Articulo 14.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (217–234). June 2010 Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols L. P. Castro1 and A. S. Silva2 Department of Mathematics, Aveiro University, 3810-193 Aveiro, Portugal email: castro@ua.pt email: anabela.silva@ua.pt ABSTRACT We will present sufficient conditions for the Fredholm property of Wiener-Hopf plus and minus Hankel operators with different Fourier matrix symbols in the C∗-algebra of semi- almost periodic elements. In addition, under such conditions, we will derive a formula for the sum of the Fredholm indices of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators. Some examples are provided to illustrate the results of the paper. RESUMEN Presentaremos condiciones suficientes para garantizar la propiedad de Fredholm de oper- adores de tipo Wiener-Hopf más y menos Hankel con diferentes śımbolos de Fourier matri- ciales en la C*-álgebra de elementos semi-casi periódicos. Además, bajo tales condiciones, obtendremos una fórmula para la suma de los ı́ndices de Fredholm de estos operadores Wiener-Hopf más Hankel y Wiener-Hopf menos Hankel. Algunos ejemplos son dados para ilustrar los resultados del art́ıculo. Key words and phrases: Fredholm property, Fredholm index, Wiener-Hopf operator, Hankel oper- ator, semi-almost periodic matrix-valued function 1Corresponding author: castro@ua.pt 2Sponsored by Fundação para a Ciência e a Tecnologia (Portugal) under grant number SFRH/BD/38698/2007. 218 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) Math. Subj. Class.: 47B35, 47A05, 47A12, 47A20, 42A75. 1 Introduction One of the objectives of the present paper is to obtain sufficient conditions for the Fredholm property of matrix Wiener-Hopf plus and minus Hankel operators of the form WΦ1 ± HΦ2 : [L 2 +(R)] N → [L2(R+)] N (N ∈ N) (1) with WΦ1 and HΦ2 being matrix Wiener-Hopf and Hankel operators defined by WΦ1 = r+F −1Φ1 · F and HΦ2 = r+F −1Φ2 · FJ , respectively. We denote by BN×N the Banach algebra of all N × N matrices with entries in a Banach algebra B, and BN will denote the Banach space of all N dimensional vectors with entries in a Banach space B. Let L2(R) be the usual space of square-integrable Lebesgue measurable functions on the real line R, and L2(R+) the corresponding one in the positive half-line R+ = (0, +∞). We are using the notation L2+(R) for the subspace of L 2(R) formed by all the functions supported on the closure of R+. In addition, r+ represents the operator of restriction from [L 2 +(R)] N into [L2(R+)] N , F denotes the Fourier transformation, J is the reflection operator given by the rule Jϕ(x) = ϕ̃(x) = ϕ(−x), x ∈ R, and (in general) Φ1, Φ2 ∈ [L ∞(R)]N×N are the so-called Fourier matrix symbols. It is well-known that for such Fourier matrix symbols (with Lebesgue measurable and essentially bounded entries) the operators in (1) are bounded. We would like to point out that the operators presented in (1) have been central objects in several recent research programmes (cf. e.g. [1]–[8]). One of the reasons for such interest is related to the fact that eventual additional knowledge about regularity properties of (1) have direct consequences in different types of applications (see [9]–[12]). In the present work, the main purpose is to obtain conditions which will characterize the situation when WΦ1 + HΦ2 and WΦ1 − HΦ2 are at the same time Fredholm operators, and to present a formula for the sum of their Fredholm indices. All these will be done for matrices Φ1 and Φ2 in the class of semi-almost periodic elements (cf. Definition 2.1). Therefore, the present work deals with a more general situation than what was under consideration in [1, 7, 13], and some of the present results can be seen as a generalization of part of the results of the just mentioned works. However, the most general situation of considering the operators WΦ1 + HΦ2 and WΦ1 − HΦ2 independent of each other (with semi-almost periodic symbols) is not considered in the present paper and remains open. 2 Preliminary