INVERTIBILITY OF MATRIX WIENER-HOPF PLUS HANKEL OPERATORS WITH APW FOURIER SYMBOLS

Operators of Wiener-Hopf plus Hankel type have been receiving an increasing attention in the last years (see [1, 2, 4, 6, 10, 12–16]). Some of the interest in their study arises directly from concrete applications where these kind of operators appear. This is the case in problems of wave diffraction by some particular rectangular geometries which originate specific boundary-transmission value problems that may be equivalently translated by systems of integral equations that lead to such kind of operators (see, e.g., [5, 7, 8]). A great part of the study in this kind of operators is concentrated in the description of their Fredholm and invertibility properties. In particular, for some classes of the so-called Fourier symbols of the operators, their invertibility properties are already known (cf. the above references). Despite those advances, for some other classes of Fourier symbols, a complete description of the Fredholm and invertibility properties is still missing. In this way, some of the ongoing researches try to achieve the best possible factorization procedures of the involved Fourier symbols in such a way that a representation of the (generalized) inverses of the Wiener-Hopf plus Hankel operators will be possible to obtain when in the presence of a convenient factorization. Within this spirit, the main aim of the present work is to provide an invertibility criterion for the Wiener-Hopf plus Hankel operators of the form


Introduction
Operators of Wiener-Hopf plus Hankel type have been receiving an increasing attention in the last years (see [1,2,4,6,10,[12][13][14][15][16]).Some of the interest in their study arises directly from concrete applications where these kind of operators appear.This is the case in problems of wave diffraction by some particular rectangular geometries which originate specific boundary-transmission value problems that may be equivalently translated by systems of integral equations that lead to such kind of operators (see, e.g., [5,7,8]).
A great part of the study in this kind of operators is concentrated in the description of their Fredholm and invertibility properties.In particular, for some classes of the so-called Fourier symbols of the operators, their invertibility properties are already known (cf. the above references).Despite those advances, for some other classes of Fourier symbols, a complete description of the Fredholm and invertibility properties is still missing.In this way, some of the ongoing researches try to achieve the best possible factorization procedures of the involved Fourier symbols in such a way that a representation of the (generalized) inverses of the Wiener-Hopf plus Hankel operators will be possible to obtain when in the presence of a convenient factorization.
Within this spirit, the main aim of the present work is to provide an invertibility criterion for the Wiener-Hopf plus Hankel operators of the form where W Φ stands for the Wiener-Hopf operator defined by H Φ denotes the Hankel operator given by and the Fourier symbol Φ belongs to the so-called APW subclass of [L ∞ (R)] N×N .This will therefore extend some of the results of [15] to the matrix case.
As for the notations in (1.1)- (1.3), and in what follows, [L 2 + (R)] N denotes the subspace of [L 2 (R)] N formed by all vectors of functions supported in the closure of R + , r + represents the operator of restriction from [L 2 (R)] N into [L 2 (R + )] N , Ᏺ stands for the Fourier transformation, and J is the reflection operator given by the rule JΦ(x In view of defining the APW functions, let us first consider the algebra of almost periodic functions, usually denoted by AP.When endowed with the usual norm and multiplicative operation, AP is the smallest closed subalgebra of L ∞ (R) that contains all the functions e λ (λ ∈ R), where e λ (x) := e iλx , x ∈ R. ( In this framework, it turns out that the elements of APW are those from AP which allow a representation by an absolutely convergent series.In fact, APW is precisely the (proper) subclass of all functions ϕ ∈ AP which can be written in an absolutely convergent series of the form Let us agree on the notation ᏳB for the group of all invertible elements of a Banach algebra B. To end with the notation, we will say that a matrix function Φ belongs to APW N×N , and write Φ ∈ APW N×N , if all entries of the matrix Φ belong to APW.
As mentioned above, the representation of the (generalized/lateral/both-sided) inverses of WH Φ based on some factorization of the Fourier symbol Φ is an important goal, and will be obtained in the final part of the paper.In this way, the main contributions of the present work are described in Theorems 5.1, 4.1, and 3.3.
In the next section we will recall some useful particular known results which are anyway presented with complete proofs for the reader's convenience.

