ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2008, Vol. 29, No. 3, pp. 119–129. c© Pleiades Publishing, Ltd., 2008.

Reduction of Singular Integral Operatorswith Flip and Their Fredholm Property

L. P. Castro* and E. M. Rojas**

(Submitted by O.E. Tikhonov)Research Unit Mathematics and Applications, Department of Mathematics,

University of Aveiro, 3810-193 Aveiro, PortugalReceived February 2, 2008

Abstract—This paper deals with singular integral operators with a reverting orientation Carlemanshift defined on the classic Lebesgue space and having essentially bounded functions as coefficients.We will use similarity relations to show that the mentioned operators are equivalent to matrix Toeplitzplus Hankel operators acting on the corresponding Hardy space. The main purpose is to extractFredholm characteristics of the initial operators (in the form of necessary and sufficient conditions).Namely, Fredholm criteria are obtained for some of the operators under study when they havecoefficients in the classes of continuous, piecewise continuous, and semi-almost-periodic functions.In addition, Fredholm index formulas are also provided in some of these cases.

2000 Mathematics Subject Classification: 47G10, 47B35, 47A53, 45E05, 45F15, 47A20DOI: 10.1134/S1995080208030025

Key words and phrases: Singular integral operator, Carleman shift, Toeplitz operator, Hankeloperator, Fredholm property.

1. INTRODUCTION

This paper is devoted to the study of singular integral operators Awith reverting orientation Carleman

shift ( ˜Jϕ)(t) =1tϕ

(

1t

)

on the space Lp(T), 1 < p < ∞, defined as

A = a0ILp(T) + a1ST + b0˜J + b1ST

˜J, (1.1)

where a0, a1, b0, and b1 belong to L∞(T). Here, ILp(T) denotes the identity operator and ST is theCauchy singular integral operator along the unit circle T which is defined by

(STf)(t) =1πi

∫

T

f(τ)τ − t

dτ, (1.2)

where the integral is understood in the Cauchy principal value. For p ∈ (1,+∞), by Lp(T) we aredenoting the usual Banach space of all Lebesgue measurable complex valued functions on T withabsolutely integrable pth power. By L∞(T) we denote the Banach algebra of all essentially bounded andLebesgue measurable complex valued functions defined on T (endowed with the essential supremumnorm).

The techniques to study singular integral operators vary, and we can find a vast quantity of worksin this subject. The singular integral operators without shift (see, for instance, [16]) are much moredeveloped than the corresponding ones with shift (cf., e.g., [15]). The reasons for such a fact are diverse.Factorization theory and operator relations are recognized to be some of the most frequent and

*E-mail address: castro@ua.pt**E-mail address: edixonr@ula.ve

119

120 CASTRO, ROJAS

powerful techniques to study singular integral operators (cf., e.g., [12, 13, 14] for factorization theoryand [6, 9, 10, 11] for operator relations). In this paper, we use operator relation techniques to derivenecessary and sufficient conditions for the Fredholm property of the above-mentioned singular integraloperators with shift (when within certain classes of coefficients).

In view of defining the exact class of singular integral operators with shift which we are interested in,we start by presenting some notation.

Let us consider the Riesz projection P := (ILp(T) + ST)/2. The Hardy space is defined as Hp(T) :=P (Lp(T)). For a Banach space X, we denote by [X]2 the direct sum of two copies of X, and the elementsin this space can be written as column vectors of length two with entries on X. L(X,Y ) denotes asusual the set of all bounded linear operators mapping the Banach space X into the Banach space Y , andL(X) ≡ L(X,X).

We recall that two bounded and linear operators T : X1 → X2 and S : Y1 → Y2 acting betweenBanach spaces are said to be equivalent [1] if there are two boundedly invertible linear operatorsE : Y2 → X2 and F : X1 → Y1 such that

T = ESF. (1.3)

When in the presence of the particular case E = F−1 in (1.3), we will say that we have a similarityrelation between operators T and S. For regularity properties of a linear and bounded operator actingbetween Banach spaces, we will mean those properties that arise from a direct influence of the kerneland image of that operator. Thus, it follows from (1.3) that, if two operators are equivalent, then theybelong to the same regularity class [7, 19].