results and notions The smallest closed subalgebra of L∞(R) that contains all the functions eλ (λ ∈ R), where eλ(x) = eiλx, x ∈ R, is denoted by AP and called the algebra of almost periodic functions: AP := algL∞(R){eλ : λ ∈ R}. In addition, we will also use the notation AP+ := algL∞(R){eλ : λ ≥ 0}, AP− := algL∞(R){eλ : λ ≤ 0} CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 219 for these two subclasses of AP (which are still closed subalgebras of L∞(R)). We will likewise make use of the Wiener subclass of AP (denoted by AP W ) formed by all those elements from AP which allow a representation by an absolutely convergent series. Therefore, AP W is precisely the (proper) subclass of all functions ϕ ∈ AP which can be written in an absolutely convergent series of the form: ϕ = ∑ j ϕj eλj , λj ∈ R , ∑ j |ϕj| < ∞ . We recall that all AP functions have a well-known mean value. The existence of such a number is provided in the next proposition. Proposition 2.1. (cf., e.g., [14, 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) := lim α→∞ 1 |Iα| ∫ Iα ϕ(x) dx exists, is finite, and is independent of the particular choice of the family {Iα}. For any ϕ ∈ AP , the number that has just been introduced M (ϕ) is called the Bohr mean value or simply the mean value of ϕ. In the matrix case the mean value is defined entry-wise. Let Ṙ := R ∪ {∞}. We will denote by C(Ṙ) the set of all continuous functions ϕ on the real line for which the two limits ϕ(−∞) := lim x→−∞ ϕ(x), ϕ(+∞) := lim x→+∞ ϕ(x) exist and coincide. The common value of these two limits will be denoted by ϕ(∞). Furthermore, C0(Ṙ) will represent the collection of all ϕ ∈ C(Ṙ) for which ϕ(∞) = 0. Let C(R) := C(R) ∩ P C(Ṙ), where C(R) is the usual set of continuous functions on the real line and P C(Ṙ) is the set of all bounded piecewise continuous functions on Ṙ. As mentioned above, we will deal with Fourier symbols from the C∗-algebra of semi-almost periodic elements which is defined as follows. Definition 2.1. The C∗-algebra SAP of all semi-almost periodic functions on R is the smallest closed subalgebra of L∞(R) that contains AP and C(R): SAP = algL∞(R){AP, C(R)}. In addition, it is possible to interpret the SAP functions in a different form due to the following characterization of D. Sarason [15]. Theorem 2.1. Let u ∈ C(R) be any function for which u(−∞) = 0 and u(+∞) = 1. If ϕ ∈ SAP , then there is ϕl, ϕr ∈ AP and ϕ0 ∈ C0(Ṙ) such that ϕ = (1 − u)ϕl + uϕr + ϕ0. (2) 220 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) The functions ϕl, ϕr are uniquely determined by ϕ, and independent of the particular choice of u. The maps ϕ 7→ ϕl, ϕ 7→ ϕr are C∗-algebra homomorphisms of SAP onto AP. This theorem is also valid in the matrix case. Let us now recall the so-called right and left AP factorizations. In such notions, we will use the notation GB for the group of all invertible elements of a Banach algebra B. Definition 2.2. 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) (3) for all x ∈ R, with Φ− ∈ GAP N×N − , Φ+ ∈ GAP N×N + , (4) and D is a diagonal matrix of the form D(x) = diag [ eiλ1x, . . . , eiλN x ] , λj ∈ R. The numbers λj are called the right AP indices of the factorization. A right AP factorization with D = IN×N is referred to be a canonical right AP factorization. If in a right AP factorization besides condition (4) the factors Φ± belong to AP W , then we say that Φ admits a right APW factorization (it being clear in such a case that Φ ∈ AP W ). It is said that a matrix function Φ ∈ GAP N×N admits a left AP factorization if instead of (3) we have Φ(x) = Φ+(x) D(x) Φ−(x) for all x ∈ R, and Φ± and D having the same property as above. Note that from the last definition it follows that if an invertible almost periodic matrix function Φ admits a right AP factorization, then Φ̃ 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 the above case k(Φ) := (λ1, . . . , λN ). If we consider the case with equal right AP indices (k(Φ) := (λ1, λ1, . . . , λ1)), then the matrix d(Φ) := M (Φ−)M (Φ+) is