . . . 83

4.1.4 Anotherlookattheinvertibilityandsemi-Fredholm riteriaof Wiener-Hopf-Hankel operatorswith

AP

symbols . . . 92

4.2 A Dudu hava-Saginashvili's typetheorem . . . 94

4.2.1 Someparti ular lasses of almost and semi-almost periodi fun tions 95 4.2.2 The Dudu hava-Saginashvili's theorem . . . 96

4.2.3 Mainresult . . . 100

4.3 Fredholmindex formula . . . 103

4.3.1 FredholmindexformulaforWiener-Hopf-Hankeloperatorswith

SAP

Fouriersymbols . . . 104

4.3.2 FredholmindexformulaforWiener-Hopf-Hankeloperatorswith

SAP

p

Fouriersymbols . . . 119

5 Wiener-Hopf-HankelOperatorswithPie ewiseAlmostPeriodi Symbols127 5.1 A semi-Fredholmand invertibility riterion . . . 128

5.2 Fredholmindex formula . . . 132

5.2.1 Aformulaforthesumoftheindi esofFredholmWiener-Hopfplus/minus Hankel operators . . . 132

5.2.2 Examplesand invertibility . . . 148

6 Matrix Wiener-Hopf-Hankel Operators with Good Hausdor Sets 157 6.1 Preliminaries . . . 157

6.4 Anexample . . . 164

Con lusion 167

A

n

olumns of length

n

with entries in

A

, 158

A

n

_×n

n × n

matri eswith entries in

A

, 158

AP

lass of almostperiodi fun tions,34

AP

p

losure of

AP W

in

M

p

_(R)

,95

AP

−

smallest losed subalgebraof

L

∞

_(R)

that ontainsallthe fun tions

e

λ

with

λ ≤ 0

, 41

AP

+

smallest losed subalgebraof

L

∞

_(R)

that ontainsallthe fun tions

e

λ

with

λ ≥ 0

, 41

AP W

set of all almost periodi fun tions whi h an be written inthe form of an absolutely onvergent series, 40

AP W

−

set of allfun tions

ψ ∈ AP W

su h that

Ω(ψ) ⊂ (−∞, 0]

, 41

AP W

+

set of allfun tions

ψ ∈ AP W

su h that

Ω(ψ) ⊂ [0, +∞)

,41

A

p

(z

1

, z

2

) =



z

1

+ (z

2

− z

1

) σ

p

(µ), µ ∈ R



, 97

arg φ

ontinuous argument of

φ

, 39

C

₊

omplex upperhalf plane

ℑm z > 0

,15

C

−

omplex lower half plane

ℑm z < 0

, 15

C( ˙

R)

set of all(bounded) ontinuous ( omplex-valued) fun tions on

˙

R

, 41

C(R)

set of all (bounded) ontinuous ( omplex-valued) fun tions on

R

with a possiblejump at

∞

, 42

C

0

( ˙

R)

set ofallfun tionsin

C( ˙

R)

su hthatthelimitsat

−∞

andat

+∞

areequal to zero, 42

C

p

( ˙

R)

losure of

C( ˙

R)

in

M

p

_(R)

, 95

C

p

(R)

losure of

C(R)

in

M

p

_(R)

, 95

Coker

T

okernel of the operator

T

, 8

D

₊

_{= {z ∈ C : |z| < 1}}

, 24

D

−

= {z ∈ C : |z| ≥ 1} ∪ {∞}

, 24

d(T )

dimension of the okernel of the operator

T

,8

d

_(φ)

geometri mean value of

φ ∈ GAP

, 40

d

_l

_(φ)

left geometri mean value of

φ ∈ GPAP

, 47

sup

F

Fourier transformation,3

F

−1

inverse of the Fouriertransformation

F

,3

φ

l

almost periodi representative of

φ

at

−∞

, 45, 47

φ

r

almost periodi representative of

φ

at

+∞

, 45, 47

GB

group of allinvertible elements of aBana h algebra

B

,17

H

φ

Hankeloperator, 7

H(Θ)

Hausdor set ornumeri al range of the omplex matrix

Θ ∈ C

n

×n

, 159

H

∞

_(C

±

)

set of allbounded and analyti fun tionsin

C

±

, 15

H

∞

_(D

±

)

set of allbounded and analyti fun tionsin

D

±

, 24

H

p

_(C

±

)

set of all fun tions

φ

whi h are analyti in

C

±

and su h that

sup

_±y>0

R

R

|φ(x + iy)|

p

_{dy < ∞}

, 15

H

p

_(D

+

)

set of all fun tions

φ

whi h are analyti in

D

+

and su h that

sup

r

∈(0,1)

R

2π

0

|φ(re

iθ

)|

p

dθ < ∞

, 24

H

p

_(D

−

)

set of all fun tions

φ(z)

(

z ∈ D

−

) for whi h

φ(

1

z

)

is a fun tion in

H

p

_(D

+

)

, 24

H

∞

±

(R)

set of all fun tions in

L

∞

_(R)

that are non-tangential limitsof elements in

H

∞

_(C

±

)

, 15

H

_±

p

(R)

set of all fun tions in

L

p

_(R)

that are non-tangential limits of elements in

H

p

_(C

±

)

,15

H

_±

p

(T)

set of all fun tions in

L

p

_(T)

that are non-tangential limits of elements in

H

p

_(D

±

)

,24

Im

T

image of the operator T, 8

ℑm(z)

imaginary part of the omplex number

z

Ind

T

Fredholm index of the operator

T

,9 ind

φ

Cau hy index of

φ

, 103, 105, 108, 135

inf

inmum

I

Z

identity operator onthe Bana h spa e

Z

, 7

J

ree tion operator on

R

, 5, 158

J

T

ree tion operator on

T

, 25 Ker

T

kernel of the operator T, 8

κ(φ)

mean motion of

φ ∈ GAP

,39

κ

l

(φ)

left mean motionof

φ ∈ GPAP

, 47

κ

r

(φ)

right mean motionof

φ ∈ GPAP

, 47

ℓ

0

zero extension operator from

L

p

_(R

+

)

into

L

p

_(R)

,2

ℓ

2

spa e of all innitesequen es

{ξ

i

}

∞

i=1

su h that

∞

X

i=1

|ξ

i

|

2

<

∞

, 6

ℓ

e

even extension operator from

L

p

_(R

+

)

into

L

p

_(R)

,7

ℓ

o

odd extensionoperatorfrom

L

p

_(R

+

)

into

L

p

_(R)

, 7

L(X, Y )

linear spa e of allbounded linear operators from the Bana h spa e

X

into the Bana hspa e

Y

L

∞

_(Ω)

