Talk Delivered In Woat Conference Lisbon 2006

Conditions for the Fredholm property of matrix
Wiener-Hopf plus/minus Hankel operators
with semi-almost periodic symbols

WOAT 2006
September 1-5, Lisbon

Giorgi Bogveradzea
(joint work with L.P. Castro)

Research Unit Mathematics and Applications
Department of Mathematics
University of Aveiro, PORTUGAL
a
¸˜ ´ ¸˜
Supported by Unidade de Investigacao Matematica e Aplicacoes of Universidade de Aveiro through the
¸˜ ˆ
Portuguese Science Foundation (FCT–Fundacao para a Ciencia e a Tecnologia).

Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm

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 of
complex-valued Lebesgue measurable functions ϕ for which |ϕ| 2 is integrable on
R 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 Fourier
transformation, J is the reflection operator given by the rule
Jϕ(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 the
following diagonal form:
¡

£
WΦ + HΦ 0
: [L2 (R)]2N → [L2 (R+ )]2N . (1.4)
DΦ := +
¢

¤
WΦ − HΦ
0

Let C− = {z ∈ C : m z < 0} and C+ = {z ∈ C : m z > 0}. As usual, let us
denote by H ∞ (C± ) the set of all bounded and analytic functions in C ± . Fatou’s
Theorem 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 that
H± (R) are closed subalgebras of L∞ (R).
∞


Almost periodic functions

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

1
M (ϕ) := lim ϕ(x)dx
|Iα |
α→∞ Iα

exists, is finite, and is independent of the particular choice of the family {I α }.
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 on
˙˙
the real line for which the two limits

ϕ(−∞) := lim ϕ(x), ϕ(+∞) := lim ϕ(x)
x→−∞ x→+∞

exist and coincide. The common value of these two limits will be denoted by
ϕ(∞). Further, C0 (R) will stand for the functions ϕ ∈ C(R) for which ϕ(∞) = 0.
˙ ˙
We denote by P C := P C(R) the C ∗ -algebra of all bounded piecewise continuous
˙
¯
functions on R, and we also put C(R) := C(R) ∩ P C, where C(R) denote the
˙
usual 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
¯
∞
(R) that contains AP and C(R) :
smallest closed subalgebra of L

¯
SAP = algL∞ (R) {AP, C(R)} .

In [4] D. Sarason proved the following theorem which reveals in a different way the
structure 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 → ϕ , ϕ → ϕr
∗
are C -algebra homomorphisms of SAP onto AP.
Remark 1.5. The last theorem is also valid for the matrix case.


AP factorization

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

(1.5)
Φ(x) = Φ− (x)D(x)Φ+ (x)

∈ R, with
for all x

Φ− ∈ GAP− ×N , Φ+ ∈ GAP+ ×N
N N

and

D(x) = diag(eiλ1 x , . . . , eiλN 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.
N ×N
We will say that a matrix function Φ ∈ GAP admits a left AP factorization if instead of (1.5)
we have the following

(1.6)
Φ(x) = Φ+ (x)D(x)Φ− (x)

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


Geometric mean

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

−1
function Φ admits a right AP factorization, then Φ admits a left AP factorization, and also Φ
admits a left AP factorization.
The vector containing the right AP indices will be denoted by k(Φ), i.e., in the
above case k(Φ) := (λ1 , . . . , λN ). If we consider the case with equal right AP
indices (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 linear
operators.


Equivalence after extension

Deﬁnition 1.8. We will say that the linear bounded operator
S : X1 → X2 (acting between Banach spaces) is equivalent after
extension with T : Y1 → Y2 (also acting between Banach spaces) if
there exist Banach spaces Z1 , Z2 and boundedly invertible linear
operators E and F, such that the following identity holds

T0 S0
(1.7)
=E F,
0 I Z1 0 I Z2

here IZi represents 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, then
T and S have same Fredholm regularity properties (i.e., the properties
that 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 extension
with W −1 .
ΦΦ

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

0
WΦΦ−1
¥

(1.8)
DΦ = E F.
0 IL2 (R)
+


Matrix-Valued SAP symbols

∈ SAP N ×N and assume that the almost periodic
Theorem 1.11. [1, Theorem 10.11] Let Φ
representatives Φ , Φr admit a right AP factorization. Then WΦ is Fredholm if and only if

Φ ∈ GSAP N ×N , k(Φ ) = k(Φr ) = (0, . . . , 0) ,

−1
sp(d (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 )]−1 d(Φ ) .



