R E S E A R C H
Open Access
Fredholmness of multiplication of a
weighted composition operator with its
adjoint on
H
2
and
A
2
α
Mahmood Haji Shaabani
**Correspondence: [email protected]
Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
Abstract
In this paper, we obtain thatCψ,ϕ∗ is bounded below onH2orA2
αif and only ifCψ,ϕis
invertible. Moreover, we investigate the Fredholm operatorsCψ1,ϕ1C∗ψ2,ϕ2and Cψ∗
1,ϕ1Cψ2,ϕ2onH
2andA2 α.
MSC: Primary 47B33; secondary 47A53
Keywords: Hardy space; weighted Bergman spaces; weighted composition operator; Fredholm operator
1 Introduction
LetDdenote the open unit disk in the complex plane. The Hardy space, denotedH2(D) =
H2, is the set of all analytic functionsf onDsatisfying the norm condition
f21=lim
r→1
2π
0
freiθ2dθ
2π <∞.
The spaceH∞(D) =H∞consists of all analytic and bounded functions onDwith supre-mum normf∞=supz∈D|f(z)|.
Forα> –1, the weighted Bergman spaceA2
α(D) =A2αis the set of functionsf analytic in
Dwith
f2α+2= (α+ 1)
D
f(z)21 –|z|2αdA(z) <∞,
wheredAis the normalized area measure inD. The case whereα= 0 is known as the (unweighted) Bergman space, often simply denoted byA2.
Letϕbe an analytic map from the open unit diskDinto itself. The operator that takes the analytic mapf to f ◦ϕ is a composition operator and is denoted byCϕ. A natural generalization of a composition operator is an operator that takesf to ψ·f ◦ϕ, where
ψ is a fixed analytic map on D. This operator is aptly named a weighted composition operator and is usually denoted byCψ,ϕ. More precisely, if z is in the unit disk, then (Cψ,ϕf)(z) =ψ(z)f(ϕ(z)). For some results on weighted composition and related operators on the weighted Bergman and Hardy spaces, see, for example, [1–14].
If ψ is a bounded analytic function on the open unit disk, then the multiplication operatorMψ defined byMψ(f)(z) =ψ(z)f(z) is a bounded operator onH2 andA2α and Mψ(f)γ≤ ψ∞fγ whenγ = 1 forH2andγ =α+ 2 forA2α. LetPdenote the orthog-onal projection ofL2(∂D) ontoH2. For eachb∈L∞(∂D), the Toeplitz operatorTb acts
onH2 byT
b(f) =P(bf). Also suppose thatPα is the orthogonal projection ofL2(D,dAα) onto A2
α. For each function w∈L∞(D), the Toeplitz operator Tw on A2α is defined by
Tw(f) =Pα(wf). SincePandPαare bounded onH2andA2α, respectively, the Toeplitz op-erators are bounded.
Letw∈D, and letHbe a Hilbert space of analytic functions onD. Letewbe the point
evaluation atw, that is,ew(f) =f(w) forf ∈H. Ifewis a bounded linear functional onH,
then the Riesz representation theorem implies that there is a function (usually denotedKw)
inHthat induces this linear functional, that is,ew(f) = f,Kw. In this case, the functions
Kw are called the reproducing kernels, and the functional Hilbert space is also called a
reproducing kernel Hilbert space. Both the weighted Bergman spaces and the Hardy space are reproducing kernel Hilbert spaces, where the reproducing kernel for evaluation atw
is given byKw(z) = (1 –wz)–γ forz,w∈D, withγ = 1 forH2andγ =α+ 2 forA2α. Letkw
denote the normalized reproducing kernel given bykw(z) =Kw(z)/Kwγ, whereKw2γ= (1 –|w|2)–γ.
Suppose thatHandHare Hilbert spaces andA:H→His a bounded operator. The operatorAis said to be left semi-Fredholm if there are a bounded operatorB:H→Hand a compact operatorKonHsuch thatBA=I+K. Analogously,Ais right semi-Fredholm if there are a bounded operatorB:H→Hand a compact operatorKonHsuch that
AB=I+K. An operatorAis said to be Fredholm if it is both left and right semi-Fredholm. It is not hard to see thatAis left semi-Fredholm if and only ifA∗is right semi-Fredholm. HenceAis Fredholm if and only ifA∗ is Fredholm. Note that an invertible operator is Fredholm. By using the definition of a Fredholm operator it is not hard to see that if the operatorsAandBare Fredholm on a Hilbert spaceH, thenABis also Fredholm onH. The Fredholm composition operators onH2were first identified by Cima et al. [15] and
later by a different and more general method by Bourdon [2]. Cima et al. [15] proved that only the invertible composition operators onH2are Fredholm. Moreover, MacCluer [16]
characterized Fredholm composition operators on a variety of Hilbert spaces of analytic functions in both one and several variables. Recently, Fredholm composition operators on various spaces of analytic functions have been studied (see [13] and [14]).