Let us also recall that a normally solvable operator T : X −→ X is called a Fredholm operator ifn(T ) := dim kerT < ∞ and d(T ) := dimX/ImT < ∞. The Fredholm index is then defined as IndT :=n(T ) − d(T ).

In the first part of this paper, we will construct a similarity transformation between the singularintegral operator A with reverting orientation Carleman shift ˜J (frequently, this last operator is alsocalled flip operator or weighted Carleman shift) on the space Lp(T), 1 < p < ∞, and a matrix Toeplitzplus Hankel operator with multipliers a and b on [Hp(T)]2 defined by

Ta + Hb := Pa · P + Pb · ˜JP, (1.4)

where the first operator P in (1.4) is understood as the Riesz projection from [Lp(T)]2 onto [Hp(T)]2, andthe last operator P in (1.4) should be understood as the embedding operator from [Hp(T)]2 to [Lp(T)]2.

Afterwards, in the second part, we will use this equivalence relation in order to characterize theFredholm characteristics of the singular integral operator with flip A for the coefficients a0, a1, b0, and b1

defined on different classes of functions on L∞(T). In addition, for some cases, Fredholm index formulaswill also be derived.

2. REDUCTION OF THE SINGULAR INTEGRAL OPERATOR WITH FLIP ATO A MATRIX TOEPLITZ PLUS HANKEL OPERATOR

In this section, we will show that the singular integral operator with shift defined by (1.1) and actingon the space Lp(T) is equivalent to some matrix Toeplitz plus Hankel operator acting on [Hp(T)]2. Thefollowing result presents such a reduction.

Theorem 2.1. Let a0, a1, b0, b1 ∈ L∞(T), 1 < p < ∞, and consider

a :=

⎛

⎝

x1 y1

˜Jx1˜Jy1

⎞

⎠ , b :=

⎛

⎝

x2 y2

˜Jx2˜Jy2

⎞

⎠ , (2.1)

where

x1 = a1 + a0, x2 = b0 − b1,

y1 = b0 + b1, y2 = a0 − a1.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

REDUCTION OF SINGULAR INTEGRAL OPERATORS 121

Then, the singular integral operator with flip

A = a0ILp(T) + a1ST + b0˜J + b1ST

˜J

(acting on the space Lp(T)) is equivalent to the matrix Toeplitz plus Hankel operator withmultipliers a and b, Ta + Hb (acting on [Hp(T)]2).

Proof. First, observe that

a0ILp(T) + a1ST + b0˜J + b1ST

˜J = 2a1P + (a0 − a1)ILp(T) + 2b1P ˜J + (b0 − b1) ˜J

= 2a1P + (a0 − a1)ILp(T) + (b0 + b1)P ˜J + (b0 − b1) ˜J − (b0 − b1)P ˜J

= (a0 + a1)P + (b0 + b1)P ˜J + (b0 − b1)Q ˜J + (a0 − a1)Q = x1P + y1P ˜J + x2˜JP + y2

˜JP ˜J,

where Q := ILp(T) − P = (ILp(T) − ST)/2. Note that, in the last line of the previous computation, we

put ˜JP ˜J instead of Q (which will be used for obtaining the conclusion). On the other hand, observe thatthe matrix Toeplitz plus Hankel operator on [Hp(T)]2, Ta + Hb, has the following matrix form:

Ta + Hb =

⎛

⎝

P P ˜J

P ˜J P

⎞

⎠

⎛

⎝

x1 y1

0 0

⎞

⎠

⎛

⎝

P 0

0 P

⎞

⎠ +

⎛

⎝

P P ˜J

P ˜J P

⎞

⎠

⎛

⎝

x2 y2

0 0

⎞

⎠

⎛

⎝

˜J 0

0 ˜J

⎞

⎠

⎛

⎝

P 0

0 P

⎞

⎠ .

