talk delivered in woat conference lisbon 2006

Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operators

with semi-almost periodic symbols

WOAT 2006September 1-5, Lisbon

Giorgi Bogveradzea

(joint work with L.P. Castro)

Research Unit Mathematics and ApplicationsDepartment of Mathematics

University of Aveiro, PORTUGAL

aSupported by Unidade de Investigacao Matematica e Aplicacoes of Universidade de Aveiro through the

Portuguese Science Foundation (FCT–Fundacao para a Ciencia e a Tecnologia).

Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-almost periodic symbols – p. 1/32

Basic definitions

We will consider Wiener-Hopf plus/minus Hankel operators of the form

Wϕ ± Hϕ : [L2+(R)]N → [L2(R+)]N , (1.1)

with Wϕ and Hϕ being Wiener-Hopf and Hankel operators defined by

Wϕ = r+F−1

ϕ · F : [L2+(R)]N → [L2(R+)]N (1.2)

Hϕ = r+F−1

ϕ · FJ : [L2+(R)]N → [L2(R+)]N , (1.3)

respectively. Here, L2(R) and L2(R+) denote the Banach spaces ofcomplex-valued Lebesgue measurable functions ϕ for which |ϕ|2 is integrable onR and R+, respectively. Additionally , L2

+(R) denotes the subspace of L2(R)

formed by all functions supported in the closure of R+ = (0,+∞), the operator r+

performs the restriction from L2(R) into L2(R+), F denotes the Fouriertransformation, J is the reflection operator given by the ruleJϕ(x) =

�

ϕ(x) = ϕ(−x), x ∈ R, and ϕ ∈ L∞(R) is the so-called Fourier symbol.


Basic definitions

The main result of this work will be obtained for the matrix operator which has thefollowing diagonal form:

DΦ :=

�� WΦ + HΦ 0

0 WΦ − HΦ

�� : [L2

+(R)]2N → [L2(R+)]2N. (1.4)

Let C− = {z ∈ C : =m z < 0} and C+ = {z ∈ C : =m z > 0}. As usual, let usdenote by H∞(C±) the set of all bounded and analytic functions in C±. Fatou’sTheorem ensures that functions in H∞(C±) have non-tangential limits on R

almost everywhere. Thus, let H∞± (R) be the set of all functions in L∞(R) that are

non-tangential limits of elements in H∞(C±). Moreover, it is well-known thatH∞

± (R) are closed subalgebras of L∞(R).


Almost periodic functions

Now consider the smallest closed subalgebra of L∞(R) that contains all thefunctions eλ with λ ∈ R (eλ(x) := eiλx, x ∈ R). This is denoted by AP and calledthe algebra of almost periodic functions:

AP := algL∞(R){eλ : λ ∈ R} .

The following subclasses of AP are also of interest:

AP+ := algL∞(R){eλ : λ ≥ 0}, AP− := algL∞(R){eλ : λ ≤ 0}

Almost periodic functions have a great amount of interesting properties. Amongthem, for our purposes the following are the most relevant.


Bohr mean value

Proposition 1.1. [1, Proposition 2.22] Let A ⊂ (0,∞) be an unbounded set and let

{Iα}α∈A = {(xα, yα)}α∈A be a family of intervals Iα ⊂ R such that |Iα| = yα − xα → ∞

as α → ∞. If ϕ ∈ AP, then the limit

M(ϕ) := limα→∞

1

|Iα| Iα

ϕ(x)dx

exists, is finite, and is independent of the particular choice of the family {Iα}.

Definition 1.2. Let ϕ ∈ AP. The number M(ϕ) given by Proposition 1.1 is called the Bohr mean

value or simply the mean value of ϕ.

In the matrix case the mean value is defined entry-wise.


