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# Talk Delivered In Woat Conference Lisbon 2006

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### Talk Delivered In Woat Conference Lisbon 2006

1. 1. 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
2. 2. 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.   Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
3. 3. 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). ∞ Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
4. 4. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
5. 5. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
6. 6. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
7. 7. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
8. 8. 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 Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
9. 9. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
10. 10. 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). Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
11. 11. 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) + Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
12. 12. 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 ) . Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
13. 13. Matrix-Valued SAP symbols 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 Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
14. 14. Matrix-Valued SAP symbols 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] = ∅ . Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
15. 15. Matrix-Valued SAP symbols ¥ 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] = ∅ . Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
16. 16. Matrix-Valued SAP symbols 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
17. 17. Matrix-Valued SAP symbols 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
18. 18. Matrix-Valued SAP symbols 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 Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
19. 19. 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 Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
20. 20. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
21. 21. 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). Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
22. 22. 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). Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
23. 23. 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). Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
24. 24. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
25. 25. 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) = Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
26. 26. 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 Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
27. 27. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
28. 28. 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)). Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
29. 29. 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 Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
30. 30. 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). Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
31. 31. 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 . Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm
32. 32. 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. Conditions for the Fredholm property of matrixWiener-Hopf plus/minus Hankel operatorswith semi-alm