Now, we will use the operators

U := (P, ˜JP ) : [Hp(T)]2 −→ Lp(T), U−1 :=

⎛

⎝

P

P ˜J

⎞

⎠ : Lp(T) −→ [Hp(T)]2,

which are inverses to each other, and establish a Banach space isomorphism between the spaces Lp(T)and [Hp(T)]2. Indeed, we have

UU−1 = (P, ˜JP )

⎛

⎝

P

P ˜J

⎞

⎠ = P + ˜JP ˜J = ILp(T), U−1U =

⎛

⎝

P

P ˜J

⎞

⎠ (P, ˜JP ) =

⎛

⎝

P 0

0 P

⎞

⎠ = I[Hp(T)]2 ,

and moreover

U(Ta + Hb)U−1 = (P, ˜JP )

⎡

⎢

⎣

⎛

⎝

P P ˜J

P ˜J P

⎞

⎠

⎛

⎝

x1 y1

0 0

⎞

⎠

⎛

⎝

P 0

0 P

⎞

⎠ +

⎛

⎝

P P ˜J

P ˜J P

⎞

⎠

⎛

⎝

x2 y2

0 0

⎞

⎠

⎛

⎝

˜J 0

0 ˜J

⎞

⎠

×

⎛

⎝

P 0

0 P

⎞

⎠

⎤

⎥

⎦

⎛

⎝

P

P ˜J

⎞

⎠ = (I, ˜J)

⎡

⎢

⎣

⎛

⎝

x1 y1

0 0

⎞

⎠

⎛

⎝

P 0

0 P

⎞

⎠ +

⎛

⎝

x2 y2

0 0

⎞

⎠

⎛

⎝

˜JP 0

0 ˜JP

⎞

⎠

⎤

⎥

⎦

⎛

⎝

P

P ˜J

⎞

⎠

= (I, ˜J)

⎛

⎝

x1 y1

0 0

⎞

⎠

⎛

⎝

P

P ˜J

⎞

⎠ + (I, ˜J)

⎛

⎝

x2 y2

0 0

⎞

⎠

⎛

⎝

˜J 0

0 ˜J

⎞

⎠

⎛

⎝

P

P ˜J

⎞

⎠ = (x1, y1)

⎛

⎝

P

P ˜J

⎞

⎠

+ (x2, y2)

⎛

⎝

˜J 0

0 ˜J

⎞

⎠

⎛

⎝

P

P ˜J

⎞

⎠ = x1P + y1P ˜J + (x2˜J, y2

˜J)

⎛

⎝

P

P ˜J

⎞

⎠ = x1P + y1P ˜J + x2˜JP + y2

˜JP ˜J.

Hence, A = U(Ta + Hb)U−1. That is, A is equivalent to the matrix Toeplitz plus Hankel operatorTa + Hb.

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122 CASTRO, ROJAS

3. FREDHOLM CHARACTERISTICS FOR THE SINGULARINTEGRAL OPERATOR WITH FLIP A

In this section, based on Theorem 2.1, we will obtain Fredholm criteria for the operator A withcoefficients belonging to different classes of functions on L∞(T). In addition, for some cases, a Fredholmindex formula will be also derived.

3.1. Case 1: Continuous and piecewise continuous coefficients. We are in the position to give aFredholm criterion for the singular integral operator with flip A for the case of continuous and piecewisecontinuous coefficients. In view of this goal, we will use the equivalence relation proved in Theorem 2.1and some known Fredholm theorems for Toeplitz plus Hankel operators with continuous and piecewisecontinuous symbols.

As usual, C(T) denotes the space of all continuous functions on T, and let PC(T) stand for the spaceof all bounded piecewise continuous functions on T, i.e., functions ϕ ∈ L∞(T) for which the one-sidedlimits

ϕ±(t) := limε→0±

ϕ(teiε) (3.1)

exist for each t ∈ T. Let in addition T+ := z : |z| = 1, 0 < arg z < π.We will also make use of the functions

s(x) := coth((

x +i

p

)

π

)

, n(x) := sinh−1

((

x +i

p

)

π

)

(3.2)

for x ∈ R := R ∪ ±∞.Theorem 3.1. For a0, a1, b0, b1 ∈ C(T) and 1 < p < ∞, the singular integral operator with flip