Essential facts

Let C(R) (R = R ∪ {∞}) represent the (bounded and) continuous functions ϕ onthe real line for which the two limits

ϕ(−∞) := limx→−∞

ϕ(x), ϕ(+∞) := limx→+∞

ϕ(x)

exist and coincide. The common value of these two limits will be denoted byϕ(∞). Further, C0(R) will stand for the functions ϕ ∈ C(R) for which ϕ(∞) = 0.

We denote by PC := PC(R) the C∗-algebra of all bounded piecewise continuousfunctions on R, and we also put C(R) := C(R) ∩ PC, where C(R) denote theusual set of continuous functions on the real line.In our operators we will deal with symbols from the C∗-algebra of SAP functions,which is defined as follows.


Semi-almost periodic functions

Definition 1.3. The C∗-algebra SAP of all semi-almost periodic functions on R is defined as the

smallest closed subalgebra of L∞(R) that contains AP and C(R) :

SAP = algL∞(R){AP, C(R)} .

In [4] D. Sarason proved the following theorem which reveals in a different way thestructure of the SAP algebra.

Theorem 1.4. Let u ∈ C(R) be any function for which u(−∞) = 0 and u(+∞) = 1. If

ϕ ∈ SAP, then there exist ϕ`, ϕr ∈ AP and ϕ0 ∈ C0(R) such that

ϕ = (1 − u)ϕ` + uϕr + ϕ0 .

The functions ϕ`, ϕr are uniquely determined by ϕ, and independent of the particular choice of u.

The maps

ϕ 7→ ϕ`, ϕ 7→ ϕr

are C∗-algebra homomorphisms of SAP onto AP.

Remark 1.5. The last theorem is also valid for the matrix case.


AP factorization

Definition 1.6. A matrix function Φ ∈ GAP N×N is said to admit a right AP factorization if it can

be represented in the form

Φ(x) = Φ−(x)D(x)Φ+(x) (1.5)

for all x ∈ R, with

Φ− ∈ GAPN×N− , Φ+ ∈ GAP

N×N+

and

D(x) = diag(eiλ1x, . . . , e

iλN x), λj ∈ R .

The numbers λj are referred as the right AP indices of the factorization. A right AP factorization

with D(x) = IN×N is referred to as a canonical right AP factorization.

We will say that a matrix function Φ ∈ GAP N×N admits a left AP factorization if instead of (1.5)

we have the following

Φ(x) = Φ+(x)D(x)Φ−(x) (1.6)

for all x ∈ R and Φ±, D having the same properties as above.


Geometric mean

Remark 1.7. It is readily seen from the above definition that if an invertible almost periodic matrix

function Φ admits a right AP factorization, then

�

Φ admits a left AP factorization, and also Φ−1

admits a left AP factorization.

The vector containing the right AP indices will be denoted by k(Φ), i.e., in theabove case k(Φ) := (λ1, . . . , λN ). If we consider the case with equal right APindices (k(Φ) = (λ, . . . , λ)), then the matrix

d(Φ) := M(Φ−)M(Φ+)

is independent of the particular choice of the right AP factorization (cf. [1,Proposition 8.4]). In this case the matrix d(Φ) is called the geometric mean of Φ.

We also need the notion of the equivalence after extension for bounded linearoperators.


Equivalence after extension

Definition 1.8. We will say that the linear bounded operatorS : X1 → X2 (acting between Banach spaces) is equivalent afterextension with T : Y1 → Y2 (also acting between Banach spaces) ifthere exist Banach spaces Z1, Z2 and boundedly invertible linearoperators E and F, such that the following identity holds

[

T 0

0 IZ1

]

= E

[

S 0

0 IZ2

]

F , (1.7)

here IZirepresents the identity operator on the Banach space Zi,

i = 1, 2.

Remark 1.9. It is clear that if T is equivalent after extension with S, thenT and S have same Fredholm regularity properties (i.e., the propertiesthat directly depend on the kernel and image of the operator).


