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Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.


Introduction and preliminaries
Our aim is an extension of certain theorems from complex analysis and operator theory to the realm of quaternions. The Wiener algebra has two well-known variants: the discrete one (defined on the unit sphere) and the continuous one (defined on the real line). Both have been extended to the quaternionic (scalar-valued) setting by D. Alpay et al. (see [1]), successfully carrying over the Wiener-Lévy theorem. We seek to gain more richness about these quaternionic algebras, namely Wiener-Hopf factorization and the Riemann-Hilbert problem in the matrix-valued case.
Let us recall that the (complex) discrete Wiener algebra W n×n consists of functions of the form where F u ∈ C n×n and u∈Z F u < ∞, · denoting the operator norm. The space W n×n with pointwise multiplication and the norm defined above is a Banach algebra. When n = 1, we denote the Wiener algebra by W.
The Wiener-Lévy theorem characterizes the invertible elements of W n×n : an element is invertible in W n×n if and only if it is pointwise invertible (in C n×n ).
A quaternionic valued function f (p) belongs to the (discrete) quaternionic Wiener algebra W H if it is of the form u∈Z p u f u , where u∈Z |f u | < ∞. The sum of two elements in W H is defined in the natural way, while their product, denoted by ⋆, is obtained by taking the convolution of the coefficients (much like the ⋆ product for polynomials and more generally slice hyperholomorphic functions). In [1], a counterpart of the Wiener-Lévy theorem was successfully obtained in the aforementioned scalar-valued case. Moreover, the theorem characterized invertibility via a slice of the function with an arbitrary complex plane. The continuous Wiener algebra W(R, C n×n ) consists of functions of the form where C ∈ C n×n , K ∈ L 1 (R, C n×n ). There is a Wiener-Lévy theorem in this case, too, with invertibility being equivalent to the condition inf t∈R | det(f (t))| > 0.
The scalar-valued case was also extended in [1] to the quaternionic setting. The initial motivation behind this paper was the desire to extend the connection between the Wiener algebras and certain functional equations (of three types: difference, convolution and more generally integro-difference) from the complex setting to the quaternionic one. Consider (over C) the integro-difference equation where f ∈ L p (0, ∞) is given, φ ∈ L p (0, ∞) is sought (we define φ(t) = 0 for t ≤ 0), k ∈ L 1 (R), and the a r ∈ C vanish for r outside of a countable set and satisfy r∈R |a r | < ∞. [6] includes a study of the solvability of (1.1) via the factorization of the symbol function The collection of such functions is denoted by B(R). Note that functions of the form r∈R a r e irt are almost periodic, and their collection is called the almost periodic Wiener algebra, being denoted by AP W . The first section of the paper extends the above definitions to quaternions, yet again carrying over the Wiener-Lévy theorem. In this general framework of Wiener algebras, we also characterize invertibility in the matrix-valued case. The second section deals with factorization in the quaternionic case, but we limit the discussion to the (matrix-valued) discrete and continuous Wiener algebras. The reason for this limitation is that, in fact, it is known that even in the complex case both B(R) n×n and AP W n×n fail to ensure factorization in general (that is, for any invertible function) for n > 1. We should note that the case of n = 1 is interesting since factorization does hold for it over C, but our tools are not enough to determine if this remains the case over H. Regardless, what is notable about the discrete and continuous algebras, is the connection between factorization and the Riemann-Hilbert problem. We will show that the theorems from the complex setting successfully carry over. The final two sections deal with applications of factorization: To difference and convolution equations (but not integro-difference ones due to the lack of a factorization theorem) and to canonical factorization of quaternionic rational matrix functions. Again, the results are direct counterparts of the known theorems in the complex cases. We follow [5] for the following definitions. We denote by S the sphere of unitary purely imaginary quaternions. Any element i ∈ S satisfies i 2 = −1, and using e 1 , e 2 , e 3 as a standard basis, we get S = {xe 1 + ye 2 + ze 3 : x, y, z ∈ R, x 2 + y 2 + z 2 = 1}.