A = a0ILp(T) + a1ST + b0˜J + b1ST

˜J

(acting on Lp(T)) is a Fredholm operator if and only if det(a(t)) = 0, t ∈ T, where a is definedin (2.1). Moreover, in this case,

IndA = −wind(det(a))

(where wind(det(a)) denotes the winding number of det(a)).In the case of a0, a1, b0, b1 ∈ PC(T), A is a Fredholm operator if and only if

det(ψ(τ, x)) = 0

for all x ∈ R and τ ∈ T+ ∪ −1, 1, where

ψ(τ, x) := (1 + s(x))a+(τ) + (1 − s(x))a−(τ) + τn(x)(b+(τ) − b−(τ))

for τ ∈ −1, 1 and

ψ(τ, x) := (1 + s(x))

⎛

⎝

a+(τ) 0

0 a+(τ)

⎞

⎠ + (1 − s(x))

⎛

⎝

a−(τ) 0

0 a−(τ )

⎞

⎠

+ n(x)

⎛

⎝

0 b+(τ) − b−(τ)

b+(τ ) − b−(τ) 0

⎞

⎠

for t ∈ T+ (and where a and b are defined in (2.1)).Proof. From Theorem 2.1, we have that A and Ta + Hb are equivalent operators. Hence, A and

Ta + Hb have the Fredholm property only at the same time. So, the conclusion will be obtained by usingtwo known results for Toeplitz plus Hankel operators.

For the continuous case, we therefore make use of, e.g., [8, Proposition 2.7]. In detail, if Ta + Hb is aFredholm operator, then a is an invertible matrix function, and therefore det(a(t)) = 0 for all t ∈ T.

Conversely, if det(a) = 0, then a is invertible. Thus, we can see that Ta−1 is a Fredholm regularizerof Ta + Hb and the Fredholm conclusion is obtained.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

REDUCTION OF SINGULAR INTEGRAL OPERATORS 123

The index formula is derived from the fact that, for a continuous matrix function b, the Hankel operatorHb is compact, and therefore

Ind(Ta + Hb) = Ind(Ta) = −wind(det(a)).

For the piecewise continuous case, we will make use of, e.g., [8, Theorem A.3]. The proof for thiscase is given by means of a symbol calculus technique and the use of Mellin convolution operators in thefollowing way. Let us start by recalling that the Mellin transformation

M : [Lp(R+)]N −→ [Lp(R)]N

is defined by

(Mf)(x) :=

∞∫

0

ξ−1+1/p−ixf(ξ)dξ, x ∈ R. (3.3)

For a ∈ [L∞(R)]N×N , the corresponding Mellin convolution operator M0(a) ∈ L([Lp(R+)]N ) is givenby

M0(a) = M−1a ·M. (3.4)

Let S := SR+ stand for the Cauchy singular integral operator acting on the space [Lp(R+)]N ,

(Sf)(x) =1πi

∞∫

0

f(y)y − x

dy, x ∈ R+, (3.5)

and let N denote the integral operator acting on [Lp(R+)]N by the rule

( Nf)(x) =1πi

∞∫

0

f(y)y + x

dy, x ∈ R+. (3.6)

The operators S and N can be identified in the following form of Mellin convolution operators:

S = M0(sIN ), N = M0(nIN ), (3.7)

where the generating functions s and n are the same as in (3.2), and IN denotes the identity matrix inC

N×N . In addition, let J stand for the following Carleman shift operator

(Jf)(t) = f(t−1), t ∈ T, (3.8)

acting on [Lp(T)]N . Notice that ˜J = t−1J .

At this point, we observe that, owing to the corresponding actions of P , J and the extension bythe complementary projector Q (of P ), it follows that the operator Ta + Hb is a Fredholm operator on[Hp(T)]2 if and only if the operator

C = P (a + bt−1J)P + Q : [Lp(T)]2 → [Lp(T)]2 (3.9)

is Fredholm. In view of this, for obtaining a characterization of the Fredholm property of C (and thereforethe one of Ta + Hb), we have just to analyze the invertibility of the operator Φτ (C) for τ ∈ T+ ∪ −1, 1;cf. Theorem A.1 in [8]. Here,