The automorphisms ofD, that is, the one-to-one analytic maps of the disk onto itself, are just the functionsϕ(z) =λ1–a–azz with|λ|= 1 and|a|< 1. We denote the class of auto-morphisms ofDbyAut(D). Automorphisms ofDtake∂Donto∂D. It is known thatCϕis Fredholm on the Hardy space if and only ifϕ∈Aut(D) (see [2]).
An analytic mapϕof the disk to itself is said to have a finite angular derivative at a point
ζ on the boundary of the disk if there exists a pointη, also on the boundary of the disk, such that the nontangential limit asz→ζof the difference quotient (η–ϕ(z))/(ζ–z) exists as a finite complex value. We writeϕ(ζ) =∠limz→ζ η–ζϕ–(zz).
In the second section, we investigate Fredholm and invertible weighted composition operators. In Theorem 2.7, we show that the operatorCψ∗,ϕis bounded below onH2orA2
α if and only ifCψ,ϕis invertible.
In the third section, we investigate the Fredholm operatorsCψ1,ϕ1C∗ψ2,ϕ2andC ∗
ψ1,ϕ1Cψ2,ϕ2 onH2andA2
2 Bounded below operatorsC∗ψ,ϕ
LetHbe a Hilbert space. The set of all bounded operators fromHinto itself is denoted by
B(H). We say that an operatorA∈B(H) is bounded below if there is a constantc> 0 such thatch ≤ A(h)for allh∈H.
Iff is defined on a setVand if there is a positive constantmsuch that|f(z)| ≥mfor all
zinV, then we say thatf is bounded away from zero onV. In particular, we say thatψis bounded away from zero near the unit circle if there areδ> 0 and> 0 such that
ψ(z)> forδ<|z|< 1.
Suppose thatTbelongs toB(H). We denote the spectrum ofT, the essential spectrum ofT, the approximate point spectrum ofT, and the point spectrum ofT byσ(T),σe(T), σap(T) andσp(T), respectively. Moreover, the left essential spectrum ofT is denoted by σle(T).
Suppose thatϕ is an analytic self-map ofD. For almost all ζ ∈∂D, we defineϕ(ζ) = limr→1ϕ(rζ) (the statement of the existence of this limit can be found in [17, Theo-rem 2.2]). Iff is a bounded analytic function on the unit disk such that|f(eiθ)|= 1 al-most everywhere, then we callf an inner function. We know that ifϕis inner, thenCϕis bounded below onH2, and thereforeC
ϕhas a closed range (see [17, Theorem 3.8]). Now we state the following simple and well-known lemma, and we frequently use it in this paper.
Lemma 2.1 Let Cψ,ϕbe a bounded operator on H2or A2α.Then Cψ∗,ϕKw=ψ(w)Kϕ(w)for all
w∈D.
In this paper, for convenience, we assume thatγ= 1 forH2andγ =α+ 2 forA2
α.
Lemma 2.2 Suppose that A and B are two bounded operators on a Hilbert space H.If AB is a Fredholm operator,then B is left semi-Fredholm.
Proof Suppose thatABis a Fredholm operator. Then there are a bounded operatorCand a compact operatorKsuch thatCAB=I+K. ThereforeBis left semi-Fredholm.
Zhao [13] characterized Fredholm weighted composition operators onH2. Also, Zhao
[14] found necessary conditions of ϕ andψ for a weighted composition operator Cψ,ϕ onA2
αto be Fredholm. In the following proposition, we obtain a necessary and sufficient condition forCψ,ϕto be Fredholm onH2andA2α. Then we use it to find whenCψ∗1,ϕ1Cψ2,ϕ2 andCψ1,ϕ1Cψ∗2,ϕ2 are Fredholm. The idea of the proof of the next proposition is different from [13] and [14].