L

p

_(Ω)

Lebesgue spa e of

p

th

-power-integrablefun tionson

Ω

, 2

L

p

+

(R)

subspa e of

L

p

_(R)

formed by all the fun tions supported in the losure of

R

₊

, 2

L

p

₋

(R)

subspa e of

L

p

_(R)

formed by all the fun tions supported in the losure of

R

−

, 2

L

p

even

(T)

set of all fun tions

φ ∈ L

p

_(T)

su h that

φ(t) = φ(t

−1

₎

, 58

L

p

_JT

(T)

=

Im

P

JT

|

L

p

(T)

, 58

M(φ)

Bohr meanvalue of

φ

(or mean value of

φ

),36

M

p

_(R)

set of all Fouriermultiplierson

L

p

_(R)

, 6

n(T )

dimension of the kernel of the operator

T

,8

N

₀

_{= {0, 1, 2, 3, . . .}}

,144

Ω

open set of

R

,2

Ω(φ)

Bohr-Fourierspe trum of

φ

, 38

P

set of all trigonometri polynomialson

T

, 30

P

orthogonal proje tion of

L

2

_(R)

onto

H

2

+

(R)

, 24

P

T

orthogonal proje tion of

L

2

_(T)

onto

H

2

+

(T)

, 24

P

+

anoni al proje tor from

L

p

_(R)

onto

L

p

+

(R)

, 2

P

₋

anoni al proje tor from

L

p

_(R)

onto

L

p

−

(R)

, 2

P

JT

=

I+JT

₂

, 58

PAP

algebraof pie ewise almost periodi fun tionson

R

,47

P C

algebraof pie ewise ontinuous fun tions on

˙

R

, 46

P C

0

lassof allpie ewise ontinuous fun tionsforwhi hboth limitsat

−∞

and at

+∞

are equal to zero, 46

P C(T)

algebraof pie ewise ontinuous fun tions on

T

, 85

ψ

#

_ψ

#

_{: ˙}

_{R × [0, 1] → C}

,

ψ

#

_{(x, µ) = (1 − µ) ψ(x − 0) + µ ψ(x + 0)}

, 96

ψ

p

_ψ

p

_{: ˙}

_{R × R → C}

,

ψ

p

(x, µ) = (1 − σ

p

(µ)) ψ(x − 0) + σ

p

(µ) ψ(x + 0)

, 98

R

set of real numbers

R

₊

positive half-line, 2

R

−

negative half-line, 2

˙

R

_{= R ∪ {∞}}

, 41

R

_{= R ∪ {±∞}}

, 42

R(φ)

essential range of

φ

, 28

ℜe(z)

real part of the omplex number

z

r

+

restri tion operator from

L

p

_(R)

into

L

p

_(R

+

)

, 2

S

Cau hy singularintegral operator on

L

2

_(R)

, 24

S

T

Cau hy singularintegral operator on

L

2

_(T)

, 24

S(R)

S hwartz spa eof all rapidly de reasing fun tionson

R

, 3

SAP

algebraof the semi-almostperiodi fun tions,42

SAP

p

smallest losed subalgebra of

M

p

_(R)

that ontains

AP

p

and

C

p

(R)

, 96

sp T

spe trumof the operator

T ∈ L(X)

, 28

sp

ess

T

essential spe trum ofthe operator

T ∈ L(X)

, 28

σ

p

σ

p

(µ) =

1

₂

+

1

₂

coth

h

π



i

p

+ µ

i

(µ ∈ R)

,97

T

unit ir le,3

T

₊

_{= {t ∈ T : ℑm t > 0}}

, 88

(T +H)

ν

Toeplitz plus Hankel operator, 25

(T −H)

ν

Toeplitz minusHankel operator, 25

W

Wiener algebra, 4

W

φ

Wiener-Hopf operator, 7

(W +H)

φ

Wiener-Hopf plus Hankel operator,7

(W −H)

φ

Wiener-Hopf minus Hankel operator,7

wind

φ

winding number of

φ

, 104, 105, 108, 135, 137, 138

k.k

A

norm in the linear spa e

A

f ∗ g

onvolution of

f

and

g

, 3

e

ϕ

a tion of the ree tion operatoron the fun tion

ϕ

, 5

∼

equivalen e relation,11

∗

∼

equivalen e after extensionrelation,11

⊕

dire t sum

∂A

boundaryof the set

A

[c

1

, c

2

]

line segment in the omplex plane between and in luding the endpoints

c

1

, c

2

∈ C

,78

Wiener-Hopf-Hankel operators (as well as their dis rete analogues based on Toeplitz

and Hankel operators) are well-known to play an important role in several applied areas.

E.g.,this isthe asein ertainwave dira tionproblems,digitalsignalpro essing,dis rete

inverses attering, andlinear predi tion. For on rete examplesofadetailedenrollmentof

thoseoperatorsinthese(andothers)appli ationswereferto[21,23, 32,49,52, 53,68,74℄.

In view of the needs of the appli ations, itis naturalto expe t a orresponding higher

interestinmathemati alfundamentalresear hforthosekindofoperators. Infa t,inre ent

yearsseveralauthorshave ontributedtothemathemati alunderstandingof

Wiener-Hopf--Hankeloperators(andtheirdis reteanalogues)underdierenttypesofassumptions( f.[6,

7,8,20,22,31,41,42,52,63,64,69℄). Asa onsequen e,thetheoryofWiener-Hopf-Hankel

operatorsisnowadayswelldevelopedforsome lassesofFouriersymbols(likeinthe aseof

ontinuousorpie ewise ontinuoussymbols). Inparti ular,the invertibilityand Fredholm

properties of su h kind of operators with pie ewise ontinuous Fourier symbols are now

well known (and of great importan efor the appli ations [23, 49, 52, 74℄). However, this

is not the ase for almostperiodi orsemi-almost periodi Fourier symbols whi h are also

importantinthe appli ations inview of their appearan e due to, e.g., (i)parti ular nite

boundaries in the geometry of physi al problems [21℄, or (ii) the needs of ompositions

withshift operatorswhi hintrodu ealmostperiodi and semi-almostperiodi elementsin

the Fourier symbols of those operators[19, 44℄.

Having this in mind, in this thesis we study the regularity properties of

periodi fun tions and a ting between

L

p

Lebesgue spa es (

1 < p < ∞

). In the ase

where both Wiener-Hopf and Hankel operators have the same Fourier symbol, we obtain

a Wiener-Hopf plus Hankel operator, and in the ase where Wiener-Hopf and Hankel

operatorshave symmetri Fourier symbols,we get a Wiener-Hopfminus Hankel operator.