independent of the particular choice of the right AP factorization. In this case, this matrix d(Φ) is called the geometric mean of Φ. In order to relate operators and to transfer certain operator properties between the related oper- ators, we will also be using the known notion of equivalence after extension relation between bounded linear operators. CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 221 Definition 2.3. Consider two bounded linear operators T : X1 → X2 and S : Y1 → Y2, acting between Banach spaces. We say that T is equivalent after extension to S if there are Banach spaces Z1 and Z2 and invertible bounded linear operators E and F such that [ T 0 0 IZ1 ] = E [ S 0 0 IZ2 ] F, (5) where IZ1 , IZ2 represent the identity operators in Z1 and Z2, respectively. This relation between T and S will be denoted by T ∗ ∼ S. Note that such operator relation between two operators T and S, if obtained, allows several consequences on the properties of these two operators. Namely, T and S will have the same Fredholm regularity properties (i.e., the properties that directly depend on the kernel and on the image of the operator). As we will realize in the next result, such kind of operator relation is valid for a diagonal operator constructed with our Wiener-Hopf plus and minus Hankel operators and a corresponding pure Wiener-Hopf operator. Lemma 2.1. Let Φ1, Φ2 ∈ G[L ∞(R)]N×N . Then DΦ1,2 := [ WΦ1 + HΦ2 0 0 WΦ1 − HΦ2 ] : [L2+(R)] 2N → [L2(R+)] 2N (6) is equivalent after extension to the Wiener-Hopf operator WΨ : [L 2 +(R)] 2N → [L2(R+)] 2N with Fourier symbol Ψ =   Φ1 − Φ2Φ̃ −1 1 Φ̃2 −Φ2Φ̃ −1 1 Φ̃−11 Φ̃2 Φ̃ −1 1   . (7) We refer to [4, Theorem 2.1] for a detailed proof of this lemma (where all the elements in the corresponding operator relation are given in explicit form and within the context of a so-called ∆- relation after extension; see [16]). 3 The Fredholm property In the present section we will work out characterizations for the Fredholm property of WΦ1 + HΦ2 and WΦ1 − HΦ2 . We start with the general case (where no dependence between the SAP matrices Φ1 and Φ2 is imposed), and in the last subsection (of the present section) we will consider a particular case where some relation between Φ1 and Φ2 will allow extra detailed descriptions. 3.1 General Case We start by recalling a known Fredholm characterization for Wiener-Hopf operators with SAP matrix Fourier symbols having lateral almost periodic representatives admitting right AP factorizations. 222 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) Theorem 3.1. (cf. e.g., [14, Theorem 10.11]) Let Φ ∈ SAP N×N and assume that the almost periodic representatives Φℓ and Φr admit a right AP factorization. Then the Wiener-Hopf operator WΦ is Fredholm if and only if: (i) Φ ∈ GSAP N×N ; (ii) The almost periodic representatives Φℓ and Φr admit canonical right AP factorizations (and therefore with k(Φℓ) = k(Φr) = (0, . . . , 0)); (iii) sp(d−1(Φr)d(Φℓ)) ∩ (−∞, 0] = ∅, where sp(d −1(Φr)d(Φℓ)) stands for the set of the eigenvalues of the matrix d−1(Φr )d(Φℓ) := [d(Φr)] −1d(Φℓ). The matrix version of Sarason’s Theorem (cf. Theorem 2.1) applied to Ψ in (7) says that if Ψ ∈ GSAP 2N×2N then this matrix function admits the following representation Ψ = (1 − u)Ψℓ + uΨr + Ψ0, (8) where Ψℓ,r ∈ GAP 2N×2N are defined for the particular Ψ in (7) by Ψℓ =   Φ1ℓ − Φ2ℓΦ̃ −1 1r Φ̃2r −Φ2ℓΦ̃ −1 1r Φ̃−11r Φ̃2r Φ̃ −1 1r   (9) and Ψr =   Φ1r − Φ2rΦ̃ −1 1ℓ Φ̃2ℓ −Φ2rΦ̃ −1 1ℓ Φ̃−11ℓ Φ̃2ℓ Φ̃ −1 1ℓ   (10) (with Φ1ℓ, Φ1r and Φ2ℓ, Φ2r being the local representatives at ∓∞ of Φ1 and Φ2, respectively), u ∈ C(R), u(−∞) = 0, u(+∞) = 1, Ψ0 ∈ [C0(Ṙ)] 2N×2N . From (9) it follows that Ψ̃−1 ℓ =   Φ̃−11ℓ Φ̃ −1 1ℓ Φ̃2ℓ −Φ2rΦ̃ −1 1ℓ Φ1r − Φ2rΦ̃ −1 1ℓ Φ̃2ℓ   . (11) Therefore, we obtain that Ψr =   0 IN IN 0   Ψ̃−1 ℓ   0 IN IN 0   . (12) These representations, and the above relation between the operator (6) and the pure Wiener-Hopf operator, lead to the following characterization in the case when Ψℓ admits a right AP factorization. Theorem 3.2. Let Ψ ∈ SAP 2N×2N and