Φτ : Y1 → Y2

is a Banach algebra homomorphism with Y1 being the smallest closed subalgebra of L([Lp(T)]2) whichcontains the multiplication operators generated by piecewise continuous 2 × 2 matrix functions, allthe compact operators acting between [Lp(T)]2, the operator J , and the projections P and Q, andY2 = L([Lp(R+)]4) if τ ∈ −1, 1 and Y2 = L([Lp(R+)]8) if τ ∈ T+.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

124 CASTRO, ROJAS

In a more detailed way, letting P be the following operator

P :=12

⎛

⎜

⎝

I[Lp(R+)]2 + S − N

N I[Lp(R+)]2 − S

⎞

⎟

⎠, (3.10)

we have that

Φτ (P ) =

⎧

⎪

⎨

⎪

⎩

P if τ ∈ −1, 1(

P 00 P

)

if τ ∈ T+.

In addition, for τ ∈ −1, 1, we have

Φτ (C) = P

⎛

⎝

a+(τ)I[Lp(R+)]2 τb+(τ)I[Lp(R+)]2

τb−(τ)I[Lp(R+)]2 a−(τ)I[Lp(R+)]2

⎞

⎠ P + (I[Lp(R+)]4 − P ). (3.11)

This operator is a Mellin convolution operator M0(c) with symbol

c(x) = p(x)

⎛

⎝

a+(τ) τb+(τ)

τb−(τ) a−(τ)

⎞

⎠ p(x) + (I4 − p(x)),

where

p(x) =12

⎛

⎝

(1 + s(x))I2 −n(x)I2

n(x)I2 (1 − s(x))I2

⎞

⎠ .

Thus, Φτ (C) is invertible if and only if c(x) is an invertible matrix for each x ∈ R.

The matrix p(x) is an idempotent matrix for each fixed x ∈ R and can be written as a product with

p1 =

⎛

⎝

αI2

βI2

⎞

⎠ , p2 = (γI2, δI2),

where α, β, γ, δ ∈ C are appropriate numbers. So, we obtain

p2

⎛

⎝

a+(τ) τb+(τ)

τb−(τ) a−(τ)

⎞

⎠ p1 = ϕ(τ, x).

From Lemma A.2 in [8], it follows that det c(x) = 0 if and only if detϕ(τ, x) = 0, and the proof for thecase τ ∈ −1, 1 is concluded.

For the case τ ∈ T+, the argumentation is similar. In such a case,

Φτ (C) =

⎛

⎝

P 0

0 P

⎞

⎠

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

a+(τ)I[Lp(R+)]2 0 0 τ b+(τ)I[Lp(R+)]2

0 a−(τ)I[Lp(R+)]2 τ b−(τ)I[Lp(R+)]2 0

0 τb+(τ )I[Lp(R+)]2 a+(τ)I[Lp(R+)]2 0

τb−(τ)I[Lp(R+)]2 0 0 a−(τ )I[Lp(R+)]2

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

×

⎛

⎝

P 0

0 P

⎞

⎠ +

(

I[Lp(R+)]8 −

⎛

⎝

P 0

0 P

⎞

⎠

)

.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

REDUCTION OF SINGULAR INTEGRAL OPERATORS 125

This is a Mellin convolution operator with the symbol

q(x)

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

a+(τ) 0 0 τ b+(τ)

0 a−(τ) τ b−(τ) 0

0 τb+(τ ) a+(τ ) 0

τb−(τ) 0 0 a−(τ)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

q(x) + (I8 − q(x)),

where

q(x) =

⎛

⎝

p(x) 0

0 p(x)

⎞

⎠ .

Following the above procedure, the desired conclusion is obtained.

3.2. Case 2: Semi-almost-periodic coefficients. In the spirit of the Allan–Douglas localizationprinciple (see, e.g., [2, 4, 5, 18]), we present in this subsection a Fredholm criterion for A on L2(T) withsemi-almost-periodic coefficients. Our criterion may be seen as an adaptation of Theorem 3.1 in [4] forthe operator A.