Proposition 2.3 The operator C∗ψ,ϕ is left semi-Fredholm on H2or A2
αif and only ifϕ∈ Aut(D)andψ∈H∞is bounded away from zero near the unit circle.Under these conditions,
Cψ,ϕis a Fredholm operator.
|xn|< 1 and |ψ(xn)|< 1/n. Then there exist a subsequence{xnm} andζ ∈∂Dsuch that
xnm→ζ asm→ ∞. Sinceψ(xnm)→0 asm→ ∞, by Lemma 2.1 we see that
lim
m→∞C
∗
ψ,ϕkxnmγ = lim
m→∞ψ(xnm)
Kϕ(xnm)γ Kxnmγ
≤lim sup
m→∞
ψ(xnm)
(1 +|xnm|)(1 –|xnm|)
(1 +|ϕ(xnm)|)(1 –|ϕ(xnm)|)
γ/2
≤2γ/2 lim
m→∞ψ(xnm)lim supm→∞
1 –|xnm|
1 –|ϕ(xnm)|
γ/2
= 0,
where the last equality follows from the fact that lim inf1–1–|ϕ|(xxnk)|
nk| = 0 (see [17,
Corol-lary 2.40]). Sincekxnm tends to zero weakly asm→ ∞(see [17, Theorem 2.17]), by [18,
Theorem 2.3, p. 350],Cψ∗,ϕis not left semi-Fredholm. This is a contradiction. Henceψis bounded away from zero near the unit circle. Denote the inner product onH2orA2
αby by ·,·γ, whereγ = 1 forH2andγ =α+ 2 forA2α. Define the bounded linear functionalFψ byFψ(f) = f,ψγ for eachf that belongs toH2orA2α, whereγ = 1 forH2andγ =α+ 2 forA2
α. We know that, for eachζ ∈∂D,Krζ/Krζγ tends to zero weakly asr→1. Then
lim
r→1Fψ
Krζ Krζγ
=lim
r→1
Krζ Krζγ
,ψ
γ = 0,
and so|ψ(rζ)|/Krζγ→0 asr→1. Now, we show thatϕis inner. For eachζ∈∂Dsuch thatϕ(ζ) :=limr→1ϕ(rζ) exists, by Lemma 2.1 we have
lim
r→1C
∗
ψ,ϕkrζγ =lim
r→1
|ψ(rζ)| Krζγ
1 1 –|ϕ(rζ)|2
γ/2
≤lim
r→1
|ψ(rζ)| Krζγ
1 1 –|ϕ(rζ)|
γ/2
.
Since|ψ(rζ)|/Krζγ →0 asr→1, ifϕ(ζ) /∈∂D, thenlimr→1Cψ∗,ϕkrζγ = 0, which is a contradiction (see [18, Theorem 2.3, p. 350]). Henceϕ is inner. SinceC∗ψ,ϕ is left semi-Fredholm, by Lemma 2.2,Tψ∗ is left semi-Fredholm. Thendim kerTψ∗ is finite. It follows from Lemma 2.1 thatψhas only finite zeroes inD. Ifϕis constant onD, thendim kerC∗ψ,ϕ= dim(ranCψ,ϕ)⊥=∞, a contradiction. Ifϕ(a) =ϕ(b) for somea,b∈Dwitha=b, then by using the idea similar to that used in [2, Lemma] there exist infinite sets {an} and{bn}
inDwhich are disjoint and such thatϕ(an) =ϕ(bn). We can assume thatψ(an)ψ(bn)= 0
becauseψ has only finite zeroes inD. By Lemma 2.1 we see that
Cψ∗,ϕ
Kan
ψ(an)
– Kbn
ψ(bn)
=Kϕ(an)–Kϕ(bn)≡0.
