On a polyanalytic a approach to noncommutative de Branges-Rovnyak spaces and Schur analysis

In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carath\'eodory multipliers.


Introduction
Two important function theories that allow to extend complex analysis and operator theory results to higher dimensions are the so-called monogenic and slice monogenic functions with values in a Clifford algebra. In the case of quaternions these two theories are known as Fueter hyperholomorphic and slice regular or slice hyperholomorphic functions, respectively, see [32,44,45,61,63]. For the necessary preliminaries on quaternions, we refer the reader to Section 2. An interesting problem is to investigate the possible relations and intersections between these two different theories. We note that it is always possible to construct Fueter hyperholomorphic functions starting from slice regular ones using different techniques such as the Fueter mapping theorem [42,43], or using the Radon and dual Radon transforms, see [41]. But in general, the slice monogenicity does not imply, nor is implied by monogenicity. However, in [14,15] the authors extended the notion of slice regular functions to higher order by considering the so-called slice polyanalytic functions. These functions can be considered from three different points of view. The first approach consists of viewing the space of quaternions H as union of complex planes and to see these functions as a subclass of null solutions of the n-th power of the Cauchy-Riemann operator with respect to each complex plane. The second approach is based on the so-called poly-decomposition which allows us to consider such functions as sums of the form with all the components f k which are slice regular functions and n is the order of poly-analyticity. The third approach consists in considering slice polyanalytic functions as subclass of the null solutions of the n-th power of a global operator with non-constant coefficients, see [13]. The study in this paper is in the quaternionic context and it is based on some polynomials (P n (x)) n≥0 where P n (x) = n j=0 L j,nx j x n−j , n ≥ 0, that are at the same time Fueter hyperholomorphic and slice polyanalytic functions of order n + 1, for suitable real coefficients L j,n (see [40,54] and Section 2). These polynomials are very special since they belong to the intersection of two different non-commutative function theories, namely the classical Fueter theory and the slice polyanalytic theory, moreover they have nice properties with respect to multiplication and derivation. Another important feature, see Theorem 3.10 in [12], is that any Fueter hyperholomorphic function f of axial type admits a power series expansion in terms of the polynomials P n of the form This fact allows to embed the space of Fueter hyperholomorphic functions of axial type, denoted by AR, into a space consisting of series of slice polynalytic functions that we denote here by where SP n denotes the set of slice polyanalytic functions of order n. More precisely we consider the subspaces of slice polyanalytic functions associated with the polynomials (P n ) n≥0 defined by P n := {P n (x)λ, λ ∈ H} and P ∞ := ∞ n=0 P n .
Then, since the P n are the unique hyperholomorphic extensions of axial type of the real valued functions (3x 0 ) n , it is possible to show that the space of hyperholomorphic functions of axial type AR corresponds to the space P ∞ , i.e. AR = P ∞ .
The previous subspaces of slice polyanalytic functions P n were considered before from a different point of view and using a different terminology, namely they were called spaces of homogeneous special monogenic polynomials of degree n, see for example Lemma 1 in [3]. Using these ideas and identifications we show that it is always possible to embed this interesting subclass of special monogenic functions in a more general framework of slice polyanalytic functions. We use techniques from slice polyanalytic function theory to prove results on such special monogenic functions. In particular, in Proposition 2.18 we prove a Representation Formula in the monogenic setting using a slice polyanalytic approach.
Furthermore, we note that these slice polyanalytic (and Fueter hyperholomorphic) polynomials (P n ) n≥0 are just a particular case of a more general interesting construction which makes use of the classical Cauchy-Kovalevskaya extension theorem as we explain here. Consider an entire real analytic and quaternionic-valued function h of the real variables x 1 , x 2 , x 3 . The Cauchy-Kovalevskaya theorem guarantees the existence of a hyperholomorphic function H, its CKextension. We have, with h n = h n and H n = H ⊙n (where ⊙ denotes the Cauchy-Kovalevskaya product) and so H n = H n⊙ 1 . We can see already here that obstructions occur; if we take quaternions u and v, the CK-product CK(h n u) ⊙ CK(h m v) will not be in general be equal to CK(h n+m uv) since h m and u do not commute. As a consequence, the CK-product will not be, in general, translated into convolution of the coefficients of the expansions along the H n . In spite of this, with this new variable H 1 it is possible to define a number of counterparts of the classical reproducing kernel Hilbert spaces, with reproducing kernel of the form K(x, y) = n∈I H n (x)H n (y) α n , α n > 0, I ⊂ N 0 , converging in some neighborhood of the origin in R 4 . We already mention at this point that the CK-product is not a law of composition for the Hardy space (defined below), and more generally, for series in the functions H n .
A corresponding Schur analysis would consist in particular of the following problems: • Characterize the contractive multipliers of this Hardy space. The definition has to be adapted to the present situation, where we lack the convolution of the coefficients and the CK-product is not a law of composition. • Study interpolation problems for these multipliers.
Here we will focus on the counterpart of H(s) and L(Φ) spaces, whose reproducing kernel are of the form respectively.
These various definitions and corresponding results need to be adapted to the present case, where we do not have a law of composition. We note that the theory can be developed easily as in the classical way when the coefficients are real, but this is of course restrictive. On the other hand, the theory using Fueter variables works well because these variables are real when restricted to x 0 = 0 where x 0 denotes the real part of a quaternion.