assume that Ψℓ admits a right AP factorization. In this case, the Wiener-Hopf plus and minus Hankel operators WΦ1 + HΦ2 and WΦ1 −HΦ2 are both Fredholm if and only if the following three conditions are satisfied: CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 223 (c1) Ψ ∈ GSAP 2N×2N ; (c2) Ψℓ admits a canonical right AP factorization; (c3) sp[Hd(Ψℓ)] ∩ iR = ∅, where H =   0 IN IN 0  . Proof. (i) Let us assume that the Wiener-Hopf plus and minus Hankel operators WΦ1 + HΦ2 and WΦ1 − HΦ2 are both Fredholm operators. Then, WΨ is also Fredholm due to the above presented equivalence after extension relation. Therefore, using Theorem 3.1 we obtain that Ψ ∈ GSAP 2N×2N , Ψℓ and Ψr admit canonical right AP factorizations and sp(d−1(Ψr)d(Ψℓ)) ∩ (−∞, 0] = ∅. (13) In particular, we realize that propositions (c1) and (c2) are already fulfilled. Additionally, the canon- ical right AP factorization of Ψℓ can be normalized into Ψℓ = θ−Λθ+, (14) where θ± have the same factorization properties as the original lateral factors of the canonical factor- ization but with M (θ±) = I, and where Λ := d(Ψℓ). Let H =   0 IN IN 0   . (15) From (12) and (14) we derive that Ψr = HΨ̃ −1 ℓ H = Hθ̃−1+ Λ −1θ̃ −1 − H which shows that d(Ψr) = HΛ −1H (16) and therefore d−1(Ψr) = HΛH. (17) In this way, we conclude that sp[d−1(Ψr)d(Ψℓ)] = sp[HΛHΛ] = sp[(HΛ)2]. Thus, (13) turns out to be equivalent to sp[(HΛ)2] ∩ (−∞, 0] = ∅ which leads to sp[HΛ] ∩ iR = ∅ . Therefore, the proposition (c3) is also satisfied. 224 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) (ii) Let us now assume that (c1), (c2) and (c3) hold true. From condition (c1) we have Ψ ∈ GSAP 2N×2N . The left and right representatives of Ψ are given by (9) and (10). Due to the fact that Ψℓ admits a canonical right AP factorization, it follows that Ψ −1 ℓ admits a canonical left AP factorization and Ψ̃−1 ℓ admits a canonical right AP factorization. Therefore,   0 IN IN 0   Ψ̃−1 ℓ   0 IN IN 0   = Ψr (18) admits a canonical right AP factorization. These two canonical right AP factorizations and condition (c3) imply that sp(d−1(Ψr)d(Ψℓ)) ∩ (−∞, 0] = ∅. All these facts together with Theorem 3.1 give us that WΨ is a Fredholm operator. Using the equiv- alence after extension relation, we obtain that the Wiener-Hopf plus and minus Hankel operators WΦ1 + HΦ2 and WΦ1 − HΦ2 are both Fredholm operators. Let us now think about the case of Ψ ∈ SAP W 2N×2N , where SAP W denotes the algebra of all semi-almost periodic functions ϕ whose almost periodic representatives ϕℓ and ϕr (cf. (2)) belong to AP W . If Ψ ∈ SAP W 2N×2N , then in Theorem 3.2 we can drop the assumption which states that Ψℓ admits an AP factorization and also simplify the corresponding conditions (c1) and (c2): Corollary 3.1. Let Ψ ∈ SAP W 2N×2N . The Wiener-Hopf plus and minus Hankel operators WΦ1 + HΦ2 and WΦ1 − HΦ2 are both Fredholm if and only if the following three conditions are satisfied: (c1′) Ψ ∈ GSAP W 2N×2N ; (c2′) Ψℓ admits a canonical right AP W factorization; (c3′) sp[Hd(Ψℓ)] ∩ iR = ∅, where H =   0 IN IN 0  . Proof. The result is derived from Theorem 3.2 and from the following known facts which apply to any Φ ∈ GAP W 2N×2N : (j) Φ has a canonical right AP factorization if and only if Φ has a canonical right AP W factorization; (jj) Φ has a canonical right AP W factorization if and only if WΦ is invertible. In fact, for our Ψ ∈ SAP W 2N×2N , note that if both operators WΦ1 + HΦ2 and WΦ1 − HΦ2 have the Fredholm property, then by the above equivalence after extension relation we also have that the Wiener-Hopf operator WΨ is a Fredholm operator. Therefore, WΨℓ and WΨr are invertible operators and from (jj) this is equivalent to Ψℓ and Ψr to admit canonical right AP W factorizations. Thus, the assertion now follows from Theorem 3.2 and proposition (j). 