Therefore, Kan/ψ(an) –Kbn/ψ(bn)∈ kerCψ∗,ϕ. It is not hard to see that {Kan/ψ(an) –
Kbn/ψ(bn)}is a linearly independent set in the kernel ofC∗ψ,ϕ, and so we have our desired contradiction. Henceϕmust be univalent. Then [17, Corollary 3.28] implies thatϕis an automorphism ofD. SinceCψ,ϕCϕ–1=Mψis a bounded multiplication operator onH2and
A2
Conversely, suppose thatϕ∈Aut(D) andψ∈H∞is bounded away from zero near the unit circle. SinceCϕis invertible,Cϕ has a closed range. Sinceψ ≡0,kerTψ = (0). We infer thatTψ has a closed range by [18, Corollary 2.4, p. 352], [20, Theorem 3], and [12, Theorem 8], so by [18, Proposition 6.4, p. 208],Tψ is bounded below. We claim thatCψ,ϕ has a closed range. This can be seen as follows. Suppose that{hn}is a sequence such that
{Cψ,ϕ(hn)}converges tof asn→ ∞. SinceTψ has a closed range,{Cψ,ϕ(hn)}converges
toTψg for somegasn→ ∞. SinceTψ is bounded below, there is a constantc> 0 such thatTψ(Cϕ(hn) –g) ≥cCϕ(hn) –g. ThereforeCϕ(hn)→gasn→ ∞. There existsh
such thatCϕ(h) =gbecauseCϕhas a closed range. Hencef =Cψ,ϕ(h), as desired. Hence ranCψ,ϕis closed andkerCψ,ϕ= (0). [20, Theorem 3] and [12, Theorem 10] imply thatTψ is Fredholm, and sokerTψ∗ is finite dimensional. Sinceϕ∈Aut(D), it is not hard to see that
kerC∗ψ,ϕ= (ranCψ,ϕ)⊥= (ranTψ)⊥=kerTψ∗.
Therefore,dim kerC∗ψ,ϕ<∞, and the conclusion follows from [18, Corollary 2.4, p. 352].
In the next proposition, we give a necessary condition of ψ for an operatorC∗ψ,ϕ to be bounded below onH2 andA2
α. Then we use Proposition 2.4 to obtain all invertible weighted composition operators onH2andA2
α.
Proposition 2.4 Letψbe an analytic map ofD,and letϕbe an analytic self-map ofD.If Cψ∗,ϕis bounded below on H2or A2α,thenψ∈H∞is bounded away from zero onD,and
ϕ∈Aut(D).
Proof Letϕ≡dfor somed∈D. SinceC∗ψ,ϕ is bounded below, there is a constantc> 0 such thatC∗ψ,ϕfγ ≥cfγ for allf. Then for eachw∈D, by Lemma 2.1,Cψ∗,ϕKwγ = |ψ(w)|Kdγ≥cKwγ. Therefore, for eachw∈D,
ψ(w)≥ c Kdγ
1 (1 –|w|2)γ/2.
It is easy to see that ψ is bounded away from zero onD. Now assume that ϕ is not a constant function. Suppose thatψis not bounded away from zero onD. Therefore, there exist a sequence{xn}inDanda∈Dsuch thatxn→aand|ψ(xn)| →0 asn→ ∞. First,
suppose thata∈D. By Lemma 2.1 we have
Cψ∗,ϕkaγ =ψ(a)
1 –|a|2
1 –|ϕ(a)|2
γ/2
= 0.
SinceCψ∗,ϕis bounded below, 0≥ckaγ =c, a contradiction. Now assume thata∈∂D. It is not hard to see that there is a subsequence{xnm}such that{ϕ(xnm)}converges. By
Lemma 2.1 we see that
lim sup
m→∞
C∗ψ,ϕkxnmγ =lim sup
m→∞
ψ(xnm)
1 –|xnm|2
1 –|ϕ(xnm)|2
γ/2
. (1)
If{ϕ(xnm)}converges to a point inD, then (1) is equal to zero. Now assume that{ϕ(xnm)}
Julia-Carathéodory theorem we have
lim sup
m→∞
1 –|xnm|2
1 –|ϕ(xnm)|2
= 1 |ϕ(a)|,
which shows that (1) is equal to zero. Ifϕdoes not have a finite angular derivative ata, then
lim sup
m→∞
1 –|xnm|
1 –|ϕ(xnm)|
= 0,
so again (1) is equal to zero. SinceC∗ψ,ϕis bounded below andkxnmγ = 1, we havec= 0, is a contradiction. Therefore,ψ is bounded away from zero onD. Since by [18, Propo-sition 6.4, p. 208], 0 /∈σap(C∗ψ,ϕ), we have thatlimr→1C∗ψ,ϕkrζγ = 0 for allζ ∈∂D. We employ the idea of the proof of Proposition 2.3 to see thatϕis a univalent inner function. Thusϕ∈Aut(D) (see [17, Corollary 3.28]). Moreover, sinceCψ,ϕ is a bounded operator, as we saw in the proof of Proposition 2.3, we conclude thatψ∈H∞, and the proposition
follows.