To study the regularity properties of Wiener-Hopf-Hankel operators, we use ertain

relations between operators that allow us derive the regularity properties of the

Wiener--Hopf-Hankeloperatorsfromthe regularitypropertiesofothersoperatorsforwhi hresults

on erning toregularity properties are known.

This thesis is organized as follows. In the rst hapter, after givingthe formal

deni-tion of the operators under study - the Wiener-Hopf-Hankel operators - several relations

between some lasses of onvolution type operators are investigated. The rst relation is

a multipli ativerelation between Wiener-Hopf plus Hankel operators, whi h is a

general-izationofthe well-known resultfor theWiener-Hopf operators. Usingthe samereasoning,

an analogue multipli ative relation between Wiener-Hopf minus Hankel operators is also

possible to derive. After this, we present a relation between Wiener-Hopf plus Hankel

operators and Wiener-Hopf operators (based on Wiener-Hopf minus Hankel and paired

operators). Sin e this relationis in fa ta

∆

relationafter extensionbetween the

Wiener--Hopf plus Hankel operatorand the Wiener-Hopfoperator,and therefore there istransfer

of the regularity propertiesfrom the Wiener-Hopf operatorto the Wiener-Hopf plus

Han-kel operator, this relation turns out to be a de isive result in the study of the regularity

properties of the Wiener-Hopf-Hankel operators. The last relations under investigation

in this hapter are two relations of equivalen e between Wiener-Hopf plus/minus Hankel

operators and Toeplitz plus/minus Hankel operators, this due to the equivalen e relation

existent between Wiener-Hopf and Toeplitz operators. Finally, we use these equivalen e

relationsbetween Wiener-Hopf-Hankel and Toeplitz-Hankel operators toderive ne essary

onditions for the semi-Fredholmproperty of Wiener-Hopf-Hankel operators.

Wiener-algebra of almost periodi fun tions, the algebra of semi-almost periodi fun tions, and

the algebra of pie ewise almost periodi fun tions. Some properties of these algebras are

presented aswell.

The main purpose of Chapter 3 is to establish an invertibility and semi-Fredholm

ri-terion for Wiener-Hopf plus Hankel operators with almost periodi Fourier symbols, and

a tingbetween

L

2

Lebesgue spa es. Westart by onsideringthe ase of Wiener-Hopfplus

HankeloperatorswithFouriersymbolsinthesubalgebraofalmostperiodi fun tions

AP W

and then we onsider the more general ase of Wiener-Hopf plus Hankel operators with

almost periodi Fourier symbols. To obtain the invertibility and semi-Fredholm riterion

weintrodu eafa torization on ept forthe almostperiodi Fouriersymbolssu h thatthe

properties of the fa tors will allow orresponding operator fa torizations. Using then the

multipli ativerelationpresented inChapter 1,a riterionfor the semi-Fredholmproperty

and for the one-sided and both-sided invertibility is therefore obtained upon ertain

in-di es of the fa torizations. Under su h onditions, the one-sided and two-sided inverses

of the operators are alsoobtained. Moreover, the introdu ed fa torizations alsoallowthe

expositionofdependen ies between theinvertibilityof Wiener-HopfandWiener-Hopfplus

HankeloperatorswiththesameFouriersymbol. AlltheseresultsholdtrueforWiener-Hopf

minus Hankel operators sin e we use an analogue multipli ative relationfor Wiener-Hopf

minusHankeloperators of that one used for Wiener-Hopf plus Hankel operators.

InChapter4,Wiener-Hopf-Hankeloperatorswithsemi-almostperiodi Fouriersymbols

are info us. Consideringinrstpla e theseoperatorsa tingbetween

L

2

Lebesgue spa es,

and motivated by the Sarason's Theorem, we obtain a generalization of the invertibility

and semi-Fredholm riteria a hieved inChapter 3 for Wiener-Hopf-Hankel operatorswith

almost periodi symbols - the so- alled Sarason's type theorem. This result is attained

by using the

∆

relation after extension and also the main idea of the approa h of R.

V. Dudu hava and A. I. Saginashvili [29℄. After settled the Sarason's type theorem, a

reformulation of the invertibility and semi-Fredholm riterionfor Wiener-Hopf plus

tors with almost periodi Fourier symbols without the assumption on the orresponding

fa torization of the Fourier symbol of the operator, being this also possible due to the

Sarason'stypetheorem. Inase ondpart ofthis hapter,we onsiderWiener-Hopf-Hankel

operators with semi-almost periodi Fourier symbols a ting between

L

p

Lebesgue spa es

(

1 < p < ∞

). Having by motivation the Dudu hava-Saginashvili's theorem, we a hieve a

generalizationof the Sarason's type theorem, the so- alledDudu hava-Saginashvili's type

theorem. The fundamental key to rea h this result is the

∆

relation after extension

be-tweentheWiener-HopfplusHankeloperatorandtheWiener-Hopfoperator. Alltheresults

mentioned above meana hara terization of the Fredholmproperty, and one-sided

invert-ibility of the Wiener-Hopf-Hankel operators, based on the mean motions and geometri

mean values of the almost periodi representatives of the Fourier symbols at minus and

plus innity. We end up this hapter with a se tion devoted to the Fredholm index of

the Wiener-Hopf-Hankeloperatorsunder studyin this hapter. Inthe rst ase wherethe

operatorsare onsidered a tingbetween

L

2

Lebesgue spa es,a Fredholmindex formula is

derived in terms of the winding number of the Fourier symbols of the operators, while in

the se ond ase where we onsider the operators a ting between

L

p

Lebesgue spa es, the

Fredholm index formulais given in terms of the winding number of ontinuous fun tions

onstru tedfromtheFouriersymbolsoftheoperators. Additionally,inthe aseof

Wiener--Hopf-Hankeloperatorsa tingbetween

L

2

Lebesguespa es, onditions fortheinvertibility

of these operatorsare alsoobtained.

Chapter 5 presents a riterion for the semi-Fredholm property and for the one-sided

invertibilityof Wiener-Hopf-Hankeloperatorswith pie ewisealmost periodi Fourier

sym-bols,a tingbetween

L

2

Lebesguespa es. This riterionisalsoobtainedupontheuseofthe

∆

relation after extension and, in virtue of the nature of pie ewise almost periodi fun -tions, it is based on the mean motionsand geometri mean values of the almost periodi

representatives of the Fourier symbols as well as on the dis ontinuities of ertain

Hankeloperators. Sin einthis hapterwearedealingwithoperatorshavingdis ontinuous

symbols,theFredholmindexformulaisalsointerpretedupondierent asesofsymmetries

of the dis ontinuities of the Fourier symbols. In ordertoexemplify the simpli ationthat

o urs in the Fredholm index formula due to the symmetries of the dis ontinuities of the

Fourier symbol, several examples are presented. Finally, the (both-sided) invertibility of

the operators instudy isalsodis ussed.