There are important differences between the present treatment of Fueter hyperholomorphic functions and the treatment using Fueter variables; in the first case, the kernel functions are eigenvectors of the backward shift, in the case of Fueter variables the kernel functions are eigenvectors of the three underlying backward-shift operator. Here the kernel functions are not eigenvectors of the backward-shift operator. However, the present approach allows to make connections with the theory of slice polyanalytic functions, in particular with slice hyperholomorphic functions, and will also allow a simpler functional calculus. Moreover, Toeplitz operators do appear in a natural way and play an important role. In both cases, it is possible to develop a Schur type analysis. On the other hand, specific choices of the approach allow to make connections with slice hyperholomorphic functions. We here consider the cases and relate the underlying analysis with the Appell polynomials setting. Note that w(0) = 1 = 0.
A key fact used in the paper is that the hyperholomorphic functions considered are of axial type, and hence uniquely determined by their values on the real line. We shall prove that the CK-extension of so that we set where Q m denotes the m-th quaternionic Appell polynomial (see [12, (3.8)] and [49]). The coefficients c m will be specified in Section 2. We are thus looking at a theory of hyperholomorphic functions of the variable equipped with the CK-product. In our discussion it is crucial that We associate in a natural way to a Schur multiplier in the present setting a slice hyperholomorphic Schur multiplier; this allows to develop Schur analysis in the present setting.
The de Branges-Rovnyak space associated with a Schur multiplier S allows in the cases considered up to now to get a coisometric realization of the multiplier. In the complex setting, this is the celebrated backward-shift realization (see [60]). Here, the situation is a bit different. We can still associate to S a coisometric operator matrix, in the form (in the current setting) of the backward-shift realization, but the realization is on the level of the coefficients (like in [56] in the finite dimensional case).
We have given, or outlined, proofs of some classical results, for instance the extension result in Theorem 2.21 and the closely related Theorem 2.22. The reason is that the results play a key role in this paper and some of the arguments are not necessarily well known in the Clifford analysis community. We apply them in the quaternionic setting in particular in Theorem 4.21 and in Step 2 in the proof of Theorem 4.15.
The paper contains twelve sections, besides this Introduction. Section 2 contains some preliminary results. Section 3 contains results on reproducing kernel spaces and Toeplitz operators.
In Section 4 we define the Hardy space in this framework, the backward-shift operator, Schur multipliers and their characterization. The Schur algorithm is presented in Section 5. Section 6 is focused on intrinsic functions, among which the polynomials P n and a description of Fueter hyperholomorphic functions of axial type which are also intrinsic. In Section 7 we consider de Branges-Rovnyak spaces while in Section 8 we show how to define Blaschke functions, and the corresponding operator of multiplication which turns out to be an isometry. In Section 9 we consider the counterpart of Herglotz functions and multipliers and their associated kernels and Hilbert spaces. The next three sections concern the half-space case of Schur and Carathéodory multipliers. In Section 13 we summarize in a table a comparison between the various quaternionic settings.

Preliminaries
This section contains three subsections: the first one introduces the map χ; the second one introduces the Fueter variables and the polynomials obtained via the Appell polynomials which will be the basis of our treatment. Finally, the third one shortly reviews positivity, analytic extensions and Toeplitz operators in the classical complex setting.

2.1.
Quaternions and the map χ. We will work in the skew field of quaternions, which is defined to be where the imaginary units satisfy the multiplication rules e 2 i = −1, i = 1, 2, 3, e 1 e 2 = −e 2 e 1 = e 3 , e 2 e 3 = −e 3 e 2 = e 1 , e 3 e 1 = −e 1 e 3 = e 2 . The conjugate and the modulus of x ∈ H are defined by The set of all imaginary units is given by S = {q ∈ H; q 2 = −1}. We note also that a domain Ω of H is called a slice domain if Ω ∩ R is nonempty and for all I ∈ S, the set Ω I := Ω ∩ C I is a domain of the complex plane C I . If moreover, for every x = u + Iv ∈ Ω, the whole sphere [q] := {u + Jv; J ∈ S}, is contained in Ω, we say that Ω is an axially symmetric slice domain. We can write a quaternion as x = z + we 2 with z = x 0 + x 1 e 1 and w = x 2 + x 3 e 1 ∈ C. The map χ defined by allows to transfer a number of problems from the quaternions to matrices in C 2×2 . We recall the following result, whose proof is immediate and will be omitted.
belongs to the range of χ if and only if it satisfies the symmetry For matrices and operators, there are various ways to define χ. Let X = A + Be 2 ∈ H r×s . We set We define for a block matrix (X jk ) with X jk ∈ H r×s , (χ(X)) jk = χ(X jk ).
For matrices M 1 , M 2 possibly infinite, with block entries in H r×s and H s×t , respectively, we have the property We note that χ will not be compatible with the CK-product. An important tool in the paper consists of bounded block Toeplitz operators, with blocks in the range of χ: . and we will need the following result, set for general operators.
if and only if the operator T defined by where the X jk are matrices in H r×s is bounded from ℓ 2 (N 0 , C 2s ) into ℓ 2 (N 0 , C 2r ), and both operators have same norm.
Proof. One direction is clear: Applying χ we get where I 2 denotes the identity matrix of order 2. Multiplying this inequality by 1 0 on the left and by its transpose on the right, we get the result. Conversely, if T is bounded there exists K > 0 such that for all u 1 , v 1 , . . . ∈ C r , On the other hand, setting Comparing with (2.8) we get and hence the result, since χ preserves order. The claim on the norms being the same follows from the previous inequality and (2.5).

2.2.
Various notions of hyperholomorphy and homogeneous polynomials. In this section we briefly review the setting of Fueter variables and the Cauchy-Kovalevskaya product. We recall that left-hyperholomorphic functions (we will usually just say hyperholomorphic in the sequel) are solutions of the equation Df = 0, where D denotes the Cauchy-Fueter operator These functions are widely studied in the literature. They are, in particular, harmonic functions in four real variables. Unfortunately, the monomials x n in the quaternionic variable x are not in the kernel of the Cauchy-Fueter operator, not even when n = 1. However, hyperholomorphic functions admit a series expansion in terms of the so-called Fueter variables, as we shall see below. We point out that, in this paper, we shall provide only the notions and results needed in the sequel, and for further information on this class of functions we refer the reader to [44,63].