3.2 The case of Φ1 = Φ̃2 For some particular cases where Φ1 and Φ2 are dependent on each other, we can simplify the statement of Theorem 3.2 by making use of consequent equivalence after extension operator relations. In the present subsection we will analyze the case of Φ1 = Φ̃2. CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 225 Let Φ2 ∈ GSAP N×N and consider Φ1 = Φ̃2. In this case, the matrix Ψ takes the form Ψ =   0 −IN Φ−12 Φ̃2 Φ̃ −1 2   and the Wiener-Hopf operator WΨ is equivalent after extension to the operator WΦ−1 2 fΦ2 . In fact, we have in this case: WΨ = r+F −1   0 −IN IN Φ̃ −1 2   Fℓ0r+F−1   Φ−12 Φ̃2 0 0 IN   F (where ℓ0 : [L 2(R+)] 2N → [L2+(R)] 2N denotes the zero extension operator). This together with the equivalence after extension relation between the operator (6) and WΨ shows that DΦ1,2 ∗ ∼ W Φ −1 2 fΦ2 (19) (due to the transitivity of the equivalence after extension relation). From Theorem 2.1 we conclude that Φ2 ∈ GSAP N×N admits the following representation Φ2 = (1 − u)Φ2ℓ + uΦ2r + Φ20 (20) (with Φ20 ∈ [C0(Ṙ)] N×N ) and Φ−12 Φ̃2 = [(1 − u)Φ2ℓ + uΦ2r + Φ20] −1[(1 − ũ)Φ̃2ℓ + ũΦ̃2r + Φ̃20]. (21) Therefore, from (21), we obtain that (Φ−12 Φ̃2)ℓ = Φ −1 2ℓ Φ̃2r, (Φ −1 2 Φ̃2)r = Φ −1 2r Φ̃2ℓ. (22) These representations and the above relation between WΨ and WΦ−1 2 fΦ2 (when Φ1 = Φ̃2), allow us to construct the following result. Theorem 3.3. Let Φ2 ∈ SAP N×N and assume that Φ−12ℓ Φ̃2r admits a right AP factorization. In this case, the Wiener-Hopf plus and minus Hankel operators W fΦ2 + HΦ2 and W fΦ2 − HΦ2 are both Fredholm operators if and only if the following three conditions are satisfied: (l) Φ2 ∈ GSAP N×N ; (ll) Φ−12ℓ Φ̃2r admits a canonical right AP factorization; (lll) sp[d(Φ−12ℓ Φ̃2r)] ∩ iR = ∅. Proof. (i) If W fΦ2 ±HΦ2 are both Fredholm operators, then from a similar reasoning as in [5, Proposition 2.6] it follows that Φ2 ∈ G[L ∞(R)]N×N and therefore Φ2 ∈ GSAP N×N . The Fredholm property of the Wiener-Hopf plus and minus Hankel operators W fΦ2 + HΦ2 and W fΦ2 − HΦ2 implies that the operator WΨ is Fredholm and due to the transitivity of equivalence 226 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) after extension relations, it follows that the operator W Φ −1 2 fΦ2 has also the Fredholm property (cf. (19)). Employing Theorem 3.1 we obtain that Φ−12 Φ̃2 ∈ GSAP N×N , (Φ−12 Φ̃2)ℓ and (Φ −1 2 Φ̃2)r admit canonical right AP factorizations and sp[d−1((Φ−12 Φ̃2)r)d((Φ −1 2 Φ̃2)ℓ)] ∩ (−∞, 0] = ∅. (23) Due to (22) we conclude that Φ−12ℓ Φ̃2r admits a canonical right AP factorization and we derive from (23) that sp[d−1(Φ−12r Φ̃2ℓ)d(Φ −1 2ℓ Φ̃2r)] ∩ (−∞, 0] = ∅. (24) A canonical right AP factorization of Φ−12ℓ Φ̃2r can be normalized into Φ−12ℓ Φ̃2r = Θ−ΛΘ+, (25) where Θ± have the same factorization properties as the original lateral factors of the canonical fac- torization but with M (Θ±) = I, and where Λ := d(Φ −1 2ℓ Φ̃2r). Thus, (25) allows Φ−12r Φ̃2ℓ = ( ˜ Φ−12ℓ Φ̃2r) −1 = Θ̃−1+ Λ −1Θ̃−1 − which shows that d(Φ−12r Φ̃2ℓ) = Λ −1 and therefore (24) turns out to be equivalent to sp[Λ2] ∩ (−∞, 0] = ∅. From the eigenvalue definition, it therefore results in sp[Λ] ∩ iR = ∅ which proves proposition (lll). (ii) Let us now consider that (l)–(lll) hold true. The property (l) implies that Φ−12 Φ̃2 is also invertible in SAP N×N . Since Φ−12ℓ Φ̃2r = (Φ −1 2 Φ̃2)ℓ admits a canonical right AP factorization, then ( ˜ Φ−12 Φ̃2)ℓ = Φ̃ −1 2ℓ Φ2r admits a canonical left AP factorization and [( ˜ Φ−12 Φ̃2)ℓ] −1 = Φ−12r Φ̃2ℓ admits a canonical right AP factorization. These last two canonical right AP factorizations and condition (lll) imply that sp[d−1((Φ−12 Φ̃2)r)d((Φ −1 2 Φ̃2)ℓ)] ∩ (−∞, 0] = sp[d −1(Φ−12r Φ̃2ℓ)d(Φ −1 2ℓ Φ̃2r)] ∩ (−∞, 0] = ∅. All these facts together with Theorem 3.1 show that W Φ −1 2 fΦ2 is a Fredholm operator. Using the equivalence after extension relations of (19) and Lemma 2.1, we obtain that the Wiener-Hopf plus and minus Hankel operators W fΦ2 + HΦ2 and W fΦ2 − HΦ2 have the Fredholm property. CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 227 4 Index formula In the present section we will be concentrated in obtaining a Fredholm index formula for DΦ1,2 , i.e., for the sum of Wiener-Hopf plus and minus Hankel operators WΦ1 ± HΦ2 with Fourier symbols Φ1, Φ2 ∈ GSAP N×N such that Ψℓ admits a right AP factorization. Within this context, let us now assume that WΦ1 + HΦ2 and WΦ1 − HΦ2 are Fredholm operators. 