We denote by C(R) the set of all complex-valued continuous functions ζ on R which have finite limitsζ(±∞) at −∞ and +∞. More precisely,

C(R) := C(R) ∩ PC(R). (3.12)

The algebra AP (R) of continuous almost-periodic functions is defined as

AP (R) := algL∞(R)eλ : λ ∈ R, eλ(x) := eiλx (x ∈ R). (3.13)

Finally, the algebra SAP (R) of the semi-almost-periodic functions is the smallest closed subalgebraof L∞(R) containing C(R) ∪ AP (R). That is,

SAP (R) := algL∞(R)(AP (R), C(R)). (3.14)

It is well known that every a ∈ SAPN×N (R) can be written in the form

a = (1 − u)al + uar + a0, (3.15)

where u ∈ C(R) is any fixed function such that 0 ≤ u ≤ 1, u(−∞) = 0, u(+∞) = 1, al and ar belongto APN×N (R), and a0 is in CN×N

0 (R), i.e., a0 belongs to the set of all continuous matrix functionsvanishing at −∞ and +∞. Moreover, al and ar are uniquely determined by a and the maps a → al

and a → ar are C∗-algebra homomorphisms of SAPN×N (R) onto APN×N (R) (see, e.g., [3]). Thematrix functions al and ar are referred to as the almost-periodic representatives of a at −∞ and +∞,respectively.

The set of all almost-periodic polynomials will be denoted by AP 0(R) (recall that the algebra AP (R)can be defined as the closure of AP 0(R) in L∞(R)). The Bohr mean value of a function a ∈ AP (R) isthe element

M(a) := limT→∞

12T

T∫

−T

a(x)dx,

and the Bohr–Fourier spectrum Ω(a) of a ∈ AP (R) is the at most countable set

Ω(a) := λ ∈ R : M(aeλ) = 0.As usual, we set

AP±(R) = a ∈ AP : Ω(a) ⊂ R±, R− := (−∞, 0], R+ := [0,+∞).

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

126 CASTRO, ROJAS

It is well known that AP±(R) are closed subalgebras of AP (R) and that AP±(R) ∩ AP 0(R) is dense inAP±(R).

Let now APW (R) stand for the set of all a ∈ AP (R) which can be written in the form

a(x) =∞∑

j=−∞ajeλj

(x),∞

∑

j=−∞|aj | < ∞,

and set

APW±(R) = APW (R) ∩ AP±(R).

It is known that APW (R) is a Banach algebra with the norm ‖a‖ =∑

|aj |.A canonical (right) APW factorization of a matrix function

a ∈ APW N×N (R)

is a representation

a = a−a+

with a± ∈ GAPW N×N± (R). If c ∈ GAPW N×N (R) has a canonical right APW factorization c = c−c+,

the so-called geometric mean is uniquely defined by

d(c) := M(c−)M(c+).

The Besicovitch space B2 is defined as the completion of AP 0(R) with respect to the norm

‖f‖B2 :=

(

∑

λ

|fλ|2)1/2

= (M(|f |2))1/2,

where f =∑

λ fλeλ ∈ AP 0(R). Let l2(R) denote the collection of all functions f : R −→ C for whichthe set λ ∈ R : f(λ) = 0 is at most countable and

‖f‖2l2(R) :=

∑

|f(λ)|2 < ∞.

Further, l∞(R) is defined as the set of all functions f : R −→ C such that

‖f‖l∞(R) := supλ∈R

|f(λ)| < ∞.

The map FB : l2(R) −→ B2 which transforms a function f ∈ l2(R) with a finite support into thefunction

(FBf)(x) =∑

λ∈R

f(λ)eλ(x), x ∈ R,

can be extended by continuity to all l2(R). It is referred to as the Bohr–Fourier transform. The operatorFB is an isometric isomorphism, and the inverse Bohr–Fourier transformation acts by the rule

F−1B : B2 −→ l2(R), (F−1

B )(λ) = M(fe−λ), λ ∈ R.

If a ∈ l∞(R), then the operator ψ(a) := FBa · F−1B is bounded.