In view of the Cauchy-Kovalevskaya theorem, a linear system of first order differential equations satisfied by the real components of f has a unique solution when the function ϕ(x 1 , x 2 , x 3 ) = f (0, x 1 , x 2 , x 3 ) is pre-assigned (and assumed real analytic). The function f with this initial condition and solution of Df = 0 is called the Cauchy-Kovalevskaya extension of ϕ (here written as CK(ϕ) and abbreviated as CK-extension).
We note that A direct proof that ζ ν given by (2.11) is hyperholomorphic (and hence is the CK-extension of x ν ) is not trivial and can be found in [46, §3], [59]. The argument works also in the split quaternion setting. See [18, p. 333-334]. It is important to note that every function hyperholomorphic in a neighborhood of the origin can be written as a convergent power series in the form of a Fueter series where the coefficients f ν belong to H. See [32]. A proof based on the Gleason problem can be found in [23].
Using the CK-extension one can define a product that preserves the hyperholomorphicity, the socalled CK-product denoted by ⊙. The idea to compute the CK-product is the following: if f and g are two hyperholomorphic functions, we take their restriction to x 0 = 0, which are real analytic functions, and consider their pointwise multiplication. Then, we take the Cauchy-Kowalevskaya extension of this pointwise product, which exists and is unique, to define see [63]. Moreover, we note that the following formula holds For power series of the form (2.12) the CK-product is a convolution on the coefficients along the basis ζ ν , and in particular (2.14) ζ ν p ⊙ ζ µ q = ζ ν+µ pq, p, q ∈ H, µ, ν ∈ N 3 0 , where ν + µ is defined componentwise, see [32,82]. We now turn to a bound for the CK-product; see also [23, pp. 132-133]. Lemma 2.3. Let ρ > 0. There exists ǫ > 0 such that: Then < ρ n , n = 1, 2, 3, . . .
Proof. It suffices to note that the function is Fueter hyperholomorphic and its restriction to x 0 = 0 is the given function.
The polynomials (Q m ) m≥0 are Fueter regular. Moreover, a generalized Fueter regular exponential function associated to these polynomials was considered in the literature, see for example [39]. Another interesting feature of the quaternionic Appell polynomials is that they can be obtained by applying the Fueter mapping applied to the standard quaternionic monomials x m . In particular, in [49] the following formula is proved , m = 0, 1, ...
We then define another kind of Fueter hyperholomorphic polynomials by We have the following relation between the polynomials P m and the CK-extension of x 1 e 1 + x 2 e 2 + x 3 e 3 in (2.17): Proposition 2.6. The following equality holds: Proof. The proof is simple and it is based on the fact that at both hand sides there are monogenic functions which coincide on x 0 = 0: Remark 2.7. The polynomials Q m are called Appell since they satisfy the Appell property the P m do not respect such a property, since however, they behave better with respect to the CK-product, as we shall see below. In particular, for even indexes of the form m = 2k, the Appell property is still satisfied by the polynomials (P 2k ) k≥0 since we have c m−1 = c m in this case.
In what follows, we are looking at a theory of hyperholomorphic functions of the variable with the CK-product. Moreover, note that The ⊙-product is not a convolution on the coefficients of the P n : in opposition to (2.14) we have, in general, (2.23) P n p ⊙ P m q = P n+m pq, n, m ∈ N, p, q ∈ H.
In particular, in general P n q n for q ∈ H in a neighborhood of the origin. This obstruction is the source of the main difficulties and new results in the present paper. Still, we have the following simple result, which plays a key role in the computations (see in particular (3.11)).
Proof. We have where we used (2.20) in the last equality. In particular, by iteration, we obtain (2.26). The last claim follows from restricting the equalities for x 0 = 0, and checking that they are equal to (x 1 e 1 + x 2 e 2 + x 3 e 3 ) n+m+k u.
Corollary 2.9. For every ρ > 0 there exists ǫ > 0 such that Proof. This follows by induction from Lemma 2.3 with f = P 1 and g = P n , n ∈ N.
Remark 2.10. One could take the CK-extension of another linear combination such as t 1 x 1 e 1 + t 2 x 2 e 2 + t 3 x 3 e 3 , namely and develop a similar theory.
Let f (x 0 ) = ∞ n=0 x n 0 a n (with a 0 , a 1 , . . . ∈ H) be a real analytic function near the origin. It does not have a unique hyperholomorphic extension of course, as seen by taking ζ 1 (x)e 1 and ζ 2 (x)e 2 , in fact both functions are equal to x 0 on the real line. However the extension becomes unique by requiring that it is of a special form: x n 0 a n , a n ∈ H be a real analytic function near the origin. It has a unique (left) hyperholomorphic extension of the form f (x) = ∞ n=0 P n (x)b n , namely Similarly, its unique right hyperholomorphic extension is Proof. The function f (x) is indeed an extension of the required form. If there is another one, sayf (x) = ∞ n=0 P n (x)d n we get when setting (3x 0 ) n d n and so b n = d n , n = 0, 1, . . .. A similar reasoning works for g.
Remark 2.12. We note that the polynomials P m and Q m are both left and right hyperholomorphic and in fact P m (x) corresponds to both the left and right CK-extension of (x 1 e 1 +x 2 e 2 + x 3 e 3 ) m .