Bourdon [21, Theorem 3.4] obtained the following corollary; we give another proof (see also [22, Theorem 2.0.1]).
Corollary 2.5 Letψ be an analytic map ofD,and letϕbe an analytic self-map ofD.The weighted composition operator Cψ,ϕis invertible on H2or Aα2 if and only ifϕ∈Aut(D)and
ψ∈H∞is bounded away from zero onD.
Proof LetCψ,ϕ be invertible. ThenCψ∗,ϕis bounded below. The conclusion follows from Proposition 2.4. The reverse direction is trivial sinceCϕandTψ are invertible.
Note that ifCψ,ϕis invertible, thenCψ∗,ϕis bounded below. Hence by Proposition 2.4 and Corollary 2.5 we can see thatCψ∗,ϕis bounded below if and only ifCψ,ϕis invertible.
The algebraA(D) consists of all continuous functions on the closure ofDthat are analytic onD. In the next corollary, we find some Fredholm weighted composition operators that are not invertible.
Corollary 2.6 Suppose that ϕ∈Aut(D)andψ ∈A(D).Assume that{z∈D:ψ(z) = 0}
is a nonempty finite set andψ(z)= 0for all z∈∂D.Then Cψ,ϕis Fredholm,but it is not
invertible.
Proof It is easy to see thatψ is bounded away from zero near the unit circle. Therefore the result follows from Proposition 2.3 and Corollary 2.5.
Theorem 2.7 Suppose thatψis an analytic map ofDandϕis an analytic self-map ofD.
The operator Cψ∗,ϕis bounded below on H2or A2
αif and only ifϕ∈Aut(D)andψ∈H∞is
bounded away from zero onD.
3 The operatorsCψ1,ϕ1C
∗
ψ2,ϕ2andC
∗
ψ1,ϕ1Cψ2,ϕ2
In this section, we find all Fredholm operatorsCψ1,ϕ1Cψ∗2,ϕ2andC ∗
A linear-fractional self-map ofDis a mapping of the formϕ(z) = (az+b)/(cz+d) with
ad–bc= 0 such thatϕ(D)⊆D. We denote the set of those maps byLFT(D). Suppose
ϕ(z) = (az+b)/(cz+d). It is well known that the adjoint ofCϕacting onH2andA2αis given byC∗ϕ=TgCσTh∗, whereσ(z) = (az–c)/(–bz+d) is a self-map ofD,g(z) = (–bz+d)–γ, and
h(z) = (cz+d)γ. Note thatgandhare inH∞([17, Theorem 9.2]). Ifϕ(ζ) =ηforζ,η∈∂D, thenσ(η) =ζ. We know thatϕ is an automorphism if and only ifσ is, and in this case,
σ=ϕ–1. The mapσis called the Krein adjoint ofϕ. We denote byF(ϕ) the set of all points in∂Dat whichϕhas a finite angular derivative.
Example 3.1 Suppose thatϕ∈LFT(D) is not an automorphism ofD. Assume thatψ ∈ H∞ is continuously extendable toD∪F(ϕ). Assume thatCψ,ϕCψ∗,ϕ is considered as an operator onH2orA2
α. Sinceϕis not an automorphism ofD,ϕ(D)⊆Dor there is only one pointζ∈∂Dsuch thatϕ(ζ)∈∂D. Ifϕ(D)⊆D, then by [17, p. 129],Cϕis compact. It is easy to see thatCψ,ϕCψ∗,ϕis a compact operator. Since compact operators are not Fredholm, we can see thatCψ,ϕCψ∗,ϕis not Fredholm.
In the other case, assume thatF(ϕ) ={ζ}. Because for each w∈∂Dsuch thatw=ζ,
σ(ϕ(w)) /∈∂D, we obtainσ◦ϕ∈/Aut(D). SinceCσ◦ϕis not Fredholm (see e.g. [2] and [16]), 0∈σe(Cσ◦ϕ). By [23, Corollary 2.2] and [4, Proposition 2.3] there is a compact operatorK such that
Cψ,ϕCψ∗,ϕ=ψ(ζ)
2
CϕCϕ∗+K.