Any two orthogonal elements i, j ∈ S (in the sense ij = −ji) form a new basis i, j, ij of H. Given a quaternion p 0 , it determines a sphere [p 0 ] consisting of all the points of the form q −1 p 0 q for q = 0. In some cases, it is useful to use a map which transforms a quaternion into a 2 × 2 complex matrix or, more generally, a quaternionic matrix into a complex matrix of double size (for example, see [8]). This will appear throughout the work (albeit often in the context of a different map to be defined later). Definition 1.1. Let i ⊥ j ∈ S. Any p ∈ H has a unique representation p = a + bj, where a, b ∈ C i := R + iR. More generally, any P ∈ H n×n has a unique representation P = A + Bj, where A, B ∈ C n×n i . We define a map χ i,j , which maps H n×n into C 2n×2n i for any n ∈ N, via We will write χ instead of χ i,j whenever i, j are clear from the context. Note that χ is injective, additive and multiplicative, hence its utility. The image χ(H n×n ) is the set of all matrices Q ∈ C 2n×2n i that satisfy the relation J n QJ T n = Q, where J n = 0 I n −I n 0 .

An almost periodic extension of the quaternionic Wiener algebra
We begin with the scalar-valued case. The following definition is analogous to the one for the discrete Wiener algebra (see [1]).
Definition 2.1. We denote by AP W (SR, H) the set of functions of the form where the f u are quaternions vanishing for all but a countable subset of u ∈ R, and u∈R |f u | < ∞.
This set can be endowed with the multiplication We also define F = u∈R |f u |.
Definition 2.2. We denote by B(SR, H) the sum of AP W (SR, H) and W(SR, H), which is to say the set of all functions of the form This set can be endowed with the multiplication where • denotes convolution. We also define In [1] it was shown that the discrete and continuous Wiener algebras are real Banach algebras. Along the same lines, we have: is countable, so we may enumerate it as (w m ) m∈N . For every m ∈ N we have that (f n,wm ) n∈N is a Cauchy sequence of quaternions. Thus lim n→∞ f n,wm exists and we denote it by f wm . It is not hard to see that m∈N e itwm f wm is the norm limit of u∈R e itu f n,u as n → ∞ (much like the proof that l 1 is a Banach space). Finally, B(SR, H) is a Banach algebra: In [1], a map ω was introduced as a way of viewing the quaternionic Wiener algebras as complex matrix-valued Wiener algebras. Using the same symbol, we analogously define a map ω = ω i,j depending on the choice of an imaginary unit i ∈ S and of a j ∈ S orthogonal to i, that is, satisfying ij = −ji.
where χ = χ i,j is defined as in the introduction.
We will write ω = ω i,j if the context is clear. It is immediate that the entries of ω(F )(t) belong to B(R, C i ), and moreover, ω maps B(SR, H) injectively into B 2×2 (R, C i ) with values in C 2×2 i . Likewise, it maps AP W (SR, H) injectively into AP W 2×2 (R, C i ). It can also be verified (using the properties of χ) that a function G ∈ B 2×2 (R, C i ) (or a subalgebra) is in the image under ω (or of the respective subalgebra) if and only if J n G(−t)J T n = G(t). As explained in [1], the reason for using ω is that χ generally does not respect ⋆-multiplication (only pointwise multiplication), while ω does. The proof barely changes under our extension, but we include it for the sake of completeness.
Proof. Using the properties of χ, we get The following theorem is inspired by the theorems proved in [1] for the discrete and continuous cases. The main difference that needs to be accounted for is the fact that almost periodic functions cannot be continuously extended to ∞, so the domain being considered is not compact.
Theorem 2.6. Let F ∈ B(SR, H). The following are equivalent: For the proof, we need two lemmas, which are analogous to formulas proved in [1] (for the discrete and continuous cases).
Lemma 2.7. Let F ∈ B(SR, H) be given by For the proof, we mimic the calculation in [1], but it is longer due to there being two components. Proof.
, and by carrying out the multiplication we see that (F ⋆ F c )(it) = det(ω(F )(t)).