Actually, the previous result is a particular case of a more general result that holds for Fueter hyperholomorphic functions of axial type, whose definition which comes from the more general case of axially monogenic functions, see [47], is the following: Definition 2.13. A Fueter hyperholomorphic function is of axial type (or axially hyperholomorphic) if it is of the form where A, B are quaternionic valued.
The condition that a function f of axial type is in the kernel of the Cauchy-Fueter operator D translates into the Vekua system Starting from any real analytic function A(x 0 ) it is possible to construct its unique Fueter hyperholomorphic extension of axial type.
We will say that a matrix-valued hyperholomorphic function is of axial type if all its entries, as matrix, are of axial type.
Remark 2.14. Functions of the form (2.30) are quaternionic special monogenic according to the terminology in [3]. Any quaternionic special monogenic function in a neighborhood of the origin is of axial type. In fact any polynomial P m (x) is the sum of terms of the form which are evidently of axial type. This fact was already noted in [39], Property 2. A Fueter regular function represented by a uniformly convergent series of the form (2.30) is such that where the pair A, B satisfy the Vekua system. Conversely, any function of axial type is of the form (2.30), by Theorem 3.10 in [12].
We recall the notion of slice polyanalytic functions, see [14].
When N = 1 the notion coincides with that one of slice hyperholomorphicity (slice regularity).
We have the following characterization, see [ where f 0 , ..., f N −1 are slice regular functions in Ω.
As a consequence: Corollary 2.17. The polynomial P m is slice polyanalytic of order m + 1.
It is clear that all f k are slice regular functions on Ω, being polynomials in the variable x. Moreover, we note that Hence, the thesis follows using Proposition 2.16.
In the definition of the polynomials P m we note that to write the monomials as x jxℓ orx ℓ x j does not make any difference since xx =xx. A Representation Formula for Fueter hyperholomorphic functions of axial type is immediately deduce from the fact that they are slice functions, see [62], so we have: where Ω is an axially symmetric slice domain. Let J ∈ S, then for any x = u + I x v ∈ Ω the following equality holds : Proof. We note that the Fueter hyperholomorphic polynomials (P m ) m≥0 are slice polyanalytic of order m + 1 thanks to Corollary 2.17. Thus, the Representation Formula is an immediate consequence.
19. An alternative proof of the previous Representation Formula in the Fueter hyperholomorphic context consists to apply Proposition 3.13 in [14] to each polynomial P m .
We conclude this part with a result which will be used in the sequel while dealing with kernels: Proof. We immediately have: P m (x)D = DP m (x) = 0, and the first assertion follows. Then we have We now note that which concludes the proof.

Positivity, analytic extension and Toeplitz operators.
This section considers the complex variable setting. Recall first that a C n×n -valued function K(z, w) defined for z, w varying in some set Ω is called positive definite if for every choice of N ∈ N 0 and w 1 , . . . , w N ∈ Ω the block matrix (K(w j , w k )) N j,k=1 is non-negative. Associated to K(z, w) is a uniquely defined Hilbert space of C n -valued functions defined on Ω, denoted here H(K), and with the properties: (1) For every c ∈ C n and w ∈ Ω, the function K w c : z → K(z, w)c belongs to H(K), and (2) For every f ∈ H(K) and w, c as above, H(K) is called the reproducing kernel Hilbert space with reproducing kernel K(z, w), and there is a one-to-one correspondence between reproducing kernel Hilbert spaces and positive definite functions; see [26,75,79]. We recall the following result, which originates with the work of Donoghue [50]. We take a real neighborhood of the origin, but it could be replaced by any other zero set in the open unit disk.
Theorem 2.21. Let s be a C r×t -valued function defined in a neighborhood (−ǫ, ǫ) of the origin, and such that the kernel . Then s has a (uniquely defined) analytic and contractive extension to the open unit disk.
Proof. The proof can be found in e.g. [6, pp. 45-46]. For completeness we outline it. We set r = s = 1 to simplify the notation. Let ρ w (z) = 1 − zw. The function 1/ρ w (z) is positive definite in the open unit disk D, with reproducing kernel Hilbert space the Hardy space of the open unit disk, denoted H 2 (D), and consisting of the power series f (z) = ∞ n=0 a n z n with complex coefficients satisfying f 2 extends linearly to a densely defined operator T from H 2 (D) into itself. The positivity of the kernel (2.33) and the definition of the inner product in the Hardy space implies that T is a contraction, and hence extends to an everywhere defined contraction, still denoted by T , from H 2 (D) into itself. Let f ∈ H 2 (D). The adjoint of T satisfies: So T * is the operator of multiplication by T * 1. Since it is bounded, the formula for the adjoint of a multiplication operator acting in a reproducing kernel Hilbert space gives Since it is contractive, writing that T * 1/ρ w ≤ 1/ρ w we get and hence T * 1 takes contractive values in the open unit disk.
We also recall (we refer to [76] for more information on Toeplitz operators): Proof. We set r = t = 1 to simplify the arguments. Assume first that s is a contraction, and let P denote the orthogonal projection from L 2 (T) onto H 2 (T). Then the Toeplitz operator f → P s * f is a contraction from H 2 (T) into H 2 (T). It admits thus a matrix representation.
Using the basis 1, z, z 2 , . . . we see that and hence the Toeplitz matrix representation. For the converse, we assume that T s is a contraction. We compute T * s e z where e z = (1, z, z 2 , . . .) t . We have (compare with (2.35)) (2.37) T * s e z = s(z)e z and hence the result.

Positive operators, reproducing kernel spaces and Toeplitz operators
We use various tools from quaternionic operator theory and in particular from the theory of linear relations and the theory of reproducing kernel spaces, as developed in [8]. We recall: Definition 3.1. Given two right (or left, or two-sided) linear spaces V, W over the quaternions, a linear relation is a linear subspace of the Cartesian product V × W.