Initial multiplicative decompositions
According to (1.1)-(1.3),we have G. Bogveradze and L. P. Castro 3 where I [L 2 + (R)] N denotes the identity operator in [L 2 + (R)] N .Furthermore, since where e : [L 2 (R + )] N → [L 2 (R)] N denotes the even extension operator, we may also rewrite the Wiener-Hopf plus Hankel operator in (1.1) as From the theory of Wiener-Hopf and Hankel operators, it is well known that where 0 : [L 2 (R + )] N → [L 2 (R)] N denotes the zero extension operator.Additionally, from the last two identities, it follows that ) , and H ∞ (C ± ) be the set of all bounded and analytic functions in C ± .Fatou's theorem ensures that functions in H ∞ (C ± ) have nontangential limits on R almost everywhere, and it is usually denoted by H ∞ ± (R) the set of all elements in L ∞ (R) that are nontangential limits of functions in H ∞ (C ± ).Below, we will use the matrix versions [H ∞ (C ± )] N×N and [H ∞ ± (R)] N×N of those Hardy spaces.
It is already interesting to observe that, due to (2.6), if we consider Φ being an even function or Ψ ∈ [H ∞ − (R)] N×N , we will then obtain the multiplicative relation of two corresponding Wiener-Hopf plus Hankel operators.A more general property in this direction is formulated in the next result.
] N×N , we may apply the first presented multiplicative relation for Wiener-Hopf plus Hankel operators; see (2.7).This leads us to (2.9) In addition, since Ξ = Ξ, it also follows from (2.7) that From (2.9) and (2.10), we have that ] N×N , we have H Ψ = 0 due to the structure of the Hankel operators.Therefore WH Ψ = W Ψ , and it follows from (2.11) that N×N and Φ e = Φ e , then WH Φe is invertible and its inverse is the operator e are even functions, we directly obtain from the above multiplicative relations of Wiener-Hopf plus Hankel operators that (2.12) This obviously shows that WH φe is invertible and its inverse is 0 WH φ −1 e 0 .

Matrix APW asymmetric factorization
The (Bohr) mean value of φ ∈ AP is defined by where The mean value of an element in AP always exists, is finite, and is independent of the particular choice of the family {I α } α∈A .
with D (1) = diag[e λ1 ,...,e λN ] and λ 1 ≥ • • • ≥ λ N , is an APW asymmetric factorization of Φ and assume additionally that with D (2) = diag[e μ1 ,...,e μN ] and where Ψ(x) = (ψ jk (x)) N j,k=1 is a matrix function with nonzero and constant determinant, having entries which are entire functions, and Proof.If Φ admits the above-mentioned two APW asymmetric factorizations, then we can write which leads to (3.9) We now define N×N , and Φ e = Φ e .From (3.9), we obtain the following identity for each ( j,k) element of that matrix: Φ − jk (x)e iλkx = e iμj x Φ e jk (x); (3.10) whence 6 Matrix Wiener-Hopf plus Hankel operators and recall that Φ e is an even function.Thus and finally we infer from (3.12) that If μ j ≥ λ k , then the element in the left-hand side of (3.13) is in the class APW − , and the function in the right-hand side belongs to APW + , which implies that there exist constants c jk such that Therefore, c jk = c jk e 2i(μj −λk )x .Thus, if μ j > λ k , we obtain c jk = 0, and in the case where μ j = λ k , we conclude that c jk are nonzero constants.Altogether, we have Let us now assume that μ j < λ k .By the hypothesis, we know that (Φ − ) jk ∈ APW − and so (Φ − ) jk can be represented in the following form: and this leads us to the following identity: In conclusion, we have in the present case G. Bogveradze and L. P. Castro 7 So, for any couple ( j,k), we will obtain only one real number (ν m ) jk , which is precisely the difference μ j − λ k and this means that in the representation of (Φ − ) jk (cf.(3.16)) we need to have (Φ − ) jk (x) = c jk e i(νm)jkx , with some constant c jk = const, for all j,k = 1,N.Thus, we arrive at the conclusion that (Φ − ) jk are entire functions when μ j < λ k .

Invertibility characterization
Let We know that W Φ− is invertible because Φ − ∈ ᏳAPW N×N − (and its inverse is given by 0 W Φ −1 − 0 ).Additionally, WH Φe is also invertible because Φ e is an even element (cf.Proposition 2.2).Thus, (4.2) shows us an operator equivalence relation between WH Φ and WH D (note that 0 : ).We will therefore analize the regularity properties of WH D .
Part (c) can be deduced from the assertion (b) by passing to adjoints.If all partial indices are zero, we have that 0 WH D is just the identity operator.This, together with the operator equivalence relation (4.2) presented in the first part of the proof, leads us to the last assertion (d).