Lemma 2.8. Let F, G ∈ B(SR, H). Then Proof. If F (it) = 0, then let i t = F (it) −1 iF (it). Then If F (it) = 0, let ǫ > 0 and define F ǫ = F + ǫ. Then Since G is bounded and ǫ is arbitrary, we get (F ⋆ G)(it) = 0. In particular, (F ⋆ G)(it) = F (it)G(jt) for any j ∈ S. Now let us prove Theorem 2.6. We follow the outline of the proof in [1] for the discrete case, modifying the arguments so as to deal with infimum values.
Proof. (i) =⇒ (iv): Denoting the inverse by F −⋆ , we have by the multiplicativity of ω Now, in general, for any G ∈ B(SR, H) the matrix ω i,j (G)(t) is of the form and thus where we have used the fact that |G(it)| 2 = |r(it)| 2 + |s(it)| 2 . In particular, then using the formula for inverting a 2 × 2 matrix we see that is a matrix function in the algebra B(R, C 2×2 ). Let (v) =⇒ (iv): By Lemma 2.7, Using Lemma 2.8, we get This shows that F (SR) ⊆ F c (SR), and the reverse follows by substituting F c instead of F using the fact that (F c ) c = F . So in particular, Going back to (2.8), we get Since F c is bounded, we can divide by sup p∈SR |F c (p)| and get 0 < inf p∈SR |F (p)|.
(vi) =⇒ (v): Assume by negation that inf p∈SR |F (p)| = 0. Then there exist (i n ) n∈N ⊂ S, (t n ) n∈N ⊂ R such that lim n→∞ F (i n t n ) = 0. Since S is compact, we can assume without loss of generality that (i n ) n∈N converges to i 0 ∈ S. Observe that for all i, j ∈ S and t ∈ R we have This implies that lim n→∞ F (i 0 t n ) = 0, which contradicts inf t∈R |F (i 0 t)| > 0. Remark 2.9. As is shown in [5] for slice hyperholomorphic functions, it is easy to derive the formula Not only is F ⋆ F c invertible if and only if F is, but also all of its coefficients (in the almost periodic and continuous components) are real. Thus, ⋆-inversion is the same as pointwise inversion. Proof. We only need to show why AP W (SR, H) is closed with respect to inversion.
This implies that We now extend the previous definitions to the case of quaternionic matrix-valued functions. To clarify, the rest of this section is mostly a rewriting of the preceding results, with the real difference being the proof of Theorem 2.18. First, let · n on H n×n denote the operator norm (with respect to the Euclidean norm on H n×1 ). We will start with the discrete Wiener case: Definition 2.11. We denote by W n×n H the set of matrix-valued functions of the form This set can be endowed with the multiplication We also define F = u∈Z F u n .
Definition 2.12. We denote by AP W (SR, H n×n ) the set of matrix-valued functions of the form where the F u ∈ H n×n are quaternionic matrices vanishing for all but a countable subset of u ∈ R, and u∈R F u n < ∞.
This set can be endowed with the multiplication We also define F = u∈R F u n .
Definition 2.14. We denote by W(SR, H n×n ) the set of all matrix-valued functions of the form where Φ F ∈ L 1 (R, H n×n ) and C F ∈ H n×n is a constant matrix. This set can be endowed with the multiplication We also define Definition 2.15. We denote by B(SR, H n×n ) the sum of AP W (SR, H n×n ) and W(SR, H n×n ), which is to say the set of all matrix-valued functions of the form This set can be endowed with the multiplication (2.13) We also define We now extend the definition of the map ω = ω i,j : Then where χ is defined as in the introduction (for matrices). Similarly, letting F ∈ W n×n H be given by F (p) = u∈Z p u F u , we have Proof. The proof of Lemma 2.5 still applies.
The following are equivalent: Proof. (i) =⇒ (iv): By the multiplicativity of ω we have that since the modulus of any entry of G(it) is at most G(it) n , which in turn is bounded by G for all i ∈ S, t ∈ R. The determinant of a 2n × 2n matrix is (up to a sign for each term) a sum of (2n)! products of 2n entries, so by the triangle inequality sup Thus it follows from (2.14) that is a matrix function in the algebra B(R, C 2n×2n ), since each entry of the adjugate matrix is a product of functions in B(R, C). The inverse matrix corresponds to a G ∈ B(SR, H n×n ), in the sense that To show (2.14), we simply observe that , and invert both sides (noting that J T n = J −1 n ). Thus, G exists and Proof. All the subalgebras are closed with respect to inversion, as can be shown similarly to the proof of Corollary 2.9.