The graph of a (possibly not everywhere defined) linear operator is a linear relation, but there are linear relations which are not graphs of operators.
We will define inner products on a quaternionic right vector space, say V, with the following convention and satisfying moreover when the quaternionic space under study is two-sided (for instance, ℓ 2 (N 0 , H)).
Let K(x, y) be a the H r×r -valued function, positive definite on Ω. We will denote by H(K) the reproducing kernel space of H r -valued functions with reproducing kernel K.
Let K 1 (z, w) and K 2 (z, w) be two H r×r -valued functions positive definite on a set Ω. We recall that K 1 ≤ K 2 means that the difference K 2 − K 1 is still positive definite in Ω. This happens if and only if the space H(K 1 ) is contractively included in the space H(K 2 ).
The following result, relating operator ranges and reproducing kernel Hilbert spaces is well known. See [9] for a discussion in the quaternionic and indefinite inner product setting.
where π is the orthogonal projection onto the kernel of Γ.
Let us set H = ℓ 2 (N 0 , H r ) in the previous proposition. Since Γ is bounded, it has a block matrix representation Γ = (Γ nm ), where Γ mn ∈ H r×r . We can write Cases of interest in the present work are: Assume now Γ to be of the form with T S as in (2.36) and where S i ∈ H r×t , i = 0, 1, . . .. In particular, the block Toeplitz operator T S is a contraction. The kernel becomes in the first case in the second case.
We note that, with the ⋆-product (see [8]): and similarly, with the CK-product, using (2.27) in Lemma 2.8, The functions are Schur multipliers, for the slice hyperholomorphic and for the present case (called Appell-like case), respectively.

The Hardy space and Schur multipliers
In this section we will introduce and study the Hardy space in this framework. To start with, we denote by E the ellipsoid 3 < 1 and we prove the following: converges and is positive definite for x, y ∈ E.
Proof. For x ∈ E we have |P 1 (x)| < 1 and the result follows from Corollary 2.9.
We point out that using (2.28), we get Remark 4.2. In [22,23,24] a different approach was used and a similar construction yields the Drury-Arveson kernel Note that the formula (4.2) is easier to work with than formula (4.3). We also note that Using the polynomials Q n one can define the kernel (see [12,Remark 5.3]) whose restriction to the real axis is different, indeed it is where the coefficients f m belong to H and are such that This expression is then the square of the norm of f in the Hardy space, i.e. f 2 = ∞ m=0 |f m | 2 . Proof. The proofs follows standard arguments, see [12,21].
From the form of the elements of the Hardy space H 2 (E) and using the fact that the polynomials P m are Fueter hyperholomorphic of axial type, see Remark 3.9 in [12], we deduce: Lemma 4.6. The operator S : f → P 1 ⊙ f is an isometry in the Hardy space, with adjoint given by Proof. The proof is a consequence of Let Cf = f (0) be the point evaluation in H 2 (E). Then C * u = k E (·, 0)u = u and we get from the previous lemma (4.10)   We can consider hyperholomorphic functions operator-valued, in particular matrix or vectorvalued. The definition of this class of functions is given by following the classical complex case, but we repeat it for the sake of completeness.
Definition 4.9. Let X be a two-sided quaternionic Banach space, and X * be its dual. A function We recall that a function is hyperholomorphic if and only if it is differentiable in a suitable sense, see [74,Theorem 3] and we follow this notion of differentiability to state the following definition, in which we identify H with Definition 4.10. Let a ∈ H 3 , U be a neighborhood of a and let F : U → X be a continuous function. Then f is called left (resp. right) strongly differentiable in a in the quaternionic sense if there exists a left (resp. right) linear map L : H 3 → X such that The definition originally considered by Malonek in [74] can be obtained from the previous one when X = H. Since, in the scalar case, the definition is also equivalent to that one of left (resp. right) hyperholomorphy, we will equivalently say that a function f as in Definition 4.12 is strongly hyperholomorphic. See also [19] for a theory of hyperholomorphic functions whose values are taken in a Banach algebra. Using the same arguments as in the complex case, see [77,Theorem VI.4], which are valid also in the quaternionic case, see [8], one can prove: The validity of this result allows to simplify the terminology and we shall say, for short, that f is hyperholomorphic with values in X . In the special case in which X = H r×s , a function is weakly hyperholomorphic if and only if all its entries are left or right hyperholomorphic there.
The next result was proved in the quaternionic setting in the context of slice hyperholomorphic functions, see e.g. [8,Section 7]; here we prove its counterpart in the present framework.
Since the CK-product is not a law of composition we cannot, a priori, define Schur multipliers (see This example is of course quite trivial. We will give in Section 8 a complete characterization of Schur multipliers, from which one can get numerous other examples. The positivity of the kernel (4.15) is equivalent to the contractive inclusion of the reproducing kernel Hilbert space with reproducing kernel (4.16) ∞ n=0 (P n ⊙ S)(x)((P n ⊙ S)(y)) * inside the Hardy space. In particular, if S is a Schur multiplier P n ⊙ S ∈ H 2 (E) for every n.
Proof. Assume first that the kernel (4.15) is positive definite in E. We divide the argument in a number of steps.
This follows from Theorem 2.22.

STEP 4:
The Toeplitz operator T s is contractive.
This follows from Proposition 2.2. We restrict the operator in Step 3 to sequences of matrices in the range of χ.
We now suppose that T is a contractive operator. We write to get the positivity of the kernel K S .
Corollary 4.16. In the above notation, the function is a slice hyperholomorphic Schur multiplier.