In the last hapter, motivated by a result due to R. G. Babadzhanyan and V. S.

Ra-binovi h that settles the (one-sided and both-sided) invertibility and the semi-Fredholm

property of Wiener-Hopf operators with matrix

AP W

Fourier symbols having Hausdor

sets bounded away from zero, we derive a generalization of the invertibility and

semi--Fredholm riteria obtained in Chapter 3. Here, we onsider Wiener-Hopf-Hankel

opera-tors a ting in

L

2

Lebesgue spa es with matrix APW Fourier symbols having a parti ular

Hausdor set bounded away from zero and obtain for these operators aninvertibility and

semi-Fredholm riteria. Similarlytothe resultobtained inTheorem3.2.1, theinvertibility

and semi-Fredholm riteria here presented is given in terms of the mean motion, in this

ase,dependingonaparti ularHausdorsetboundedawayfromzero(insteadofthemean

motion of the Fouriersymbol of the operatoras inTheorem 3.2.1).

As far as the author knows, the results presented in this thesis are new. Most of the

material is published or a epted for publi ation in journals or onferen e pro eedings

[58, 59, 60, 61℄, and the material whi h is not published or a epted for publi ation is

submitted for publi ationin journals [17, 57℄. Auxiliary results of other authors in luded

Convolution Type Operators

The study of relations between dierent lasses of operators is an important subje t

in Operator Theory sin e, in parti ular, itallows the transfer of properties from one lass

of operators to the other ones. Due to this transfer of properties and in order to a hieve

the invertibilityand semi-Fredholm riteriapresented in Chapters 3,4, 5and 6,the main

purpose of this hapter isthe study of some relationsbetween some lassesof onvolution

type operators. In rst pla e, we present a multipli ative relation between Wiener-Hopf

plus Hankel operators (whi h in fa tis a orresponding result for Wiener-Hopf plus

Han-kel operators of the well-known result for the Wiener-Hopf operators). In se ond pla e,

it is exhibited a relation between Wiener-Hopf plus Hankel operators and Wiener-Hopf

operators (based on Wiener-Hopf minus Hankel and paired operators). This relation will

perform a signi ant role in the obtainment of several results be ause it allows to study

the regularity propertiesof Wiener-Hopf operators, and then transferthe regularity

prop-erties to Wiener-Hopf-Hankel operators. Finally, in what on erns to relations between

onvolution type operators, sin e Wiener-Hopf and Toeplitz operators in Hilbert spa es

areequivalentoperators,itisalsopresented awaytorelateWiener-Hopf-Hankeloperators

with Toeplitz-Hankel operators. Here and in what follows, we will simply all

Wiener--Hopf-HankeloperatorstobothWiener-Hopfplus Hankel,and Wiener-HopfminusHankel

Toeplitz plus Hankel, and Toeplitz minus Hankel operators. Using equivalen e relations

between Wiener-Hopf-Hankel and Toeplitz-Hankel operators, we end up this hapter by

deriving ne essary onditions for the semi-Fredholm property of Wiener-Hopf-Hankel

op-erators.

Before presenting the results mentioned above, we will rst introdu e the formal

de-nitions of the operators under study as well as some of the notation that will be used

throughout this thesis.

1.1 Some denitions, notations, and histori al notes

Let

Ω

be an open set of

R

. For

1 ≤ p < ∞

,

L

p

_(Ω)

represents the Bana h spa e of

Lebesgue measurable omplex-valued fun tions

ϕ

on

Ω

su h that

|ϕ|

p

is integrable. This

spa e isendowed with the norm

kϕk

L

p

_(Ω)

:=

Z

Ω

|ϕ(η)|

p

_dη



1

p

.

(1.1.1) Furthermore,let

L

p

+

(R)

denotethesubspa eof

L

p

_(R)

formedbyallthefun tionssupported

in the losure of

R

+

:= (0, +∞)

, and

L

p

−

(R)

represent the subspa e of

L

p

_(R)

formed by

all the fun tions supported in the losure of

R

−

:= (−∞, 0)

. We will use the anoni al proje tors

P

+

and

P

−

that map

L

p

_(R)

onto

L

p

+

(R)

, and

L

p

_(R)

onto

L

p

−

(R)

, respe tively. Considering

r

+

being the restri tion operator from

L

p

_(R)

into

L

p

_(R

+

)

and

ℓ

0

the zero extension operator from

L

p

_(R

+

)

into

L

p

_(R)

, wehave

P

+

= ℓ

0

r

+

.

(1.1.2) Consideralso

L

∞

_(Ω)

astheBana hspa eofLebesguemeasurableandessentiallybounded

( omplex-valued)fun tions on

Ω

. The norm in

L

∞

_(Ω)

isgiven by

kφk

L

∞

_(Ω)

:=

ess

sup |φ(η)|,

(1.1.3)

where

For the ase of Lebesgue spa es dened on the unit ir le

T := {z ∈ C : |z| = 1}

, let

L

p

_{(T) (1 ≤ p < ∞)}

stand for the Bana h spa e of Lebesgue measurable omplex-valued

fun tions

ϕ

on

T

su h that

|ϕ|

p

is integrable. The norm in

L

p

_(T)

is given by

kϕk

L

p

_(T)

:=

1

2π

Z

2π

0

|ϕ(e

iθ

)|

p

_dθ



1

p

.

(1.1.5) Moreover, let

L

∞

_(T)

denote the Bana h spa e of Lebesgue measurable and essentially

bounded ( omplex-valued) fun tions dened on

T

.

L

∞

_(T)

is endowed with the essential

supremum norm

kφk

L

∞

_(T)

:=

ess

sup |φ(η)|,

(1.1.6)

where, inthis ase,

ess

sup |φ(η)| := inf{α : |{η ∈ T : |φ(η)| > α}| = 0}.