Also, [23, Theorem 3.1], [23, Proposition 3.6], and [24, Theorem 3.2] imply that there is a compact operatorKsuch that
Cψ,ϕCψ∗,ϕ=ψ(ζ)
2
ϕ(ζ)–γCσ◦ϕ+K. (2)
From the fact that 0∈σe(Cσ◦ϕ) and equation (2) we can infer that 0∈σe(Cψ,ϕCψ∗,ϕ). Then
Cψ,ϕCψ∗,ϕis not Fredholm.
By the preceding example it seems natural to conjecture that ifCψ,ϕCψ∗,ϕis Fredholm, thenϕ∈Aut(D). We will prove our conjecture in Theorem 3.2 and show that ifCψ1,ϕ1Cψ∗2,ϕ2 is Fredholm onH2orA2α, thenCψ1,ϕ1andCψ2,ϕ2are Fredholm.
Theorem 3.2 The operator Cψ1,ϕ1Cψ∗2,ϕ2 is Fredholm on H
2 or A2
α if and only ifϕ1,ϕ2∈
Aut(D),ψ1,ψ2∈H∞,andψ1andψ2are bounded away from zero near the unit circle.
Proof LetCψ1,ϕ1C∗ψ2,ϕ2 be Fredholm. ThereforeC ∗
ψ2,ϕ2 is left semi-Fredholm. By Proposi-tion 2.3 we see thatϕ2∈Aut(D) andψ2∈H∞is bounded away from zero near the unit
circle. SinceCψ2,ϕ2Cψ∗1,ϕ1 is Fredholm, again we can see thatϕ1is an automorphism ofD andψ1∈H∞is bounded away from zero near the unit circle.
Conversely, letϕ1,ϕ2∈Aut(D) andψ1,ψ2∈H∞be bounded away from zero near the
unit circle. By Proposition 2.3,Cψ1,ϕ1andC∗ψ2,ϕ2are Fredholm, so the result follows.
In the following theorem for functionsψ1,ψ2∈A(D), we find all Fredholm operators
Theorem 3.3 Suppose thatψ1,ψ2∈A(D).Letϕ1andϕ2be univalent self-maps ofD.The
operator C∗ψ1,ϕ1Cψ2,ϕ2is Fredholm on H2or A2αif and only if Cψ1,ϕ1and Cψ2,ϕ2are Fredholm
on H2or A2
α,respectively.
Proof LetCψ∗1,ϕ1Cψ2,ϕ2 be Fredholm onH2 orA2α. ThenCψ∗2,ϕ2Cψ1,ϕ1 is also Fredholm. It is easy to see thatCϕ2 andCϕ1 are left semi-Fredholm. Therefore, 0 /∈σle(Cϕ1) and 0 /∈
σle(Cϕ2). Sincedim kerC∗ψ1,ϕ1Cψ2,ϕ2<∞anddim kerCψ∗2,ϕ2Cψ1,ϕ1<∞,ψ1≡0,ψ2≡0, and
ϕ1andϕ2are not constant functions. By the open mapping theorem we know that 0 /∈
σp(Cϕ1) and 0 /∈σp(Cϕ2). Now [18, Proposition 4.4, p. 359] implies that 0 /∈σap(Cϕ1) and 0 /∈σap(Cϕ2). Hence by [18, Proposition 6.4, p. 208],ranCϕ1 andranCϕ2 are closed. By [1, Theorem 5.1],ϕ1,ϕ2∈Aut(D). SinceC∗ϕ1 andCϕ2 are invertible, (Cϕ∗1)
–1 andC–1
ϕ2 are Fredholm. Then Tψ∗1Tψ2 is Fredholm, and so 0 /∈σe(Tψ1ψ2). We get from [25] and [20, Theorem 2] thatψ1andψ2never vanish on∂D. Sinceψ1≡0 andψ2≡0,ψ1andψ2have
only finite zeroes onD. This implies that there isr< 1 such that for eachwwithr<|w|< 1,
ψ1(w)= 0 andψ2(w)= 0. Therefore,ψ1andψ2are bounded away from zero near the unit
circle. Therefore, by Proposition 2.3,Cψ1,ϕ1 andCψ2,ϕ2 are Fredholm onH
2orA2
α. The reverse implication follows from the fact stated before Theorem 3.2.
Acknowledgements
The author would like to thank the referees for their valuable comments and suggestions.
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
The author read and approved the final manuscript.
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Received: 28 August 2017 Accepted: 16 January 2018
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