Factorization and the Riemann-Hilbert problem
We limit the discussion to the matrix-valued discrete and continuous algebras. First, let us reiterate definitions of subalgebras discussed in [1] in the scalar-valued case.
(i) We denote by W n×n H,+ (resp. W n×n H,− ) the set of elements  (i) Given F ∈ W n×n H , we say that F admits a factorization if there exist a diagonal matrix D(p) = diag p k1 , . . . , p kn (where k 1 ≥ k 2 ≥ · · · ≥ k n are integers) and invertible (with respect to their respective subalgebras)

Note that any function in W n×n
(ii) Given F ∈ W(SR, H n×n ), we say that F admits a factorization if there exist a diagonal matrix (where k 1 ≥ k 2 ≥ · · · ≥ k n are integers) and invertible (with respect to their respective subalgebras) F ± ∈ W ± (SR, H n×n ) and such that In both cases k 1 ≥ k 2 ≥ · · · ≥ k n are called the factorization indices, and the factorization is called canonical if D = I.
It is well known (see [4]) that invertible functions in the complex-valued counterparts of the aforementioned algebras, admit factorization. To be clear, the diagonal matrix in W(R, C n×n ) is of the form and in the quaternionic case we set p = it. It is also known that in the complex case the factorization indices are unique, which is to say that the diagonal elements are uniquely determined by the function once the order along the diagonal is fixed (multiplication by the same elementary matrices from the left and right can alter the order). In order to establish these facts in the quaternionic setting, we need to first recall the complex Riemann-Hilbert problem and its connection to factorization. For our purposes it suffices to consider the problem in the context of the Wiener algebras, but it should be noted that there is a more general theory (studied in [4]).
The associated barrier problem is to describe all piecewise holomorphic vector functions Ψ(z) given by and satisfying where Ψ + ∈ W C,+ , Ψ − (z) − r(z) ∈ W C,− and r is a polynomial. We define ord(Ψ) to be the (possibly negative) order of ∞ as a pole of Ψ. (ii) Let F ∈ W(R, C n×n ). The associated barrier problem is to describe all piecewise holomorphic vector functions Ψ(t) given by and satisfying and r is a polynomial. We define ord(Ψ) to be the (possibly negative) order of −i as a pole of Ψ.
Note that Theorem 3.8 remains valid if we choose a different order of indices along the diagonal D, as long as we use the respective order for the indices of the solution set. In this case, a standard solution set is defined with respect to the corresponding lexicographic order. By s 1 → s 2 → s 3 → · · · → s n we refer to the lexicographic order first determined by comparing the s 1 -th entries of two index sets, then their s 2 -th entries, and so on until the entries are not equal. Rather than immediately extending these results to the quaternionic setting, we use them for the proof of the following theorem regarding the existence of factorization: Proof. Fix i ⊥ j ∈ S. We will prove uniqueness first: If a factorization exists with an index set k 1 ≥ k 2 ≥ · · · ≥ k n , then it induces a factorization . It is known that (up to the order of indices), the diagonal element of a factorizable matrix function in W n×n (W(R, C n×n )) is unique. Thus D is unique. If n = 1, then so the components are real). Thus, if we have two factorizations meaning that it has a bounded, slice hyperholomorphic continuation to H (rendering it constant by the quaternionic version of Liouville's theorem). c = 0 since G −⋆ − ⋆F − must be invertible. So G + = c ⋆ F + and G − = F − c −1 , and it is obvious that any = 0 satisfies the factorization.
where the indices are ordered as follows: Let us consider the set of piecewise holomorphic functions Ψ 1 , . . . , Ψ 2n : C i → C 2n By Theorem 3.8, it is a complete and standard solution set for the barrier problem Ψ − = F ′ Ψ + with respect to the lexicographic order 1 → n + 1 → 2 → n + 2 → · · · → n → 2n.