(1.1.7)

Let

F

denotetheFouriertransformation dened inthe S hwartz spa e

S(R)

ofrapidly

de reasing fun tions on

R

by

Fϕ(ξ) :=

Z

R

e

iξη

ϕ(η) dη,

_{ξ ∈ R ,}

(1.1.8) and let

F

−1

denote the inverse of the Fourier transformation

F

, also dened on

S(R)

,

given by

F

−1

ψ(η) :=

1

2π

Z

R

e

−iξη

ψ(ξ) dξ,

_{η ∈ R .}

(1.1.9)

A onvolution operator is dened by

Cϕ(ξ) := (K ∗ ϕ)(ξ) =

Z

R

K(ξ − η)ϕ(η) dη,

ξ ∈ R,

(1.1.10)

where

K

is alled the onvolution kernel of

C

. Convolution operators may be onsidered

a ting in several fun tions spa es as well as their onvolution kernels that may also be

onsidered in dierent fun tions spa es. For instan e, we may onsider the onvolution

operator

C

a tingin

L

1

_(R)

with

K ∈ L

1

_(R)

. The onvolutionoperator

C

anbewrittenin

the form

with

φ

C

= FK

. From this, we see why

φ

C

is alled the Fourier symbol of the onvolution operator

C

. If in (1.1.10) instead of

R

we onsider the open set

Ω ⊂ R

, then we obtain a

onvolution type operator

Aϕ(ξ) := (K ∗ ϕ)(ξ) =

Z

Ω

K(ξ − η)ϕ(η) dη,

ξ ∈ Ω.

(1.1.12)

The probably best known onvolution type operators are the Wiener-Hopf operators

a ting between Lebesgue spa es on the half-line. We re all that the name Wiener-Hopf

operatorsisduetotheinitialworkofN.WienerandE.Hopfin1931[76℄whereareasoning

to solve integral equations whose kernels depend only on the dieren e of the arguments

was provided:

cf (x) +

Z

+∞

0

k(x − y)f(y)dy = g(x), x ∈ R

+

,

(1.1.13)

i.e.theso- alledintegralWiener-Hopfequations. Here

c ∈ C, k ∈ L

1

_(R)

and

f, g ∈ L

2

_(R

+

)

, where

c

and

k

are xed,

g

is given and

f

is the unknown element.

From those Wiener-Hopfequations arise the ( lassi al)Wiener-Hopf operators dened

by

W

φ

f (x) = cf (x) +

Z

+∞

0

k(x − y)f(y)dy , x ∈ R

+

,

(1.1.14)

where

φ = c + Fk

belongs tothe Wiener algebra. The Wiener algebrais dened by

W := {φ : φ = c + Fk, c ∈ C, k ∈ L

1

(R)}

(1.1.15)

and itis a Bana halgebra when endowed with the norm

kc + Fkk

W

:= |c| + kkk

_L

1

_(R)

(1.1.16)

and the usual multipli ation operation. The Wiener algebra is a subalgebra of

L

∞

_(R)

.

Having in mind the onvolution operation, the denition of the lassi al Wiener-Hopf

operators gives rise tothe following representation of the Wiener-Hopf operators

W

φ

:= r

+

F

−1

φ · F : L

2

+

(R) → L

2

(R

+

) .

(1.1.17) Making use of the anoni al proje tor on

L

2

+

(R)

, the Wiener-Hopf operators on

L

2

+

(R)

may alsobe writtenin the form

where

A

is the translation invariant operator

F

−1

_{φ · F}

. Lookingnow to the stru ture of

the operators in (1.1.17) and (1.1.18), we re ognize that possibilities other than only the

Wiener algebra an be onsidered for the so- alled Fourier symbols

φ

of the Wiener-Hopf

operators. Namely,we may onsider to hoose

φ

among the

L

∞

_(R)

elements.

Withinthe ontextof (1.1.13)and(1.1.14),theHankelintegral operatorshavetheform

Hf (x) =

Z

+∞

0

k(x + y)f (y)dy ,

_{x ∈ R}

+

(1.1.19) for some

k ∈ L

1

_(R)

. In this ase, itis well-known that

H

, asan operatordened between

L

2

spa es, is a ompa t operator. However, as seen above, itis also possible to provide a

rigorousmeaning tothe expression (1.1.19) when the kernel

k

isa temperate distribution

whose Fourier transform belongs to

L

∞

_(R)

. In su h ase, Hankel operators admit the

representation

H

φ

:= r

+

F

−1

φ · FJ : L

2

+

(R) → L

2

(R

+

) .

(1.1.20) Here and inwhat follows,

J

isthe ree tion operator on

R

given by the rule

Jϕ(x) = e

ϕ(x) = ϕ(−x), x ∈ R .

(1.1.21)

Likein (1.1.18),we may write Hankel operatorson

L

2

+

(R)

as

P

+

AJ = ℓ

0

r

+

F

−1

φ · FJ : L

2

+

(R) → L

2

+

(R) ,

(1.1.22) being

A = F

−1

_{φ · F}

. We would like to mention that the dis rete analogue of

H

has its

roots in the year of 1861 with the Ph.D. thesis of H. Hankel [39℄. There the study of

nite matri es with entries depending only on the sum of the oordinates was proposed.

Determinantsof innite omplexmatri eswith entries dened by

a

jk

= a

j+k

(for

j, k ≥ 0

, and where

a = {a

j

}

j

≥0

is a sequen e of omplex numbers) were also studied. For these (innite) Hankel matri es, one of the rst main results was obtained by L. Krone ker in

1881 [47℄ when hara terizing the Hankel matri es of nite rank as the ones that have

orresponding power series,

a(z) =

∞

X

j=0

a

j

z

j

, whi h are rational fun tions. In 1906, D. Hilbert proved that the operator(indu ed by the famous Hilbert matrix),

H : ℓ

2

→ ℓ

2

, {b

j

}

j

_≥0

7→

(

_∞

X

k=0

b

k

j + k + 1

)

j

≥0

,

(1.1.23)

is bounded on the spa e

ℓ

2

of all innite sequen es

{ξ

i

}

∞

i=1

su h that

∞

X

i=1

|ξ

i

|

2

< ∞

. This resultmaybeviewed astheoriginof(dis rete)Hankeloperators,asnaturalobje tsarising

fromHankelmatri es. Lateron,in1957,Z.Neharipresenteda hara terizationofbounded

Hankel operators on

ℓ

2

[55℄. Due tothe importan eof su h hara terization, we may say

that itmarks the beginningof the ontemporary periodof the study of Hankel operators.

After a brief note on the origins of Wiener-Hopf and Hankel operators and before

presentthedenitionofWiener-HopfplusHankeloperatorsandWiener-HopfminusHankel

operators,let usre all the denition of Fourier multiplieron

L

p

_(R)

.