4.1 General situation Let GSAP0,0 denote the set of all functions ϕ ∈ GSAP for which k(ϕℓ) = k(ϕr ) = 0. To define the Cauchy index of ϕ ∈ GSAP0,0 we need the lemma presented below. Lemma 4.1. (See e.g. [14, 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 2π lim α→∞ 1 |Iα| ∫ Iα ((argϕ)(x) − (argϕ)(−x))dx (26) exists, is finite, and is independent of the particular choices of {(xα, yα)}α∈A and arg ϕ. The limit (26) is denoted by indϕ and is usually called the Cauchy index of ϕ. Moreover, following [7, Section 4.3] we can generalize the notion of Cauchy index for SAP presented in Lemma 4.1 for functions with k(ϕℓ) + k(ϕr) = 0. The following theorem provides a formula for the Fredholm index of matrix Wiener-Hopf operators with SAP Fourier symbols. Theorem 4.1. (Cf. e.g. [14, Theorem 10.12]) Let Φ ∈ SAP N×N . If the almost periodic representa- tives Φℓ, Φr admit right AP factorizations, and if WΦ is a Fredholm operator, then Ind WΦ = −ind[det Φ] − N∑ k=1 ( 1 2 − { 1 2 − 1 2π arg ξk }) (27) where ξ1, . . . , ξN ∈ C\(−∞, 0] are the eigenvalues of the matrix d −1(Φr)d(Φℓ) and {·} stands for the fractional part of a real number. Additionally, when choosing arg ξk in (−π, π), we have Ind WΦ = −ind [det Φ] − 1 2π N∑ k=1 arg ξk. We will now be concerned with the question of finding a formula for the sum of the Fredholm indices of WΦ1 + HΦ2 and WΦ1 − HΦ2 (i.e., Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ]). Using the equivalence after extension relation presented in Lemma 2.1, we conclude that Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] = Ind WΨ. 228 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) Observing that WΨ is a Fredholm operator and using (27), we obtain IndWΨ = −ind[det Ψ] − 2N∑ k=1 ( 1 2 − { 1 2 − 1 2π arg ηk }) (28) where ηk ∈ C\(−∞, 0] are the eigenvalues of the matrix of d −1(Ψr)d(Ψℓ) = (Hd(Ψℓ)) 2, with H =   0 IN IN 0   (cf. (16)–(17)). Therefore, (28) can be rewritten as IndWΨ = −ind[det Ψ] − 2N∑ n=1 ( 1 2 − { 1 2 − 1 π arg ζk }) (29) where ζk ∈ C\iR are the eigenvalues of the matrix Hd(Ψℓ). Moreover, formula (28) is reduced to IndWΨ = −ind[det Ψ] − 1 π 2N∑ k=1 β(ζk) (30) where β(ζk) := { arg(ζk) if ℜe ζk > 0 arg(−ζk) if ℜe ζk < 0 (31) when choosing the argument in (− π 2 , π 2 ). These conclusions are assembled in the following corollary. Corollary 4.1. Let Ψ ∈ GSAP 2N×2N and assume that Ψℓ admits a right AP factorization. If WΦ1 ± HΦ2 are Fredholm operators, then Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] = −ind[det Ψ] − 2N∑ k=1 ( 1 2 − { 1 2 − 1 π arg ζk }) (32) where ζk ∈ C\iR are the eigenvalues of the matrix Hd(Ψℓ). Moreover, making use of (31), formula (32) simplifies to the following one: Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] = −ind[det Ψ] − 1 π 2N∑ k=1 β(ζk). (33) 4.2 The case of Φ1 = Φ̃2 For the particular case where Φ1 = Φ̃2 we can simplify formula (33) even further. In fact, when Φ1 = Φ̃2, employing the equivalence after extension relation (19), we deduce that Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] = −ind[det(Φ −1 2 Φ̃2)] − 1 π N∑ k=1 β(δk), (34) where δk ∈ C\iR are the eigenvalues of the matrix d(Φ −1 2ℓ Φ̃2r) and β(δk) = { arg(δk) if ℜe δk > 0 arg(−δk) if ℜe δk < 0 CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 229 with the argument in both cases being chosen in (− π 2 , π 2 ). In addition, let us now simplify the form of ind[det(Φ−12 Φ̃2)]. Observing that the matrix Φ −1 2ℓ Φ̃2r has a canonical right AP factorization, it holds k(Φ−12ℓ Φ̃2r) = (0, . . . , 0) and consequently k(det(Φ−12ℓ Φ̃2r)) = 0. Taking this into consideration, it follows that k((det Φ−12 )ℓ) + k((det Φ −1 2 )r) = k(det(Φ −1 2ℓ )) + k(det(Φ −1 2r )) = k(det(Φ−12ℓ )) + k(det(Φ2r) −1) = k(det(Φ−12ℓ )) + k[( ˜det(Φ2r)−1) −1] = k(det(Φ−12ℓ )) + k( ˜det(Φ2r)) = k(det(Φ−12ℓ )) + k(det(Φ̃2r)) = k(det(Φ−12ℓ ) det(Φ̃2r)) = k(det(Φ−12ℓ Φ̃2r)) = 0 (35) also