Remark 3.11. It is known that invertibility is not sufficient for factorizability in AP W (R, C n×n ) for n ≥ 2 (although it is sufficient for n = 1). This makes it problematic to go about a proof for the quaternionic scalar case, since it is not clear that for any invertible F ∈ AP W (SR, H) the 2 × 2 matrix-valued function ω(F ) is factorizable.
We continue this section by exploring the connection between factorization and the Riemann-Hilbert problem in the quaternionic setting. Our aim is to extend Theorem 3.8 and the preceding definitions.
(i) Let F ∈ W n×n H . The associated barrier problem is to describe all piecewise slice hyperholomorphic vector functions Φ(p) given by and satisfying and r is a polynomial. We define ord(Φ) to be the (possibly negative) order of ∞ as a pole of Φ. (ii) Let F ∈ W(SR, H n×n ). The associated barrier problem is to describe all piecewise slice hyperholomorphic vector functions Φ(p) given by and satisfying and r is a polynomial. We define ord(Φ) to be the (possibly negative) order of 1 as a pole of Φ.
Proof. Fix i ⊥ j ∈ S. In the first direction, for 1 ≤ k ≤ n define We want to show that the corresponding F − , D, ω(F + ) = [Ψ 1,+ , . . . , Ψ 2n,+ ] −⋆ . By Theorem 3.8, it suffices to show that {Ψ 1 , . . . , Ψ 2n } is a standard solution set for the barrier problem associated with ω(F ), since then it will follow that . . , Ψ 2n are linearly independent over C i at 0 (at i) follows from the relation −1), . . . , Ψ 2n (−1)]) and the fact that the columns of a matrix A ∈ H n×n are linearly independent over H (with respect to right multiplication) if and only if the columns of χ(A) are linearly independent over C i . To show minimality of the index set, let us recall that by the proof of Theorem 3.10, any standard solution set (with respect to the lexicographic order 1 → n + 1 → · · · → n → 2n) Γ 1 , . . . , Γ 2n for the barrier problem associated with ω(F ) leads to another standard solution set of the form {Γ m1 , . . . ,Γ mn , Γ m1 , . . . , Γ mn }. Thus there exist Λ 1 , . . . , Λ n such that ω(Λ k ) = [Γ m k Γ m k ] for all 1 ≤ k ≤ n, from which it follows that Λ 1 , . . . , Λ n form an admissible solution set for the barrier problem associated with F . If the index set of Φ 1 , . . . , Φ n is [k 1 , . . . , k n ] and that of Λ 1 , . . . , Λ n is [s 1 , . . . , s n ], then by the minimality of [k 1 , . . . , k n ] we have form a complete and standard solution set for the barrier problem associated with ω(F ). Then Φ 1 , . . . , Φ n given by As already seen, Ψ 1 , . . . , Ψ 2n being a standard set implies that Φ 1 , . . . , Φ n are a standard set. Completeness follows similarly: Let Φ be a solution of the barrier problem associated with F and define Ψ = ω(Φ). There exist polynomials r 1 , . . . , r 2n (with coefficients in C i ) such that This implies that Corollary 3.18. Any standard solution set of eq. (3.13) (eq. (3.15)) is complete.
Remark 3.19. It will be interesting to consider more general quaternionic Riemann-Hilbert problems. Which kinds of surfaces in H may work other than spheres and hyperplanes? Which kinds of algebras of slice hyperholomorphic functions? A reasonable requirement is that the surface should split H into two axially symmetric s-domains.

Solvability of two classes of quaternionic functional equations
The aim is to use factorization to characterize solvability of two classes of quaternionic functional equations. See [6] for the results in the complex case.
where the a n ∈ H satisfy ∞ n=−∞ |a n | < ∞ (we define φ(t) = 0 for t ≤ 0). Let us associate with each operator A ∈ B d the symbol function where c ∈ H, k(t) ∈ L 1 (R, H). Let us associate with each operator B ∈ W d the symbol function (W (SR, H)). It is linear (with respect to right multiplication) and multiplicative in the following sense:

Proposition 4.2. A one-to-one correspondence exists between operators in W d (W c ) and their symbols in W H
Proof. The correspondence is well-defined, since an operator A ∈ W d (B ∈ W c ) is uniquely determined by {a j } j∈Z (c ∈ H and k ∈ L 1 (R, H)). Indeed, taking the function (for r > 0) we get (Ae)(t) = ∞ n=−∞ a n e r (t − n) for t > 0. Then Ae is piecewise differentiable with one-sided derivatives at N satisfying for all n ∈ N. Assuming A = 0, we get a n = 0 for n ∈ N and 0 n=−∞ a n e −r(t−n) = 0 ∀r > 0.