Given two multipliers, the bounded operator M S 1 M S 2 will not be a multiplier in general. P n ((P n ⊙ S)(a)) * u, a ∈ E, u ∈ H r , defines a contraction from (H 2 (E)) s into (H 2 (E)) r , with adjoint given by Proof. Let ∞ n=0 P n u n ∈ H 2 (E). We can write:  is analytic and contractive in D; by the previous theorem this will hold if and only if S is a H t×r -valued Schur multiplier.
The operator T * cannot be written as S ⊙ ( ∞ n=0 P n u n ), i.e. it is not the ⊙ multiplication by S. We now introduce the counterpart of this latter operator here.
Definition 4.20. The operator T * will be denoted by M S : We have the following extension result, counterpart of Theorem 2.21. The proof is slightly different. Proof. We consider the scalar case to simplify the notation. The preceding argument still holds and, setting a = 0 and b ∈ N , we get But T * v ∈ H 2 (E) and in particular is hyperholomorphic in all of E. More generally, still for b ∈ N , but for f = ∞ n=0 P n f n ∈ H 2 (E) we have Writing (T * f )(b) = ∞ n=0 P n h n and S = ∞ n=0 P n s n , the previous equality is equivalent (since v varies in an open set; it would be enough to have an interval such that where T S is the lower triangular Toeplitz operator based on the coefficients of S. So T S is a contraction, and so I − T S T * S ≥ 0, and so (I − T S T * S c, c ≥ 0 for every c ∈ ℓ 2 (N 0 , H). The choice (4.18) allows to conclude that S is a Schur multiplier. Proof. Let f = ∞ n=0 P n u n . We can write: It is useful to rewrite (4.24) as

Schur algorithm
The Schur algorithm is based on Schwarz lemma and on the fact that if two numbers u and v are in the open unit disk so is It reads (see [80,81]): defines a family of Schur functions; it stops at rank n 0 if |f (n 0 ) (0)| = 1.
The numbers ρ n = f (n) (0) are called the Schur parameters associated with the Schur function f .
This recursion cannot be considered directly in the matrix-valued case. One needs to take into account that if E 1 and E 2 are strictly contractive matrices, say in C p×q , the matrix (E 1 + E 2 )(I q + E * 1 E 2 ) −1 need not be contractive, but the matrix The matricial Schur algorithm was studied in [48] and, in the next result, we repeat the statement taking into account the matrix symmetry is in force.
Theorem 5.2. Let s be a C p×q -valued Schur function satisfying (5.3). Assume s 0 strictly contractive. Then the function If s 0 < 1 one can iterate, and one gets the matricial Schur algorithm.
The condition s(0) < 1 is quite restrictive. A tangential Schur algorithm was developed in [16]. On the other hand, when p = q = 2 and s(0) is in the range of χ both (I 2 − s 0 s * 0 ) −1/2 and (I 2 − s * 0 s 0 ) 1/2 are scalar matrices and (5.4) reduces to We now turn to the setting of hyperholomorphic functions of axial type. For simplicity of exposition we first consider scalar valued Schur multiplier. From the analysis in the previous sections, S = ∞ n=0 P n S n is a Schur multiplier if and only if the block Toeplitz operator In the matrix-valued case it is not true anymore that I − χ(S 0 )χ(S 0 ) * is a scalar matrix.
Theorem 5.4. Let S be a H r×t -valued Schur multiplier, and assume S(0) < 1. The function The question whether the tangential Schur algorithm developed in [16] can lead to functions satisfying the required symmetry property in the matrix-valued case remains to be considered.

Intrinsic functions
In this section we study quaternionic intrinsic Fueter hyperholomorphic functions. Let us recall that, given an hyperholomorphic function f on some axially symmetric open set Ω, we say that f is quaternionic intrinsic if it satisfies the relation Proposition 6.1. The family of polynomials (P n ) n≥0 consists of axially hyperholomorphic quaternionic intrinsic functions on H.
Proof. We know that for all n ≥ 0 the polynomials P n are axially hyperholomorphic functions on H. Furthermore, using the relation with the n-th quaternionic Appell polynomials Q n , see [12, (3.8)], we have Proof. We know by Theorem 3.10 in [12] that f admits a power series with respect to (P n ) n≥0 .
So, we can write f = ∞ n=0 P n f n with f n ∈ H for all n ≥ 0. We assume that f is intrinsic, thus the formula (6.1) and Proposition 6.1 imply that The equivalence between the second and the third lines holds because P n is the unique axially hyperholomorphic extension of (3x 0 ) n . This ends the proof. Proposition 6.3. Let S 1 and S 2 be two hyperholomorphic functions of axial type, defined on some axially symmetric open set Ω. If S 1 is quaternionic intrinsic, then S 1 ⊙ S 2 admits a power series expansion with respect to the polynomials (P n ) n≥0 .
Proof. We note that S 1 and S 2 have power series expansions in terms of (P n ) n≥0 that we can write S 1 = ∞ n=0 P n a n and S 2 = ∞ n=0 P n b n . Since S 1 is quaternionic intrinsic we know by Proposition 6.2 that the coefficients (a n ) n≥0 are real. Thus, we apply also Lemma 2.8 to get Proposition 6.4. Let S be a hyperholomorphic function of axial type. If S is quaternionic intrinsic, then the operator M S coincides with the multiplication operator f → S ⊙ f .