A fun tion

φ ∈ L

∞

_(R)

is alledaFouriermultiplier on

L

p

_(R)

if the operator

F

−1

_{φ · F}

, a tingon

L

2

_{(R) ∩ L}

p

_(R)

, extends by ontinuity toa bounded operator on

L

p

_(R)

. The set

of all Fouriermultiplierson

L

p

_(R)

is denoted by

M

p

_(R)

and it is a Bana h algebrawhen

endowed with the norm

kφk

M

p

_(R)

:= kF

−1

φ · Fk

_L(L

p

_(R))

(1.1.24)

andpointwise multipli ation. Further,the set ofallFouriermultiplierson

L

2

_(R)

oin ides with

L

∞

_(R)

, i.e.,

M

2

(R) = L

∞

(R).

(1.1.25)

We are now in position to present the main obje ts of this thesis - the Wiener-Hopf

plus Hankel operators and the Wiener-Hopf minus Hankel operators. As the names

sug-gest,theseoperatorsresultfrom ombinationsbetweenWiener-HopfandHankeloperators.

Combinationsofsu hkindappearforthersttimein1979inthe lassi alworkofS.Power

[62℄, wherea study of the spe traand essentialspe tra of Hankel operatorswas presented

by investigating the

C

∗

algebra generated by Toeplitz and Hankel operators (in the two

ases ofpie ewise ontinuous symbols and almostperiodi symbols). Althoughinthe

ma-jorpartofthisthesiswewill onsider operatorsa tingbetween

L

2

Lebesguespa es,wewill

now dene Wiener-Hopf plus Hankel operators and Wiener-Hopf minusHankel operators

a tingbetween

L

p

For

φ ∈ M

p

_(R)

, let

W

φ

and

H

φ

be the Wiener-Hopf and Hankel operators dened by

W

φ

:= r

+

F

−1

φ · F : L

p

+

(R) → L

p

(R

+

)

(1.1.26)

H

φ

:= r

+

F

−1

φ · FJ : L

p

+

(R) → L

p

(R

+

),

(1.1.27)

respe tively. The Wiener-Hopf plus Hankel operator is given by

(W +H)

φ

:= W

φ

+ H

φ

: L

p

+

(R) → L

p

(R

+

),

(1.1.28)

and the Wiener-Hopf minus Hankeloperatoris dened by

(W −H)

φ

:= W

φ

− H

φ

: L

p

+

(R) → L

p

(R

+

).

(1.1.29) ForaBana hspa e

Z

,let

I

Z

representtheidentityoperatoron

Z

. A ordingto(1.1.26), (1.1.27)and (1.1.28), we have

(W +H)

φ

= r

+

(F

−1

φ · F + F

−1

φ · FJ) = r

+

F

−1

φ · F(I

L

p

₊

(R)

+ J).

(1.1.30) Furthermore, sin e

I

L

p

₊

(R)

+ J = ℓ

e

r

+

,

(1.1.31) where

ℓ

e

denotes the even extension operator from

L

p

_(R

+

)

into

L

p

_(R)

, we may write the

Wiener-Hopf plus Hankel operator as

(W +H)

φ

= r

+

F

−1

φ · Fℓ

e

r

+

.

(1.1.32)

Combiningnow (1.1.26),(1.1.27) and (1.1.29), we have

(W −H)

φ

= r

+

(F

−1

φ · F − F

−1

φ · FJ) = r

+

F

−1

φ · F(I

L

p

₊

(R)

− J).

(1.1.33) Denoting by

ℓ

o

the odd extension operator from

L

p

_(R

+

)

into

L

p

_(R)

,it follows that

I

L

p

₊

(R)

− J = ℓ

o

r

+

.

(1.1.34)

In this way, we may write the Wiener-Hopf minusHankeloperator as

From(1.1.32)and (1.1.35),weobservethatWiener-Hopf-Hankeloperatorsmaybewritten

as onvolutiontypeoperatorswith symmetry. Noti e that onvolution type operators with

symmetry (CTOS) are operatorsof the form

T = r

+

Al

c

: L

p

(R

+

) → L

p

(R

+

) ,

(1.1.36)

where

A = F

−1

_{φ · F : L}

p

_{(R) → L}

p

_(R)

is atranslation invariantoperator and supposed to

bebounded invertible,and

ℓ

c

denotes even or oddextension asa ontinuous operatorfrom

L

p

_(R

+

)

into

L

p

_(R)

. CTOS may also be onsidered in Bessel potential spa es. Operators

of this type were rst studied in[51℄and then in[22℄.

1.2 Regularity properties

Let

X

and

Y

be two Bana hspa es and onsider

T ∈ L(X, Y )

. The kernel Ker

T

and

the image Im

T

of the operator

T

are dened by

Ker

T := {x ∈ X : T x = 0},

Im

T := {T x : x ∈ X}.

(1.2.37)

Ker

T

and Im

T

are linearsubspa es of

X

and

Y

, respe tively. Moreover, Ker

T

isa losed

subspa e. IfIm

T

is alsoa losedsubspa e, the operator

T

issaid tobenormally solvable.

In this ase, the okernel of T isdened as

Coker

T := Y /

Im

T.

(1.2.38)

Fora normallysolvable operator

T

, the de ien y numbersof

T

are dened as

n(T ) := dim

Ker

T,

d(T ) := dim

Coker

T.

(1.2.39)

Con erning the kernel of

T

, it holds that Ker

T

satises (at least) one of the following

onditions:

(1)

n(T ) = 0

;

(3) Ker

T

is omplementable in

X

;

(4) orKer

T

is losed in

X

.

A similar hara terization an be donefor the image of

T

, thatis, Im

T

satises(at least)

one of the following onditions:

(i)

d(T ) = 0

;

(ii)

d(T ) < ∞

;

(iii) Im

T

is omplementablein

Y

;

(iv) orIm

T

is losed in

Y

.

Combiningthese properties on erning to the kernel of

T

with the properties referring to

the image of

T

, we obtain several lasses that are alled regularity lasses [16, 18, 73℄. In

what follows,we willdene some of these regularity lasses.

The operator

T

is said to be a Fredholm operator if it is normally solvable and

n(T )

and

d(T )

are nite. In this ase, the Fredholm index of

T

is dened by

Ind

T := n(T ) − d(T ).

(1.2.40)

The operator

T

is said to be a semi-Fredholm operator if it is normally solvable and at

least one of the de ien y numbers

n(T )

and

d(T )

is nite. A semi-Fredholm operator

is said to be

n

-normal if

n(T )

is nite, and

d

-normal if

d(T )

is nite. In the ase where

only one of the de ien y numbers is nite, the operator

T

is said to be a properly

semi--Fredholm operator. In this ase, the operator

T

is said to be properly

n

-normal if

n(T )

is nite and

d(T )

is innite, and properly

d

-normal if

d(T )

is nite and

n(T )

is innite.