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

(1.9)
Φ = (1 − u)Φ + uΦr + Φ0 ,

¯
where Φ ,r ∈ GAP N ×N , u ∈ C(R) , u(−∞) = 0 , u(+∞) = 1 and
˙
Φ0 ∈ [C0 (R)]N×N .
From (1.9) it follows that
¥

¦

¦

¦
Φ−1 = [(1 − u)Φ + uΦr + Φ0 ]−1 .
˜ ˜

Thus
¥

¦

¦

¦
ΦΦ−1 = [(1 − u)Φ + uΦr + Φ0 ][(1 − u)Φ + uΦr + Φ0 ]−1 . (1.10)
˜ ˜

Therefore, from (1.10), we obtain that
¥

¥
¥

¥

(ΦΦ−1 ) = Φ Φ−1 , (ΦΦ−1 )r = Φr Φ−1 . (1.11)
r



These representations, and the above relation between the operator D Φ and the
pure Wiener-Hopf operator lead to the following characterization.
∈ SAP N ×N . Then DΦ is Fredholm if and only if
Theorem 1.12. Let Φ

Φ ∈ GSAP N ×N ,

Φ Φ−1 admits canonical right AP factorization and
r

−1
(Φr Φ−1 )d(Φ Φ−1 )] ∩ (−∞, 0] = ∅
sp[d (1.12)
r

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

¥

¥

¥
will obtain that ΦΦ−1 ∈ GSAP N ×N , (ΦΦ−1 ) and (ΦΦ−1 )r admit a canonical
right AP factorizations and
¥

¥

−1
sp[d (1.13)
((ΦΦ−1 )r )d((ΦΦ−1 ) )] ∩ (−∞, 0] = ∅ .



¥
By virtue of (1.11) we conclude that Φ Φ−1 admits a right canonical AP
r
factorization. Once again by virtue of (1.11), from (1.13) we have that

¥

¥
−1
(Φr Φ−1 )d(Φ Φ−1 )] ∩ (−∞, 0] = ∅ ,
sp[d r

hence (1.12) is satisﬁed. The proof of quot;ifquot; part is completed.
Now we will proof the quot;only ifquot; part. From the hypothesis that Φ ∈ GSAP N ×N we
¥

can consider ΦΦ−1 and therefore this is also invertible in SAP N ×N . The left and
¥

right representatives of ΦΦ−1 are given by the formula (1.11). Since
¥

¥

Φ Φ−1 = (ΦΦ−1 ) admits a right canonical AP factorization, then
r
¥

¦

(ΦΦ−1 ) = Φ Φ−1 admits a left canonical AP factorization and
r
¥
¥

[(ΦΦ−1 ) ]−1 = Φr Φ−1 admits a right canonical AP factorization. Moreover,
condition (1.12) implies that
¥

¥

−1
sp[d ((ΦΦ−1 )r )d((ΦΦ−1 ) )] ∩ (−∞, 0] = ∅ .



All these facts together, with Theorem 1.11 give us that WΦΦ−1 is 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 index
formula for the sum of Wiener-Hopf plus/minus Hankel operators W Φ ± HΦ with
Fourier symbols Φ ∈ GSAP N ×N . Due to this reason, let us assume from now on
that WΦ + HΦ and WΦ − HΦ are Fredholm operators.
Let GSAP0,0 denotes the set of all functions ϕ ∈ GSAP for which
k(ϕ ) = k(ϕr ) = 0. To deﬁne the Cauchy index of ϕ ∈ GSAP0,0 we need the next
lemma.
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 1
(1.14)
((arg ϕ)(x) − (arg ϕ)(−x))dx
lim
2π α→∞ |Iα | Iα

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



The limit (1.14) is denoted by indϕ and usually called the Cauchy index of ϕ. The
following theorem is well-known.
∈ SAP N ×N . If the almost periodic representatives
Theorem 1.14. [1, Theorem 10.12] Let Φ
Φ , Φr admit right AP factorizations, and if WΦ is a Fredholm operator, then
N

¨

©

1 1 1
IndWΦ (1.15)
= −ind det Φ − − − arg ξk
2 2 2π
k=1

∈ C (−∞, 0] are the eigenvalues of the matrix d−1 (Φr )d(Φ ) and {·}
where ξ1 , . . . , ξN
stands for the fractional part of a real number. Additionally, when choosing arg ξ k in (−π, π), we
have

N
1
IndWΦ = −ind det Φ − arg ξk .
2π
k=1


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)), we
deduce that

IndDΦ = IndWΦΦ−1 .

§
Consequently we have:

Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = IndWΦΦ−1 .

§
Using (1.15) for WΦΦ−1 (which is a Fredholm operator because of the assumption
§

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

©

1 1 1
¥

IndW = −ind[det(ΦΦ−1 )] − − − arg ζk ,
§

ΦΦ−1 2 2 2π
k=1


Index formula

¥

where ζk ∈ C (−∞, 0] are the eigenvalues of the matrix d−1 (Φr Φ−1 )d(Φ Φ−1 ),
r
cf. (1.11). In the case when arg ζk are chosen in (−π, π), we have
N
1

¥
IndW = −ind[det(ΦΦ−1 )] − arg ζk .
§
2π
ΦΦ−1
k=1

In addition we can perform the following computations:
¥

¥

¥
ind[det(ΦΦ−1 )] ind[det Φ det Φ−1 ] = ind det Φ + ind det Φ−1
=
= ind det Φ − ind[det Φ−1 ]
ind det Φ + ind[det Φ−1 ]
=
−1
ind det Φ − ind[det Φ] = ind det Φ + ind det Φ
=
2 ind det Φ .
=

Consequently we arrive at the following result.