because for any f ∈ GAP we have k(f ) = k(f̃ −1) and [det Φ]ℓ = det Φℓ. Applying a similar reasoning to Φ̃2, we obtain k((det Φ̃2)ℓ) + k((det Φ̃2)r) = k(det(Φ̃2ℓ)) + k(det(Φ̃2r)) = k( ˜ det(Φ̃2ℓ)−1) + k(det(Φ̃2r)) = k(det(Φ−12ℓ )) + k(det(Φ̃2r)) = k(det(Φ−12ℓ ) det(Φ̃2r)) = k(det(Φ−12ℓ Φ̃2r)) = 0. (36) Employing now (26), (35) and (36), the following computation holds true: ind[det(Φ−12 Φ̃2)] = ind[det Φ −1 2 det Φ̃2] = ind[det Φ−12 ] + ind[det Φ̃2] = ind[det Φ2] −1 + ind[d̃et Φ2] = ind[det Φ2] −1 − ind[det Φ2] = −ind[det Φ2] − ind[det Φ2] = −2 ind[det Φ2]. Thus, we have just concluded the following corollary. Corollary 4.2. Let Φ1, Φ2 ∈ GSAP N×N such that Φ1 = Φ̃2 and assume that Φ −1 2ℓ Φ̃2r admits a right AP factorization. If WΦ1 ± HΦ2 are Fredholm operators, then Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] = 2 ind[det Φ2] − 1 π N∑ k=1 β(δk) (37) 230 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) where δk ∈ C\iR are the eigenvalues of the matrix d(Φ −1 2ℓ Φ̃2r) and β(δk) = { arg(δk) if ℜe δk > 0 arg(−δk) if ℜe δk < 0 (38) with the argument in both cases being chosen in (− π 2 , π 2 ). 5 Examples In the present section we exemplify the above theory with two particular cases of corresponding Fourier symbol matrices Φ1 and Φ2. 5.1 First example Let Φ1 = Φ̃2, with Φ2(x) = (1 − u(x))   eix 0 0 e−ix   + u(x)   e−ix 0 0 eix   +   0 − 1 x−i 1 x+i 0   (39) and where u is the real-valued function defined by u(x) = { 1 2 ex if x < 0 1 − 1 2 e−x if x ≥ 0. (40) From (39) and Theorem 2.1, it becomes clear that Φ2 ∈ SAP 2×2. In addition, we will show that Φ2 ∈ GSAP 2×2. To this purpose, let us compute the determinant of Φ2: det Φ2(x) = det   (1 − u(x))eix + u(x)e−ix − 1 x−i 1 x+i (1 − u(x))e−ix + u(x)eix   = 1 + (2u(x) − 2u2(x))(cos(2x) − 1) + 1 x2+1 . Recalling that u is a real-valued function given by (40), we obtain det Φ2(x) =    1 + (ex − e2x)(cos(2x) − 1) + 1 x2+1 if x < 0 1 + (e−2x − e−x)(cos(2x) − 1) + 1 x2+1 if x ≥ 0 (41) Let us first show that det Φ2(x) 6= 0 for x ∈ (−∞, 0). In this domain ex − e2x belongs to (0, 1 4 ] and cos(2x) − 1 ∈ [−2, 0]. Therefore, − 1 2 < (ex − e2x)(cos(2x) − 1) ≤ 0 and hence, 1 2 < 1 + (ex − e2x)(cos(2x) − 1) ≤ 1 CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 231 1.25 10 20 1.75 0−10 1.5 2.0 1.0 −20 Figure 1: The range of det Φ2 in the first example. (cf. Figure 1). Observing that 1 x2+1 ∈ (0, 1) (when x < 0), we conclude that for x < 0: det Φ2 > 1 2 . (42) Let us now consider x ∈ [0, +∞). In this case, we have e−2x − e−x ∈ [− 1 4 , 0]. This implies that 0 ≤ (e−2x − e−x)(cos(2x) − 1) < 1 2 . Hence, 1 ≤ 1 + (e−2x − e−x)(cos(2x) − 1) < 3 2 . Observing that 1 x2+1 ∈ (0, 1] (x ≥ 0) we conclude that for x ≥ 0: det Φ2 > 1. (43) From (42) and (43), it follows that Φ2 ∈ GSAP 2×2. Now, a direct computation yields that Φ−12ℓ Φ̃2r =   1 0 0 1   which obviously admits a canonical right AP factorization and d(Φ−12ℓ Φ̃2r) = I2×2. Hence, sp[d(Φ−12ℓ Φ̃2r)] ∩ iR = {1} ∩ iR = ∅. 232 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) This allows us to conclude that the operators W fΦ2 ± HΦ2 have the Fredholm property. Thus, by using the above theory (cf. Corollary 4.2) we are now in a position to compute the sum of their Fredholm indices. For this case, we have Ind[W fΦ2 + HΦ2 ] + Ind[W fΦ2 − HΦ2 ] − 2 ind det(Φ2) − 1 π 2∑ k=1 β(δk) where δk ∈ C\iR are the eigenvalues of the matrix d(Φ −1 2ℓ Φ̃2r) and β is given by (38). In addition, we have already seen that det Φ2 is a real-valued positive function, and therefore its argument is zero. Altogether, we have: Ind[W fΦ2 + HΦ2 ] + Ind[W fΦ2 − HΦ2 ] = 0 (since the eigenvalues of d(Φ−12ℓ Φ̃2r) are also real and positive, and therefore their arguments are also zero). 