Now, let B2± be the Hilbert spaces consisting of the functions in B2 with the Bohr–Fourier spectra

in R±. A canonical generalized right AP factorization of a matrix function a ∈ GAPN×N (R) is arepresentation

a = a−a+, (3.16)

where

a− ∈ G[B2−]N×N , a+ ∈ G[B2

+]N×N , a− ˜Pa−1− I ∈ L(B2

N ). (3.17)

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REDUCTION OF SINGULAR INTEGRAL OPERATORS 127

Here ˜P is the projection ˜P := FBχ+F−1B ∈ L(B2

N ), with χ+ being the characteristic function of R+.Moreover, although B2

± is not a Banach algebra, we denote by G[B2±]N×N the class of matrix functions

that belong to [B2±]N×N and have an inverse also in [B2

±]N×N .Remark 1. Notice that the above-mentioned C∗-algebras can be considered on T by using the

isometric isomorphism B0 from L∞(R) onto L∞(T), defined by

(B0υ)(t) := υ

(

i1 + t

1 − t

)

, t ∈ T \ 1. (3.18)

Now, we are in the position to establish a explicit Fredholm criterion for A acting on L2(T) and withSAP (T) coefficients.

Theorem 3.2. Let a0, a1, b0, b1 ∈ SAP (T). The operator

A = a0ILp(T) + a1ST + b0˜J + b1ST

˜J

is a Fredholm operator on the space L2(T) if and only if the following three conditions hold:(a) a, b ∈ GSAP 2×2(T), where a and b are given in (2.1).(b) The almost-periodic representatives al, ar, bl, br have canonical generalized right AP fac-

torizations al = a−l d(al)a+l , ar = a−r d(ar)a+

r , bl = b−l d(bl)b+l , and br = b−r d(br)b+

r , where M(a±l ) =M(a±r ) = M(b±l ) = M(b±r ) = I2.

(c) det(A(µ)) = 0 for µ ∈ [0, 1], where

A(µ) = P (1, µ)

⎛

⎝

d(al) 0

0 d(ar)

⎞

⎠P (1, µ) + P (1, µ)

⎛

⎝

d(bl) 0

0 d(br)

⎞

⎠

⎛

⎝

0 −i

i 0

⎞

⎠ P (1, µ),

with P (1, µ) = I4 − S(1, µ), and

S(1, µ) :=

⎛

⎝

2µ − 1 2√

µ(1 − µ)

2√

µ(1 − µ) 1 − 2µ

⎞

⎠ ⊗ I2.

Proof. In virtue of Theorem 2.1, we need to study the Fredholm theory for the Toeplitz plus Hankeloperator Ta + Hb on [H2(T)]2.

On the other hand, Theorem 3.1 in [4] is based on the local principle of Allan and Douglas ([4,Theorem 2.10]) and a symbol calculus at infinity. To achieve this goal, one uses the C∗-algebra

B := alg(S, SAP (R)) ⊂ L(L2(R))

generated by the multiplication operators aIL2(R) with a ∈ SAP (R) and the Cauchy singular integraloperator S (over R). The first step is to study the C∗-algebra

U := alg(PC(R),W 0(PC(R))) ⊂ L(L2(R))

generated by the operators aW 0(b) with a, b ∈ PC(R), where

W 0(b) = F−1b · F ,

with F being the Fourier transformation given by

(Ff)(x) :=∫

R

f(t)eixtdt, x ∈ R.

This algebra U contains the C∗-algebra

Z := alg(C(R),W 0(C(R)))

which also contains the set K(L2(R)) of all compact operators on L2(R). Therefore, by means of aDuduchava theorem (cf. [4, Lemma 5.1] or [2, Theorem B]), the corresponding localization techniquescan be used and in this case Zπ := Z/K is a central C∗-subalgebra of Uπ = U/K.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

128 CASTRO, ROJAS

At this point, by virtue of [15, Theorem 5], we can replace the C∗-algebra B by the C∗-algebra

C := alg(S,R, SAP (R)),

where R is the reflection operator in R, (Rf)(x) = f(−x), and as in [4] the symbol A(t, x, µ), forA = aW 0(b) (a, b ∈ PC(R)) and