Equivalence after extension

Lemma 1.10. Let Φ ∈ GL∞(R). Then DΦ is equivalent after extensionwith W

ΦΦ−1 .

Proof sketch.This lemma has its roots in the Gohberg-Krupnik-Litvinchukidentity, from which with additional equivalence afterextension operator relations it is possible to find invertibleand bounded linear operators E and F such that

DΦ = E

[

WΦ

�

Φ−1 0

0 IL2+(R)

]

F . (1.8)


Matrix-Valued SAP symbols

Theorem 1.11. [1, Theorem 10.11] Let Φ ∈ SAP N×N and assume that the almost periodic

representatives Φ` , Φr admit a right AP factorization. Then WΦ is Fredholm if and only if

Φ ∈ GSAPN×N

, k(Φ`) = k(Φr) = (0, . . . , 0) ,

sp(d−1(Φr)d(Φ`)) ∩ (−∞, 0] = ∅ ,

where sp(d−1(Φr)d(Φ`)) stands for the set of the eigenvalues of the matrix

d−1(Φr)d(Φ`) := [d(Φr)]

−1d(Φ`) .



The matrix version of Sarason’s Theorem (cf. Theorem 1.4) says that ifΦ ∈ GSAP N×N then this function has the following representation

Φ = (1 − u)Φ` + uΦr + Φ0 , (1.9)

where Φ`,r ∈ GAP N×N , u ∈ C(R) , u(−∞) = 0 , u(+∞) = 1 andΦ0 ∈ [C0(R)]N×N .

From (1.9) it follows that

�

Φ−1 = [(1 − u)

�

Φ` + u�

Φr +

�

Φ0]−1

.

Thus

Φ

�

Φ−1 = [(1 − u)Φ` + uΦr + Φ0][(1 − u)

�

Φ` + u

�

Φr +

�

Φ0]−1

. (1.10)

Therefore, from (1.10), we obtain that

(Φ�

Φ−1)` = Φ`

�

Φ−1r , (Φ

�

Φ−1)r = Φr

�

Φ−1` . (1.11)



These representations, and the above relation between the operator DΦ and thepure Wiener-Hopf operator lead to the following characterization.

Theorem 1.12. Let Φ ∈ SAP N×N . Then DΦ is Fredholm if and only if

Φ ∈ GSAPN×N

,

Φ`Φ−1r admits canonical right AP factorization and

sp[d−1(ΦrΦ−1` )d(Φ`Φ

−1r )] ∩ (−∞, 0] = ∅ (1.12)

(where, as before, Φ` and Φr are the local representatives at ∓∞ of Φ, cf. (1.9)).

Proof sketch. If DΦ is Fredholm operator, then by the equivalence after extensionW

Φ

�

Φ−1is also a Fredholm operator (cf. (1.8)). Employing now Theorem 1.11 we

will obtain that Φ

�

Φ−1 ∈ GSAP N×N , (Φ

�

Φ−1)` and (Φ

�

Φ−1)r admit a canonicalright AP factorizations and

sp[d−1((Φ

�

Φ−1)r)d((Φ

�

Φ−1)`)] ∩ (−∞, 0] = ∅ . (1.13)



By virtue of (1.11) we conclude that Φ`

�

Φ−1r admits a right canonical AP

factorization. Once again by virtue of (1.11), from (1.13) we have that

sp[d−1(Φr

�

Φ−1` )d(Φ`

�

Φ−1r )] ∩ (−∞, 0] = ∅ ,

hence (1.12) is satisfied. The proof of "if" part is completed.Now we will proof the "only if" part. From the hypothesis that Φ ∈ GSAP N×N wecan consider Φ

�

Φ−1 and therefore this is also invertible in SAP N×N . The left andright representatives of Φ

�

Φ−1 are given by the formula (1.11). Since

Φ`

�

Φ−1r = (Φ

�

Φ−1)` admits a right canonical AP factorization, then˜

(Φ

�

Φ−1)` =

�

Φ`Φ−1r admits a left canonical AP factorization and

[˜

(Φ

�

Φ−1)`]−1 = Φr

�

Φ−1` admits a right canonical AP factorization. Moreover,

condition (1.12) implies that

sp[d−1((Φ

�

Φ−1)r)d((Φ

�

Φ−1)`)] ∩ (−∞, 0] = ∅ .