Inverses representation
We now reach to our final goal: the representation of generalized/lateral/both-sided inverses of WH Φ based on a factorization of the Fourier symbol.This result extends the scalar version obtained in [15,Theorem 7].
Let us first recall that a bounded linear operator S − : Y → X (acting between Banach spaces) is called a reflexive generalized inverse of a bounded linear operator S : X → Y if (i) S − is a generalized inverse (or an inner pseudoinverse) of S, that is, SS − S = S; (ii) S − is an outer pseudoinverse of S, that is, S − SS − = S − .G. Bogveradze and L. P. Castro 9 Theorem 5.1.Suppose Φ admits an APW asymmetric factorization and (c) the both-sided inverse of WH Φ , if λ i = 0 for all i = 1,N.In the case when there exist partial indices with different signs, the operator WH Φ is not Fredholm but T is still a (reflexive) generalized inverse of WH Φ .
Proof.We start with the cases (a), (b), and (c).Since Φ admits an APW asymmetric factorization, we can write (with the corresponding factor properties).Consequently, from (2.3), it follows that where where the term 0 r + was omitted due to the fact that r + 0 r + = r + .Since A −1 e preserves the even property of its symbol, we may also drop the first e r + term in (5.4), and obtain Additionally, due to the definition of E and E −1 in the present case (λ i ≤ 0 for all i = 1,N), we have 0 r + E e r + E −1 e r + = 0 r + ; also because A − is a minus type factor it follows r + A − = r + A − 0 r + .Therefore, from (5.5), we have In the present case, due to the definition of E −1 , it follows e r + E −1 e r + = e r + E −1 .The same reasoning applies to the minus type factor A −1 − , and therefore the equality (5.7) takes the form where we have used the fact that e r + A e e r + = A e e r + .
(c) From the last two cases (a) and (b), it directly follows that in the case of λ i = 0 for all i = 1,N, the operator T is the both-sided inverse of WH Φ (cf.(5.6) and (5.8)).
Let us now turn to the more general case: assume now that there exist partial indices with different signs.
In this case, we recall that the assertion about the non-Fredholm property was already provided in Theorem 4.1, assertion (a).
As about the generalized inverse, we will start by rewriting the operator E in the following new form: where for j = 1,N.We will then directly compute WH Φ TWH Φ , in the following way: where in (5.14) we omitted the first term e r + of (5.13) due to the factor (invariance) property of A −1 e that yields A e e r + A −1 e e r + = e r + .Similarly we dropped the term r + in e r + A −1 − r + A − due to a factor property of A −1 − .Analogous arguments apply to the factors E −1 1 and E −1 2 .In a more detailed way: (i) if one of the factors E 1 or E 2 equals I, then it is clear that E 2 ( e r + E −1 2 e r + )E 2 = E 2 ( e r + E −1 2 )E 2 = E 2 e r + or 0 r + E 1 e r + E −1 1 e r + E 1 = 0 r + E 1 holds, respectively; (ii) in the general diagonal matrix case, the situation is identical just because in each place of the main diagonal we have the last situation.This justifies the simplification made in obtaining (5.15) from (5.14).
G. Bogveradze and L. P. Castro 11 As about the composition TWH Φ T, it follows that where the third e r + is unnecessary in (5.18) due to the factor (invariance) property of A e that yields A e e r + A −1 e e r + = e r + , and we also can omit the term r + in (5.18) since A −1 − is a minus type.Additionally, a similar reasoning as above was also used for obtaining equality (5.19) since due to the definitions of E 1 and E 2 it holds e r + E 1 e r + = e r + E 1 , and e r + E −1 2 e r + E 2 e r + E −1 2 = e r + E −1 2 .We end up by mentioning that almost all the above methods also work-without crucial changes-in the case of matrix Wiener-Hopf plus Hankel operators with almost periodic Fourier symbols.However, a corresponding version of Theorem 3.3 for invertible AP N×N elements is an open problem.This has to do with the difficulties in substituting the arguments in the part of the proof of Theorem 3.3 where some representations of APW elements are used.

) where Φ −1 e and Φ − 1 −
are the inverses of the corresponding factors of an APW asymmetric factorization of Φ, Φ = Φ − DΦ e , and the operator :[L 2 (R + )] N → [L 2 + (R)]N denotes an arbitrary extension operator (i.e., T is independent of the particular choice of the extension ).Then the operator T is a reflexive generalized inverse of WH Φ and, in the following special cases, T is additionally (a) the right inverse of WH Φ , if λ i ≤ 0 for all i = 1,N; (b) the left inverse of WH Φ , if λ i ≥ 0 for all i = 1,N; [e λ1 ,...,e λN ] • Ᏺ and A e = Ᏺ −1 Φ e • Ᏺ.(a) If λ i ≤ 0 for all i = 1,N, consider WH Φ T = r + A − EA e e r + 0 r + A −1 e e r + E −1 e r + A −1 − = r + A − EA e e r + A −1 e e r + E −1 e r + A −1 − ,

1 e r + A − 1 −r 1 e 1 e
+ A − E 1 E 2 A e e r + 0 r + A −A −1 − r + A − E 1 E 2 A e e r + A −