Forsemi-Fredholmoperators,the indexformula(1.2.40)isalsowell-dened. In parti ular,

forproperlysemi-Fredholmoperators,we have Ind

T = −∞

if

T

isproperly

n

-normaland

Ind

T = +∞

if

T

is properly

d

-normal.

We point out that in German and Russian literature, (semi-)Fredholm operators are

whowasthersttodis overthatsingularintegraloperatorswithnonvanishing ontinuous

symbols are normally solvable, and have nite kernel and okernel dimensions.

Still on erning to semi-Fredholm operators, we present next a property about the

index of these operators whi h willbe used later on.

Theorem 1.2.1. ( f. [35, Ÿ6.7℄) If

T ∈ L(X, Y )

is a Fredholm operator (resp. properly

n

-normal, properly

d

-normal), thenthere existsa number

δ > 0

su hthat, forall operators

M ∈ L(X, Y )

with

kMk < δ

,

A + M ∈ L(X, Y )

is a Fredholm operator (resp. properly

n

-normal, properly

d

-normal)and Ind

(A + M) =

Ind

A

.

Weend up this se tion by introdu ingthe denition of reexive generalized

invertibil-ity. As we will see in a moment, it is possible to hara terize the regularity lass of all

operators

T

su hthat Ker

T

is a omplementable subspa e of

X

and Im

T

is a losed and

omplementable subspa e of

Y

in terms of the reexive generalizedinvertibilityof

T

.

Let

T : X → Y

be abounded linear operatora ting between Bana h spa es.

T

is said

tobe reexive generalized invertible if thereexists abounded linear operator

T

−

_{: Y → X}

su h that

T T

−

T = T

and

T

−

_{T T}

−

_{= T}

−

_.

(1.2.41)

Inthis ase,theoperator

T

−

isreferredtoasthereexivegeneralizedinverse (or

pseudoin-verse)of

T

. From the denition ofreexive generalizedinvertibleoperator,it follows that

a linear bounded one-sided or two-sided invertible operator is also a reexive generalized

invertibleoperatorandoneofitsreexivegeneralizedinversesisthe one-sidedortwo-sided

inverse, respe tively. Ingeneral,reexive generalized inverses are not unique. However, in

the ase where

T

is invertible, the reexive generalized inverses are unique and oin ide

with the inverse of

T

.

Finally,and about the hara terization (mentionedabove)of the regularity lass of all

operators

T

su hthat Ker

T

is a omplementable subspa e of

X

and Im

T

is a losed and

omplementable subspa e of

Y

, wehavethe following:

T

isreexivegeneralized invertible

if and only if Ker

T

is a omplementable subspa e of

X

and Im

T

is a losedand

generalized invertible.

1.3 Relations between onvolution type operators

1.3.1 Relations between bounded linear operators

In what follows, onsider

T : X

1

→ X

2

and

S : Y

1

→ Y

2

two bounded linear operators a ting between Bana h spa es.

The operators

T

and

S

are said to be equivalent, and we will denotethis by

T ∼ S

, if

there are two boundedly invertible linear operators,

E : Y

2

→ X

2

and

F : X

1

→ Y

1

, su h that

T = E S F.

(1.3.42)

It dire tly follows from (1.3.42) that if two operators are equivalent, then they belong to

thesameregularity lass. Namely,oneofthese operatorsisinvertible,one-sidedinvertible,

Fredholm, (properly)

n

-normal, (properly)

d

-normal or normally solvable, if and only if

the other operator enjoys the same property.

Anoperatorrelationthatgeneralizestheoperatorequivalen erelationistheequivalen e

after extension relation. The operators

T

and

S

are said tobe equivalent after extension,

and we will denote this by

T

∗

∼ S

, if there exist two Bana h spa es

W

and

Z

su h that

T ⊕ I

W

and

S ⊕ I

Z

are equivalent operators, i.e.,





T

0

0 I

W



 = E





S

0

0 I

Z



 F,

(1.3.43)

for invertible bounded linearoperators

E : Y

2

× Z → X

2

× W

and

F : X

1

× W → Y

1

× Z

. As we an easily see, the operator equivalen e relation orresponds to the ase where the

extension spa es

W

and

Z

are hosen to be the trivial spa e (in the equivalen e after

extension relation). Like inthe equivalen e ase, two equivalentafter extension operators

belong to the same regularity lass.

Another known relation between bounded linear operators is the matri ial oupling.

linear operators





T

T

2

T

1

T

0



 : X

1

× Y

2

→ X

2

× Y

1

,

(1.3.44)





S

0

S

1

S

2

S



 : X

2

× Y

1

→ X

1

× Y

2

(1.3.45) su h that





T

T

2

T

1

T

0





−1

=





S

0

S

1

S

2

S



 .

(1.3.46)

In [3℄, H. Bart, I. Gohberg, and M. A. Kaashoek proved that matri ial oupling implies

equivalen e after extension. Later on, in [4℄, H. Bart and V. E. Tsekanovshii proved that

the onverse is alsotrue, and therefore,matri ial ouplingand equivalen eafter extension

amount tothe same. That is tosay that twooperators a ting between Bana h spa es are

equivalent after extensionif and only if they are matri ially oupled.

Con erning additional operators relations, we nd the

∆

relation and the

∆

relation

afterextension thatwereintrodu edbyL.P.CastroandF.-O.Spe k in[15,18℄. Inregard

to the

∆

relation, that an be viewed as a generalization of the equivalen e relation, we

saythat

T

is

∆

related with

S

ifthereisabounded linearoperatora tingbetween Bana h

spa es

T

∆

: X

1∆

→ X

2∆

and two invertible bounded linear operators

E : Y

2

→ X

2

× X

2∆

and

F : X

1

× X

1∆

→ Y

1

su h that





T

0

0 T

∆



 = ESF.

(1.3.47)

As we an see here, in general the extension is not made with the identity operator, like

in the equivalen e after extension relation ( f. (1.3.43)), but with a third new operator

T

∆

. Therefore, due to the presen e of three operators,

T

,

T

∆

and

S

, the relation (1.3.47) is alled

∆

relation.

In its turn, the

∆

relation after extension appears a generalizationof the

∆

relation.

Thus,

T

issaid tobe

∆

related afterextension with

S

ifthereisaboundedlinearoperator

linear operators

E : Y

2

× Z → X

2

× X

2∆

and

F : X

1

× X

1∆

→ Y

1

× Z

su h that





T

0

0 T

∆



 = E





S

0

0 I

Z



 F.