All these facts together, with Theorem 1.11 give us that WΦ

�

Φ−1is a Fredholm

operator. Now employing the equivalence after extension relation presented in(1.8) we obtain that DΦ is a Fredholm operator.



In the present section we will be concentrated in obtaining a Fredholm indexformula for the sum of Wiener-Hopf plus/minus Hankel operators WΦ ± HΦ withFourier symbols Φ ∈ GSAP N×N . Due to this reason, let us assume from now onthat WΦ + HΦ and WΦ − HΦ are Fredholm operators.Let GSAP0,0 denotes the set of all functions ϕ ∈ GSAP for whichk(ϕ`) = k(ϕr) = 0. To define the Cauchy index of ϕ ∈ GSAP0,0 we need the nextlemma.

Lemma 1.13. [1, Lemma 3.12] Let A ⊂ (0,∞) be an unbounded set and let

{Iα}α∈A = {(xα, yα)}α∈A

be a family of intervals such that xα ≥ 0 and |Iα| = yα − xα → ∞, as α → ∞. If

ϕ ∈ GSAP0,0 and arg ϕ is any continuous argument of ϕ, then the limit

1

2πlim

α→∞

1

|Iα| Iα

((arg ϕ)(x) − (arg ϕ)(−x))dx (1.14)

exists, is finite, and is independent of the particular choices of {(xα, yα)}α∈A and arg ϕ.



The limit (1.14) is denoted by indϕ and usually called the Cauchy index of ϕ. Thefollowing theorem is well-known.

Theorem 1.14. [1, Theorem 10.12] Let Φ ∈ SAP N×N . If the almost periodic representatives

Φ`, Φr admit right AP factorizations, and if WΦ is a Fredholm operator, then

IndWΦ = −ind detΦ −N

k=1

�

1

2−

1

2−

1

2πarg ξk

�

(1.15)

where ξ1, . . . , ξN ∈ C \ (−∞, 0] are the eigenvalues of the matrix d−1(Φr)d(Φ`) and {·}

stands for the fractional part of a real number. Additionally, when choosing arg ξk in (−π, π), we

have

IndWΦ = −ind detΦ −1

2π

N

k=1

arg ξk .


Index formula

It follows from the definition of the operator DΦ (cf. formula (1.4)) that

IndDΦ = Ind[WΦ + HΦ] + Ind[WΦ − HΦ] (1.16)

Now, employing the above equivalence after extension relation (cf. (1.8)), wededuce that

IndDΦ = IndWΦ

�

Φ−1.

Consequently we have:

Ind[WΦ + HΦ] + Ind[WΦ − HΦ] = IndWΦ

�

Φ−1.

Using (1.15) for WΦ

�

Φ−1(which is a Fredholm operator because of the assumption

made in the beginning of the present section), we obtain that

IndWΦ

�

Φ−1= −ind[det(Φ

�

Φ−1)] −N

k=1

�

1

2−

1

2−

1

2πarg ζk

�

,


Index formula

where ζk ∈ C \ (−∞, 0] are the eigenvalues of the matrix d−1(Φr

�Φ−1

` )d(Φ`

�

Φ−1r ),

cf. (1.11). In the case when arg ζk are chosen in (−π, π), we have

IndWΦ

�

Φ−1= −ind[det(Φ

�

Φ−1)] −1

2π

N

k=1

arg ζk .