Index formula

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

N

¨

©

1 1 1
Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = −2 ind det Φ − − − arg ζk ,(1.17)
2 2 2π
k=1

and we also have

N
1
Ind[WΦ + HΦ ] + Ind[WΦ − HΦ ] = −2 ind det Φ − (1.18)
arg ζk ,
2π
k=1

for arg ζk ∈ (−π, π).


Index formula

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

N

¨

©

1 1 1 1
Ind[WΦ Ind[WΦ − HΦ ] = −ind det Φ − − −
+ HΦ ] = arg ζk ,
4 2 2 2π
k=1

N
1
Ind[WΦ Ind[WΦ − HΦ ] = −ind det Φ −
+ HΦ ] = arg ζk ,
4π
k=1

for arg ζk ∈ (−π, π).
˙
∈ GSAP N ×N ∩ [C(R) + H∞ ]N×N . Then the Hankel operator HΦ is compact,
Proof. Let Φ −
and therefore the corollary follows from (1.17) and (1.18).


Example 1

Let us assume that

¡

£
e−i(1+α)x 0
(1.19)
(1 − u(x))
Φ(x) =

¢

¤
−iαx ix i(1+α)x
−1+e
e e

¡

£
ei(1+α)x 0
+ u(x) ,

¢

¤
−ix −i(1+α)x
iαx
−1+e
e e
√
where α = and u is a following real-valued function
5−1
,
2
1 1
(1.20)
u(x) = + arctan(x) .
2 π
In this case we have that the Wiener-Hopf operator with symbol Φ is not Fredholm
because the matrix function
¡

£
e−i(1+α)x 0
∈ GAP
¢

¤
−iαx ix i(1+α)x
−1+e
e e

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

Example 1

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

¡

£
1 0

¥
Φ Φ−1 = = I2×2
r

¢

¤
0 1

and

¡

£
1 0
¥
−1
Φr Φ = = I2×2 .

¢

¤
0 1
¥

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

¥

d−1 (Φr Φ−1 )d(Φ Φ−1 ) = I2×2 .
r

Thus the eigenvalues of this matrix are equal to 1 ∈ (−∞, 0]. This allows us to
conclude that the operator DΦ is a Fredholm operator (cf. Theorem 1.12). This
means 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 to
compute the determinant of Φ :

¡
(1 − u(x))e−i(1+α)x + u(x)ei(1+α)x
det Φ(x) = det ¢

(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 + e2i(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 that
2u(x)(1 − u(x)) ∈ 0, 1 , moreover 1 − cos 2(1 + α)x ∈ [0, 2]. Combining these

2
inclusions we have that 2u(x)(1 − u(x))(1 − cos 2(1 + α)x) ∈ [0, 1]. From here we
conclude that det Φ ∈ [0, 1], but Φ is invertible and therefore det Φ ∈ (0, 1]. Thus
we 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).

1.0

0.8

0.6

0.4

−20 −16 −12 −8 −4 0 4 8 12 16 20
x


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), their
arguments are also zero.


Example 2

Let us take the following function:

¡

£

¡

£

¡

£
e−ix
eix x−i
−1
0 0 0 x+i
Ψ(x) = (1 − u(x)) + u(x) + ,

¢

¤

¢

¤

¢

¤
−ix ix x−i
−1
0 0 0
e e x+i

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

¡

£
eix 0
Ψ=
¢

¤
e−ix
0

and
¡

£
e−ix 0
Ψr =
¢

¤

eix
0

do not have a canonical right AP factorization (since k(Ψ ) = (1, −1) and
k(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 . Moreover
we have:

¥

¥
Ψ Ψ−1 = Ψr Ψ−1 = I2×2 .
r

From here we also obtain that
¥

¥
−1
(Ψr Ψ−1 )d(Ψ Ψ−1 )] = {1} ∩ (−∞, 0] = ∅ .
sp[d r

These are sufﬁcient conditions for the Wiener-Hopf plus/minus Hankel operators
to have the Fredholm property (cf. Theorem 1.12). To calculate the index of the
sum of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators
we need the determinant of the function Ψ :
2

¨

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


Example 2

2

1

−3 −2 −1 0 1
0

−1

−2

From here it follows that the winding number of the det Ψ is equal to 1 (the graph
of the det Ψ is given above and it is closed because lim x→±∞ det Ψ(x) = 1,
moreover in the next page is shown the oscillation of the function det Ψ at inﬁnity).


Example 2

10−6

8

4

0

0.992 0.994 0.996 0.998 1.0

−4

−8

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 of
delta-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, Duke
Math. J. 44 (1977), 357–364.


Talk Delivered In Woat Conference Lisbon 2006

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