(t, x, µ) ∈ (R × 0, 1 × [0, 1]) ∪ (∞ × R × [0, 1]) ∪ ((∞,∞) × 0, 1),is given by

A(t, x, µ) =

⎛

⎝

A11(t, x, µ) A12(t, x, µ)

A21(t, x, µ) A22(t, x, µ)

⎞

⎠

withA11(t, x, µ) = a(t + 0)(b(x + 0)µ + b(x − 0)(1 − µ)),

A12(t, x, µ) = a(t + 0)(b(x + 0) − b(x − 0))√

µ(1 − µ),

A21(t, x, µ) = a(t − 0)(b(x + 0) − b(x − 0))√

µ(1 − µ),A22(t, x, µ) = a(t − 0)(b(x − 0)µ + b(x + 0)(1 − µ))

(where by convention a(∞± 0) = a(∓∞), b(0 ± 0) = b(∓∞), and√

µ(1 − µ) denotes any function : [0, 1] −→ R such that 2(µ) = µ(1 − µ)).

The matrix A(t, x, µ) can be simplified at the points t ∈ R for the generating operators A = aW 0(b)(a ∈ C(R), b ∈ PC(R)),

A(t,∞, µ) =

⎛

⎝

a(t)b(−∞) 0

0 a(t)b(+∞)

⎞

⎠ .

As in [4], we set

S(∞, µ) =

⎛

⎝

2µ − 1 2√

µ(1 − µ)

2√

µ(1 − µ) 1 − 2µ

⎞

⎠ ⊗ I2.

In this way, we have that all the results in [4] hold also for the present case (i.e., also with the inclusionof the reflection operator R). To conclude the proof, we pass from R to T as usual. In detail, using theisometric isomorphism of L2(T) onto L2(R) given by

(Bϕ)(x) =√

2x + i

ϕ(x − i

x + i

)

, x ∈ R,

with inverse

(B−1ψ)(t) =i√

21 − t

ϕ(

i1 + t

1 − t

)

, t ∈ T \ 1,

we have that

PT = B−1PB, −B−1RB = ˜J,

where, as in [17], the symbol of ˜J is

˜J(1, µ) =

⎛

⎝

0 i

−i 0

⎞

⎠ .

Now, we can rewrite condition (c) of Theorem 3.1 in [4], which is

A(µ) = P (∞, µ)

⎛

⎝

d(al) 0

0 d(ar)

⎞

⎠ P (∞, µ) + Q(∞, µ) + Q(∞, µ)

⎛

⎝

M(cl) 0

0 M(cr)

⎞

⎠ H+(∞, µ)

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 29 No. 3 2008

REDUCTION OF SINGULAR INTEGRAL OPERATORS 129

+ H−(∞, µ)

⎛

⎝

M(dl) 0

0 M(dr)

⎞

⎠H+(∞, µ) + P (∞, µ)

⎛

⎝

M((a−l )−1cl) 0

0 M((a+r )−1cr)

⎞

⎠H+(∞, µ)

+ H−(∞, µ)

⎛

⎝

M(dl(a+l )−1) 0

0 M(dr(a+r )−1)

⎞

⎠ P (∞, µ) − H−(∞, µ)

⎛

⎝

M(ηl) 0

0 M(ηr)

⎞

⎠ H+(∞, µ)

for the operator Ta + Hb acting on [H2(T)]2 satisfying conditions (a) and (b) in this theorem, as

A(µ) = P (1, µ)

⎛

⎝

d(al) 0

0 d(ar)

⎞

⎠P (1, µ) + P (1, µ)

⎛

⎝

d(bl) 0

0 d(br)

⎞

⎠

⎛

⎝

0 −i

i 0

⎞

⎠ P (1, µ),

where the point ∞ is transformed into 1+ ∈ T in the usual way—which concludes the proof.

ACKNOWLEDGMENTSThis work was supported in part by the Portuguese Science Foundation (FCT–Fundacao para

a Ciencia e a Tecnologia) through the Unidade de Investigacao Matematica e Aplicacoes of theUniversity of Aveiro, Portugal. E.M. Rojas is sponsored by Fundacao para a Ciencia e a Tecnologia(Portugal) under grant number SFRH/BD/30679/2006.

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