In addition we can perform the following computations:

ind[det(Φ

�

Φ−1)] = ind[det Φ det

�Φ−1] = ind det Φ + ind det

�

Φ−1

= ind det Φ + ind ˜[det Φ−1] = ind detΦ − ind[det Φ−1]

= ind det Φ − ind[det Φ]−1 = ind detΦ + ind det Φ

= 2 ind det Φ .

Consequently we arrive at the following result.


Index formula

Corollary 1.15. If WΦ ± HΦ are Fredholm operators for some Φ ∈ GSAP N×N , then

Ind[WΦ + HΦ] + Ind[WΦ − HΦ] = −2 ind det Φ −N

k=1

�

1

2−

1

2−

1

2πarg ζk

�

,(1.17)

and we also have

Ind[WΦ + HΦ] + Ind[WΦ − HΦ] = −2 ind det Φ −1

2π

N

k=1

arg ζk , (1.18)

for arg ζk ∈ (−π, π).


Index formula

Corollary 1.16. If Φ ∈ GSAP N×N ∩ [C(R) + H∞− ]N×N and WΦ ± HΦ are Fredholm

operators, then

Ind[WΦ + HΦ] = Ind[WΦ − HΦ] = −ind det Φ −N

k=1

�

1

4−

1

2

1

2−

1

2πarg ζk

�

,

Ind[WΦ + HΦ] = Ind[WΦ − HΦ] = −ind det Φ −1

4π

N

k=1

arg ζk ,

for arg ζk ∈ (−π, π).

Proof. Let Φ ∈ GSAP N×N ∩ [C(R) + H∞− ]N×N. Then the Hankel operator HΦ is compact,

and therefore the corollary follows from (1.17) and (1.18).


Example 1

Let us assume that

Φ(x) = (1 − u(x))

�� e−i(1+α)x 0

e−iαx − 1 + eix ei(1+α)x�

� (1.19)

+ u(x)

�� ei(1+α)x 0

eiαx − 1 + e−ix e−i(1+α)x

�� ,

where α =√

5−12

, and u is a following real-valued functionu(x) =

1

2+

1

πarctan(x) . (1.20)

In this case we have that the Wiener-Hopf operator with symbol Φ is not Fredholmbecause the matrix function�

� e−i(1+α)x 0

e−iαx − 1 + eix ei(1+α)x

�� ∈ GAP

does not have a right AP factorization (cf. [3, pages 284-285] for the details aboutthis matrix function).


Example 1

However, the Wiener-Hopf plus Hankel operator with the same symbol Φ will havethe Fredholm property. Indeed:

Φ`

�

Φ−1r =

�� 1 0

0 1

�� = I2×2

and

Φr

�

Φ−1` =

�� 1 0

0 1�

� = I2×2 .

Consequently, Φ`

�

Φ−1r obviously admits a canonical right AP factorization and

d−1(Φr

�Φ−1

` )d(Φ`

�

Φ−1r ) = I2×2 .

Thus the eigenvalues of this matrix are equal to 1 6∈ (−∞, 0]. This allows us toconclude that the operator DΦ is a Fredholm operator (cf. Theorem 1.12). Thismeans that the WΦ ± HΦ are Fredholm operators.


Example 1

Let us now calculate the sum of their indices. To this end, we need first of all tocompute the determinant of Φ :

det Φ(x) = det

�� (1 − u(x))e−i(1+α)x + u(x)ei(1+α)x

(1 − u(x))(e−iαx − 1 + eix) + u(x)(eiαx − 1 + e−ix)

0

(1 − u(x))ei(1+α)x + u(x)e−i(1+α)x

��

= [(1 − u(x))e−i(1+α)x + u(x)ei(1+α)x] × (1.21)

[(1 − u(x))ei(1+α)x + u(x)e−i(1+α)x]

= (1 − u(x))2 + u2(x) + u(1 − u(x))[e−2i(1+α)x + e

2i(1+α)x]