Proof. We note that since S is quaternionic intrinsic, it has real coefficients. Thus, we have P n ⊙ S = S ⊙ P n for all n ≥ 0. Then, starting from Definition 4.20, for any f = Proposition 6.5. Let S 1 and S 2 be two hyperholomorphic functions of axial type such that S 1 is quaternionic intrinsic. Then, we have Proof. We know by Proposition 6.3 that S 1 ⊙ S 2 is well defined and admits a power series expansion in terms of (P n ) n≥0 since S 1 is intrinsic. Therefore, using Proposition 6.4, we have

de Branges-Rovnyak spaces
The reproducing kernel Hilbert space with reproducing kernel (4.15) will be called the de Branges-Rovnyak space associated with the Schur multiplier S and denoted by H(S). The treatment using the Appell-like approach allows to prove results naturally extending the corresponding ones in the classical complex case. For example, we have the following characterization: Theorem 7.1. We have We recall the following, valid for f, g ∈ H 2 (E) (the first equality is a special case of the second one): Using the quaternionic version of [37] or [55, Theorem 4.1] (we do not give proofs of these since we will have more general results than the theorems below in the next section) we have the following results, for matrix-valued Schur multipliers f + M S g 2 2 − g 2 2 < ∞.
Using this characterization we can prove the following: Theorem 7.3. Let S be a H r×s -valued Schur multiplier. Let R 0 be defined by (4.8). Then: Recall that S denotes the forward-shift operator, and that the latter is an isometry (see Lemma 4.6). Using (4.9) and (4.24) we can write for f, g ∈ H 2 (E): since P 1 ⊙ M S g has no constant term. Theorem 7.4. Let R 0 be defined by (4.8). Then R 0 Su ∈ H(S) for every u ∈ H s and Proof. We already know that Su ∈ H 2 (E). We write In the above, to go from the third to fourth line we used that (since S is a Schur multiplier) and since M S (u + P 1 ⊙ g) has no constant term in its expansion along the P n . Similarly we used that u, P 1 ⊙ g 2 = 0 to go from the fourth to the last line.
The operators defined in the previous theorems are part of a coisometric operator matrix. In [37] (see also [35]) it is obtained using the theory of complementation. In the next section we use a different method.

The coisometric colligation and Blaschke functions
8.1. The lurking isometry. Let us denote by ⊙ r the right CK-product. Using (2.27) we note that (4.15) can be rewritten as from which we get where h, k ∈ H s and x, y ∈ E.
This last equality, called the lurking isometry (see [4,28]), can be the tool to get a co-isometric realization of S (see [2] for an application in the quaternionic setting). We will choose a different (and closely related) avenue, namely the theory of relations, which originates with the work of Krein and Langer (the ǫ z method; see e.g. [69,70]) and was developed further in [11]. We will use the lurking isometry method in Section 11 to characterize Carathéodory and Schur multipliers in the setting of the counterpart of Hardy space of the right half plane. 8.2. The co-isometric realization. We use the method of isometric relations, as in [11], suitably adapted to the present setting, and follow [22, §2]. We set Γ = I − M S M * S , and define w a via w a q = ΓM * P 1 k E (·, a)p ∈ H(S), q ∈ H r , and introduce: Proof. We want to check: where p, q ∈ H r , u, v ∈ H s , and a, b ∈ E. We divide the proof into three steps.
Then, only terms involving the directions p and q appear. In the following sequence of equality we use (4.25) to go from the second to the third line, and (4.10) to go from the fourth to the fifth line. We also note that To pursue we note that Thus Furthermore, (K S (·, a) − K S (·, 0))q , (K S (·, b) − K S (·, 0))p H(S) We now need to check that but this is straightforward.

STEP 3: Mixed terms.
By symmetry it is enough to consider the case where p and u appears. We need to verify that: but this is equivalent to which clearly holds.
We now compute the adjoint of the above isometric operator. We write Theorem 8.2. V is densely defined, extends to an everywhere defined isometry and its adjoint is given by Proof. By definition of the operator range inner product we have: and so f = 0 if the above vanishes for all u and a since M P 1 is an isometry.
We have on the one hand On the other hand, where V is as in (8.4). Then, for f = ∞ n=0 P n f n ∈ H(S) (8.7) f n = CA n f, n = 0, 1, . . . Writing S = ∞ n=0 P n S n we conclude by applying (8.7) to R 0 Su for u ∈ H s . Remark 8.4. A deep difference with respect to the classical case is that the kernel functions of H 2 (E) are not eigenvectors for R 0 .
Remark 8.5. Following linear system theory (see [31,60,67,78]) we will call the representation (8.8) a realization of S. The associated matrix V will be called the realization matrix or the Rosenbrock matrix. The case where V * is a matrix can be seen as the definition of rational functions. Unfortunately, the CK-product of two such functions will not be rational in this sense. The next section deals with an important example of rational functions. 8.3. Blaschke functions. Equation (8.8) allows us to give a family of Schur multipliers, which we call Blaschke functions, namely those corresponding to the operator-matrix (8.6) to be a unitary matrix. The definition then extends the classical case, also in the matrix-valued and possibly indefinite case; see [17] for the latter. In general there will not be ⊙-multiplicative decompositions of such a Blaschke "product" into elementary factors, hence the term function rather than product. Proposition 8.6. Let B be H r×r -valued Blaschke function, with corresponding realization matrix V * ∈ H (N +r)×(N +r) . The corresponding multiplication operator M B is an isometry from (H 2 (E)) r into itself and the corresponding space H(B) is finite dimensional.
Proof. We first remark that the assumed unitarity is equivalent to the equations Let now n 0 , m 0 ∈ N 0 , and u, v ∈ H r . We have, with b n given by (8.7), We now compute ∞ n,m=0 n 0 +n=m 0 +m b * m b n taking into account that the matrix V is unitary.
where we first used (8.9) and then (8.10). Assume now n 0 < m 0 (the case n 0 > m 0 is obtained by symmetry). We can write: in view of (8.11). We thus have an isometry on the linear span of P 0 , P 1 , . . . and on the whole of (H 2 (E)) r by continuity. To show the finite dimensionality of the space we restrict x = x 0 , y = y 0 ∈ (−1/3, 1/3). We then have It follows that the linear span of the functions x 0 → K S (3x 0 , 3y 0 )h (h ∈ H r and y 0 ∈ (−1/3, 1.3) is finite dimensional. By the uniqueness of the axially hyperholomorphic extension the linear span of the functions x → K S (x, 3y 0 )h is finite dimensional We claim that they span H(S).