5.2 Second example Let Φ1 = 1 + e −x 2 and Φ2 = −(1 − u(x))e −ix + u(x)e−2ix, where u is the real-valued function defined by u(x) = 1 2 + 1 2 tanh(x). Consequently, observing that ũ(x) = 1 − u(x) we have (cf. (7)) Ψ =   1 + e−x 2 + ( u(x)e − ix 2 −(1−u(x))e ix 2 ) 2 1+e−x 2 (1−u(x))e−ix−u(x)e−2ix 1+e−x 2 −u(x)eix+(1−u(x))e2ix 1+e−x 2 1 1+e−x 2   . From Theorem 2.1, it becomes clear that Φ1 and Φ2 ∈ SAP and thus, the matrix Ψ belongs to SAP 2×2. Since det Ψ = 1, we conclude that Ψ ∈ GSAP 2×2. Following (9), we obtain Ψℓ =   1 + eix e−ix e2ix 1   . Moreover, observing that Ψℓ =   e−ix 1 1 0     e2ix 1 1 0   , we conclude that Ψℓ admits a canonical right AP factorization and d(Ψℓ) =   M (e−ix) M (1) M (1) 0     M (e2ix) M (1) M (1) 0   . Since M (e−ix) = M (e2ix) = 0 and M (1) = 1, we obtain that d(Ψℓ) = I2×2 and therefore, Hd(Ψℓ) =   0 1 1 0   , H =   0 1 1 0   . CUBO 12, 2 (2010) Fredholm Property of Matrix Wiener-Hopf plus and minus Hankel Operators with Semi-Almost Periodic Symbols 233 Hence, sp[Hd(Ψℓ)] ∩ iR = {−1, 1} ∩ iR = ∅. These are sufficient conditions for these operators WΦ1 ± HΦ2 to have the Fredholm property (cf. The- orem 3.1). Let us now calculate the sum of their Fredholm indices. For this case, we have Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] − ind[det Ψ] − 1 π 2∑ k=1 β(ζk) where ζk ∈ C\iR are the eigenvalues of the matrix Hd(Ψℓ) and β is given by (31). In addition, we have previously seen that det Ψ = 1, therefore having a zero argument. Altogether, we have Ind[WΦ1 + HΦ2 ] + Ind[WΦ1 − HΦ2 ] = 0. Acknowledgement. This work was supported in part by Unidade de Investigação Matemática e Aplicações of Universidade de Aveiro through the Portuguese Science Foundation (FCT–Fundação para a Ciência e a Tecnologia). Received: March 2009. Revised: May 2009. References [1] G. Bogveradze, Fredholm Theory for Wiener-Hopf plus Hankel Operators, PhD Thesis, Uni- versity of Aveiro, Aveiro, 2008. [2] G. Bogveradze and L. P. Castro, Wiener-Hopf plus Hankel operators on the real line with unitary and sectorial symbols, Contemp. Math. 414 (2006), 77–85. [3] L. P. Castro, F.-O. Speck and F. S. Teixeira, A direct approach to convolution type operators with symmetry, Math. Nachr. 269-270 (2004), 73–85. [4] L. P. Castro and A. S. Silva, Invertibility of matrix Wiener-Hopf plus Hankel operators with symbols producing a positive numerical range, Z. Anal. Anwend. 28 (2009), 119–127. [5] T. Ehrhardt, Factorization Theory for Toeplitz plus Hankel Operators and Singular Integral Operators with Flip, Habilitation Thesis, Technischen Universitität Chemnitz, Chemnitz, 2004. [6] A. B. Lebre, E. Meister, F. S. Teixeira, Some results on the invertibility of Wiener-Hopf- Hankel Operators, Z. Anal. Anwend. 11 (1992), 57–76. [7] A. P. Nolasco, Regularity Properties of Wiener-Hopf-Hankel Operators. PhD Thesis, University of Aveiro, Aveiro, 2007. [8] A. P. Nolasco and L. P. Castro, A Duduchava-Saginashvili’s type theory for Wiener-Hopf plus Hankel operators, J. Math. Anal. Appl. 331 (2007), 329–341. [9] L. P. Castro and D. Kapanadze, Exterior wedge diffraction problems with Dirichlet, Neumann and Impedance boundary conditions, Acta Appl. Math., 110 (2010), 289-311. 234 L. P. Castro and A. S. Silva CUBO 12, 2 (2010) [10] L. P. Castro, F.-O. Speck and F. S. Teixeira, Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip, J. Integral Equations Appl. 15 (2003), 359–383. [11] L. P. Castro, F.-O. Speck and F. S. Teixeira, On a class of wedge diffraction problems posted by Erhard Meister, Oper. Theory Adv. Appl. 147 (2004), 211–238. [12] E. Meister, F.-O. Speck and F. S. Teixeira, Wiener-Hopf-Hankel operators for some wedge diffraction problems with mixed boundary conditions, J. Integral Equations Appl. 4 (1992), 229– 255. [13] G. Bogveradze and L. P. Castro, On the Fredholm index of matrix Wiener-Hopf plus/minus Hankel operators with semi-almost periodic symbols, Oper. Theory Adv. Appl. 181 (2008), 143–158. [14] A. Böttcher, Yu. I. Karlovich and I. M. Spitkovsky, Convolution Operators and Fac- torization of Almost Periodic Matrix Functions, Birkhäuser, Basel, 2002. [15] D. Sarason, Toeplitz operators with semi-almost periodic symbols, Duke Math. J. 44 (1977), 357–364. [16] L. P. Castro and F.-O. Speck, Regularity properties and generalized inverses of delta-related operators, Z. Anal. Anwend. 17 (1998), 577–598.