Equivalently, the power series ∞ m=0 p m a −m (which defines a slice hyperholomorphic function on B) vanishes for p ∈ (0, 1), implying a −m = 0 for all m ≥ 0. In the continuous case we have for t > 0 which implies that Be r is differentiable almost everywhere and satisfies Thus, if B = 0, then k(t) = 0 for t > 0 and c + 0 −∞ k(s)e rs ds = 0 for all r > 0. Taking r → ∞, the Riemann-Lebesgue lemma gives us c = 0. Then the Laplace transform ofk(t) = k(−t) max{0, t} vanishes on (0, ∞), which implies k(t) = 0 for t ≤ 0. A → A is a bijection, since {a j } j∈Z uniquely determines a function in W H . Likewise, B → B is a bijection since c ∈ H and k ∈ L 1 (R, H) uniquely determine a function in W(SR, H). Linearity of the correspondence is obvious, so it remains to check multiplicativity (in the sense defined). In the discrete case, we write and get for t > 0, since for l ≥ 0 the n-shift of the (right) l-shift is the same as the (n + l)shift, and for m ≤ 0 the (left) m-shift of the n-shift is the same as the (m + n)-shift. Thus the corresponding symbol is The continuous case is checked similarly.
By the results of the preceding section (looking at the scalar-valued case), an invertible function in W H admits a factorization, as does an invertible function in W(SR, H). While an operator is not necessarily invertible if its symbol is (due to the lack of general multiplicativity), Proposition 4.2 shows that it is invertible if its symbol is invertible in W H,± in the discrete case, or in W ± (SR, H) in the continuous case. The factorization A(p) = A − ⋆ p k ⋆ A + induces the factorization is the k-shift operator whose symbol is p k . More precisely, for k ≥ 0 we have U (k) = U k , where U is the right shift operator given by is the left shift operator given by  Proof. Going case by case: (ii) If k = 0, then A = A − A + is invertible as the composition of invertible operators.
(iii) If k < 0, then U (k) being right invertible implies that A is right invertible.
Theorem 4.4. Let B ∈ W c be such that the symbol is invertible and (thus) admits a factorization Then there are three cases: where B + is the operator whose symbol is B + and the c j are arbitrary quaternions.
For the proof, we need to characterize the image of V m for m > 0, and the kernel of V (m) for m < 0.  t j e −t c j , where the c j are arbitrary quaternions.
Remark 4.6. To calculate the factorization index k of f ∈ W H , we may fix i ⊥ j ∈ S and use the formula where g(z) = det(ω i,j (f )(z)). This follows from the fact that ω i,j (f ) has index set [k, k]. Similarly, to the calculate the factorization index m of h ∈ W(SR, H), we use the formula m = 1 4π [arg(r(t))] ∞ −∞ , where r(t) = det(ω i,j (h)(t)), and arg(r(t)) is a continuous function describing the argument of r(t) in the complex plane C i .
where c u ∈ H vanish outside a countable set. While the correspondence between an operator C and its symbol is one-to-one, linear and multiplicative in a certain sense, the difficulty lies in the lack of a (proven) factorization theorem for B(SR, H). The characterization of the complex case relies on the factorization for B(R, C). It may be possible to show that the subalgebra ω(B (SR, H)) fulfills a factorization theorem, even though B 2×2 (R, C) does not.