= (1 − u(x))2 + u2(x) + 2u(x)(1 − u(x)) cos 2(1 + α)x

= 1 − 2u(x)(1 − u(x))(1 − cos 2(1 + α)x)


Example 1

Recalling that u is a real-valued function given by (1.20) we have that2u(x)(1 − u(x)) ∈

�

0, 12

, moreover 1 − cos 2(1 + α)x ∈ [0, 2]. Combining theseinclusions we have that 2u(x)(1 − u(x))(1 − cos 2(1 + α)x) ∈ [0, 1]. From here weconclude that det Φ ∈ [0, 1], but Φ is invertible and therefore det Φ ∈ (0, 1]. Thuswe have that det Φ is a real-valued positive function, and so the argument is zero(the graph of the det Φ is given below).

4−16

x

2016

0.8

128−8−12 −4

0.6

−20

0.4

0

1.0


Example 1

Altogether we have:

ind[WΦ + HΦ] + ind[WΦ − HΦ] = 0 ,

since the eigenvalues are also real (recall that they are equal to 1), theirarguments are also zero.


Example 2

Let us take the following function:

Ψ(x) = (1 − u(x))

�� eix 0

0 e−ix

�� + u(x)

�� e−ix 0

0 eix

�� +

�� 0 x−i

x+i− 1

x−ix+i

− 1 0

�� ,

here u is as above. It is easily seen that the Wiener-Hopf operator with the symbolΨ is not Fredholm, because the matrix functions

Ψ` =

�� eix 0

0 e−ix�

�

and

Ψr =�

� e−ix 0

0 eix

��

do not have a canonical right AP factorization (since k(Ψ`) = (1,−1) andk(Ψr) = (−1, 1)).


Example 2

However the Wiener-Hopf plus/minus Hankel operators with the same symbol Ψ

are Fredholm. Indeed, first of all observe that Ψ is invertible in SAP 2×2. Moreoverwe have:

Ψ`

�

Ψ−1r = Ψr

�

Ψ−1` = I2×2 .

From here we also obtain that

sp[d−1(Ψr

�

Ψ−1` )d(Ψ`

�

Ψ−1r )] = {1} ∩ (−∞, 0] = ∅ .

These are sufficient conditions for the Wiener-Hopf plus/minus Hankel operatorsto have the Fredholm property (cf. Theorem 1.12). To calculate the index of thesum of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operatorswe need the determinant of the function Ψ :

det Ψ(x) = (1 − u(x))2 + u2(x) + u(x)(1 − u(x)) cos 2x −

�

x − i

x + i− 1

� 2

.


Example 2

1

0

0

1

−1

−3

−2

−2

2

−1

From here it follows that the winding number of the det Ψ is equal to 1 (the graphof the detΨ is given above and it is closed because limx→±∞ det Ψ(x) = 1,

moreover in the next page is shown the oscillation of the function det Ψ at infinity).


Example 2

0.992

−4

1.0

8

0

0.996

4

0.994

−8

10−6

0.998

Therefore, from formula (1.18), we obtain that

ind[WΨ + HΨ] + ind[WΨ − HΨ] = −2 .


References

[1] A. Böttcher, Yu. I. Karlovich, I. M. Spitkovsky: Convolution Operators and

Factorization of Almost Periodic Matrix Functions, Oper. Theory Adv. Appl. 131,Birkhäuser Verlag, Basel, 2002.

[2] L. P. Castro, F.-O. Speck: Regularity properties and generalized inverses ofdelta-related operators, Z. Anal. Anwendungen 17 (1998), 577–598.

[3] Yu. I. Karlovich, I. M. Spitkovsky: Factorization of almost periodic matrix functions and

the Noether theory of certain classes of equations of convolution type, (Russian) Izv.Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 276–308; translation in Math.USSR-Izv. 34 (1990), no. 2, 281–316

[4] D. Sarason: Toeplitz operators with semi-almost periodic symbols, DukeMath. J. 44 (1977), 357–364.


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