(1.3.48)

From (1.3.47)and (1.3.48)it follows that if we have

T

being

∆

relatedwith

S

or

T

being

∆

related after extension with

S

, then the transfer of regularity properties an only be guaranteed in one dire tion, that is, from operator

S

to operator

T

, as stated in [18,

Theorem 2.1℄. It is lear that this restri tion o urs only here and in ontrast to what

happenswiththe transferofregularitypropertiesbetween twoequivalent(after extension)

operators,where the transfer an bedone in both dire tions.

1.3.2 Fa torization of Wiener-Hopf plus Hankel operators

Inordertoobtaininvertibilityand semi-Fredholm riteriaforWiener-HopfplusHankel

operators with almost periodi Fourier symbols, and a ting between

L

2

Lebesgue spa es,

we need a fa torization theory for these operators. In this sense, in what follows, we

will onsider Wiener-Hopf, Hankel and Wiener-Hopf-Hankel operators a ting between

L

2

Lebesgue spa es. Wepointout that similar resultshold true for operatorsa ting between

L

p

Lebesgue spa es.

We will start by re alling two well-known relations between Wiener-Hopf and Hankel

operators that arise fromthe Wiener-Hopf and Hankeloperator theory.

Proposition 1.3.1. ([12, Proposition 2.10℄,[75℄) Let

φ, ϕ ∈ L

∞

_(R)

. Then

W

φϕ

= W

φ

ℓ

0

W

ϕ

+ H

φ

ℓ

0

H

ϕ

e

,

(1.3.49)

H

φϕ

= W

φ

ℓ

0

H

ϕ

+ H

φ

ℓ

0

W

ϕ

e

.

(1.3.50)

Proof. Takingintoa ount that

I

L

2

_(R)

= P

₊

+ P

₋

= P

₊

+ P

₋

JJP

₋

(1.3.51)

and

itholds

I

L

2

_(R)

= P

₊

+ JP

₊

P

₊

J = P

₊

+ JP

₊

J.

(1.3.53) Therefore, it follows

W

φϕ

= r

+

F

−1

(φϕ) · F

= r

+

F

−1

φ · FF

−1

ϕ · F

= r

+

F

−1

φ · F(P

+

+ JP

+

J)F

−1

ϕ · F

= r

+

F

−1

φ · FP

+

F

−1

ϕ · F + r

+

F

−1

φ · FJP

+

JF

−1

ϕ · F

= r

+

F

−1

φ · F ℓ

0

r

+

F

−1

ϕ · F + r

+

F

−1

φ · FJ ℓ

0

r

+

F

−1

ϕ · FJ

e

= W

φ

ℓ

0

W

ϕ

+ H

φ

ℓ

0

H

ϕ

e (1.3.54) and

H

φϕ

= r

+

F

−1

(φϕ) · FJ

= r

+

F

−1

φ · FF

−1

ϕ · FJ

= r

+

F

−1

φ · F(P

+

+ JP

+

J)F

−1

ϕ · FJ

= r

+

F

−1

φ · FP

+

F

−1

ϕ · FJ + r

+

F

−1

φ · FJP

+

JF

−1

ϕ · FJ

= r

+

F

−1

φ · F ℓ

0

r

+

F

−1

ϕ · FJ + r

+

F

−1

φ · FJ ℓ

0

r

+

F

−1

ϕ · F

e

= W

φ

ℓ

0

H

ϕ

+ H

φ

ℓ

0

W

ϕ

e

.

(1.3.55)

HavinginmindProposition1.3.1andthinkingonthe lassofWiener-HopfplusHankel

operators, it is natural to expe t a orresponding formula for this lass of operators. In

fa t,forWiener-HopfplusHankeloperatorsalsoholdsananalogueof formula(1.3.49)like

itis stated inthe following proposition.

Proposition 1.3.2. Let

φ, ϕ ∈ L

∞

_(R)

. Then

(W +H)

φϕ

= (W +H)

φ

ℓ

0

(W +H)

ϕ

+ H

φ

ℓ

0

(W +H)

ϕ

e

−ϕ

.

Proof. Adding (1.3.49)and (1.3.50),we obtain

(W +H)

φϕ

= W

φ

ℓ

0

(W +H)

ϕ

+ H

φ

ℓ

0

(W +H)

ϕ

e

.

(1.3.57)

Addingand subtra ting

H

φ

ℓ

0

(W +H)

ϕ

onthe right-handsideof the lastidentity,itholds

(W +H)

φϕ

= (W +H)

φ

ℓ

0

(W +H)

ϕ

+ H

φ

ℓ

0

(W +H)

ϕ

e

−ϕ

.

(1.3.58) Let

C

+

:= {z ∈ C : ℑm z > 0}

and

C

−

:= {z ∈ C : ℑm z < 0}

. Asusual,let

H

∞

_(C

±

)

denote the set of all bounded and analyti fun tions in

C

±

. Fatou's Theorem asserts that fun tions in

H

∞

_(C

±

)

have non-tangential limits on

R = ∂C

±

almost everywhere. In this sense, let

H

∞

±

(R)

be the set of all fun tions in

L

∞

_(R)

that are non-tangential

limitsof elementsin

H

∞

_(C

±

)

.

H

∞

+

(R)

and

H

∞

−

(R)

are losed subalgebras of

L

∞

_(R)

. For

0 < p < ∞

,

H

p

_(C

±

)

denote the set of all fun tions

φ

whi h are analyti in

C

±

and su h that

sup

±y>0

Z

R

|φ(x + iy)|

p

dy < ∞ .

(1.3.59)

Like in the ase of

H

∞

_(C

±

)

, by Fatou's theorem, it also holds that fun tions in

H

p

_(C

±

)

have non-tangential limits almost everywhere on

R

. The set of all these non-tangential

fun tions isdenoted by

H

p

±

(R)

. For

1 < p < ∞

,

H

p

±

(R)

is a losed subspa e of

L

p

_(R)

. It is well-known that: if

φ ∈ H

∞

+

(R)

, then

H

e

φ

= 0;

(1.3.60) if

φ ∈ H

∞

−

(R)

, then

H

φ

= 0.

(1.3.61)

Thesetwosimplefa tsareveryimportantinthetheoryofWiener-Hopfoperators. Namely,

it is possible to fa torize the Wiener-Hopf operator if its Fourier symbol admits a

fa tor-ization wherethe left fa tor belongsto

H

∞

−

(R)

and the rightfa tor belongs to

H

∞

+

(R)

. Proposition 1.3.3. [14, Ÿ9.5℄ If

ϕ

±

∈ H

∞

±

(R)

and

ψ ∈ L

∞

_(R)

, then

W

ϕ−

ψ ϕ+

= W

ϕ−

ℓ

0

W

ψ

ℓ

0

W

ϕ+

.

(1.3.62)