Rational matrix functions and canonical factorization
As defined in [2], an n × n slice hyperholomorphic rational matrix function is obtained by a finite number of addition, ⋆-multiplication and ⋆-division operations on quaternionic polynomial matrix functions. Equivalently, the function is of the is a polynomial matrix function with quaternionic coefficients, and r is a polynomial with real coefficients. For brevity, we will simply use the term rational matrix functions.
where the p i lie on distinct spheres and the α j are distinct real numbers. If any of the p i or α j lies on ∂B, then each entry of Q(p) must be divisible by the corresponding polynomial (p 2 − 2 Re(p i )p + |p i | 2 ) ni or (p − α j ) tj , and we may divide both Q(p) and r(p) accordingly, getting rid of the pole. So we may assume that r(p) does not vanish on ∂B, which by the Wiener-Lévy theorem, implies 1/r(p) ∈ W H and consequently F ∈ W n×n H . The other direction is obvious since functions in W n×n H are continuous on ∂B.
For our main theorem, we need to first show the existence of a certain type of realization for a general rational matrix function. The complex case (see [3]) is naturally extended as follows: Theorem 5.2. Let F be a rational n × n matrix function. Given a constant matrix D ∈ H n×n , F admits a realization of the form where B ∈ H n×m , C ∈ H m×n , G, A ∈ H m×m are constant matrices.
Remark 5.3. The realization discussed in [2], of the form assumes that F is slice hyperholomorphic in a neighborhood of the origin (but need not be a square matrix). The theorem above has the restriction of F being a square rational matrix function, but nothing else is assumed.
Proof. The proof is largely unchanged from that of the complex case, but we will make sure that the arguments are still valid. First, F (p) admits the following decomposition: where K(p) is slice hyperholomorphic at ∞ (that is, a proper rational matrix function, all of whose entries having a greater degree in the denominator than in the nominator) and L(p) is a polynomial matrix. This follows from applying Euclid's division algorithm to each entry of F (p); it should be noted that each entry admits the form q(p) r(p) , where r is a polynomial with real coefficients, and thus the algorithm may be applied as in a field (as any polynomial commutes with r(p)).
Without loss of generality, K(∞) = D (we can subtract a constant matrix from K(p) and add it to L(p)). Let G(p) := K(p −1 ). Since G(0) = D and G is defined at 0, it admits a realization of the form which yields As for L, being a polynomial matrix we can write it as where L 0 , . . . , L q are constant matrices. We introduce The matrix G L is square of size l = m(q + 1), and is nilpotent of order q + 1. It is easy to see that It follows that Finally, we can obtain the realization (5.1) for F (p) by taking Here I is of the same size as A K . (i) (pG − A × ) −⋆ has no poles on ∂B.
(ii) H m = Im Q ⊕ Ker Q × and H m = Im P ⊕ Ker P × .
Here m is the order of the matrices G and A, and It should be noted that the known theorem (see [7]) for complex-valued rational matrix functions is almost the same (with ⋆-multiplication replaced by regular matrix multiplication, H n replaced by C n , and F being a rational n × n matrix function without poles on ∂D). To reduce the quaternionic case to the complex one, we need two lemmas: Lemma 5.6. Given i ⊥ j ∈ S, define ρ : H n → C n by ρ(w) = [u − v] T , where u, v ∈ C n i are such that w = u + vj. Then for any A ∈ H n×n , w ∈ H n we have ρ(Aw) = χ(A)ρ(w).
Let A ′× = A ′ − B ′ C ′ . The theorem in the complex case stipulates that F ′ admits a canonical factorization if and only if the following two conditions are satisfied: (iii) (zG − A ′× ) −1 has no poles on ∂B ∩ C i .
Assuming (i), (pG−A) −⋆ is of the form ∞ k=−∞ p k B k , where the series is absolutely summable entry-wise. Then is also absolutely summable, and we can interchange integration and summation as follows: The other cases are shown similarly; the above calculation also shows that formulae (5.4) do not depend on the choice of i ∈ S. Now, it follows that Im Q ′ = ρ(Im Q), since Q ′ ρ(w) = χ(Q)ρ(w) = ρ(Qw) by Lemma 5.4 (and it is easy to see that ρ is a bijection). Similarly, Im P ′ = ρ(Im P ), Ker P ′× = ρ(Ker P × ), Ker Q ′× = ρ(Ker Q × ).