The above computations show equivalently that: Corollary 8.7. Let V * given by (8.6). Then the corresponding Toeplitz operator defined by the sequence (8.12) is unitary from ℓ 2 (N 0 , H r ) into itself.
More generally, one can take V * to be C (N +s)×(N +s) -valued and co-isometric. Then (8.9)-(8.11) still hold and the same proof as above leads to: where the realization matrix (8.6) is coisometric. Then the corresponding operator M S is an isometry from (H 2 (E)) s into (H 2 (E)) r . Remark 8.9. Take S 1 , . . . , , S N to be N Schur multipliers (say, H-valued) with associated finite dimensional H(S j ) spaces, j = 1, . . . , N , and let t 1 , . . . , t N to be real numbers such that is a Schur multiplier from (H 2 (E)) N into H 2 (E), and the associated reproducing kernel space is finite dimensional, but will not be isometrically included in H 2 (E) when N > 1; see [16, p. 71] for the complex setting. The argument is the same here.
Remark 8.10. In fact the results in the present section still hold when V * is isometric, but not necessarily a matrix. Then the corresponding multiplier is called inner. The study of these multipliers will be considered elsewhere. Similarly, one could assume unitarity with respect to an indefinite metric. This aspect of the theory will also be treated in a separate publication.
8.4. Rational functions. We now define rational functions in the present setting. We first recall that any C n×m -valued rational function, say M (z), with no pole at the origin can be written in the form where H, G, T, F are matrices of appropriate sizes. Expression (8.13) is called a realization (centered at the origin). See Remark 8.5 above for references. We also recall the formulas for the product and inverse of rational functions. Note that, since we consider possibly non-square functions, the sum will be a special case of the product since where M 1 and M 2 are C n×m -valued.
Assuming in (8.13) that n = m and H invertible, one has the formula (8.14) where T × = T − GH −1 F , and with M j (z) = H j + zG j (I − zT j ) −1 F j , j = 1, 2, two rational functions of compatible sizes, Definition 8.11. The H r×s -valued function R(x) hyperholomorphic of axial type is called rational if its restriction to the real axis is a rational function of the real variable, with quaternionic coefficients.
Theorem 8.12. The H r×s -valued function R(x) hyperholomorphic of axial type, and defined at the origin, is rational if and only if x 0 → R(3x 0 ) can be written as where A, B, C, B are quaternionic matrices of appropriate sizes. Equivalently: Theorem 8. 13. The H r×s -valued function R(x) hyperholomorphic of axial type, and defined at the origin, is rational if and only if it can be written as where A, B, C, B are quaternionic matrices of appropriate sizes.
We will not give the proofs of the above results, which follow easily from the previous analysis in the paper. One still has the formulas (8.14) and (8.15) for a real variable x 0 , but not for the CK-product. So the product of rational hyperholomorphic functions of axial type is not compatible with the CK product. To emphasize this point, we now make the connection with rational functions of the Fueter variables, as studied in [10,20,23] (see also [71], and see [18] for the split quaternionic case). There, rational functions are characterized by the formula (we do not specify the sizes of the various quaternionic matrices) and the variable here is ζ = ζ 1 ζ 2 ζ 3 . We look at the special case where where A and B are matrices of appropriate sizes and with quaternionic coefficients. Since P 1 (x) = ζ 1 (x)e 1 + ζ 2 (x)e 2 + ζ 3 (x)e 3 we can then rewrite (8.18) as which will not be in general a series in the P n , but is a series in the ζ α . We can define such elements as the rational functions associated with the polynomials P n . Then, things make sense in terms of realizations, with the usual formulas, but we do not get power series in P n , even when A is nilpotent. and has reproducing kernel equal to 1−j(z)j(w) 1−zw . One motivation for the theory of de Branges-Rovnyak is to characterize reproducing kernel Hilbert spaces of functions which are contractively included in H 2 (D), and R 0 -invariant. Rather than the latter, one assumes that the inequality This inequality implies contractive inclusion in the Hardy space, (8.20) f ∈ M =⇒ f ∈ H 2 (D) and f H 2 (D) ≤ f M and that in particular the space is a reproducing kernel Hilbert space since We refer to the notes in [11, p. 206] for some history on Theorem 8.14, but mention the papers [72]. Guyker characterized the spaces for which the inclusion is isometric; see [11, p. 187], [64,65]. For an illustration of the contractive inclusion, see Remark 8.9 above.
A general version of Theorem 8.14, in the operator-valued and Pontryagin space case, has been proved in [11, Theorem 3.1.2, p. 85]. We will present in a subsequent paper the general version of the result, in the Pontryagin and operator-valued function case. The purpose of this section is to illustrate the power of the methods used here on a simple case. Note that (8.21) is a weakening of (4.10).
Theorem 8.15. Let H be a Hilbert space of H r -valued functions axially hyperholomorphic in E, and R 0 -invariant and satisfying Then there exist a right quaternionic Hilbert space C and a L(C, H r )-valued function S hyperholomorphic of axial type such that H = H(S).
Proof. It follows from (8.21) that H is contractively included in (H 2 (E)) r , and that the operator R 0 is bounded (another argument, still valid in the quaternionic Pontryagin space setting would be to prove that R 0 is closed, thanks to the reproducing kernel property, and use the closed graph theorem; see [8,Theorem 5.1.16,p. 74] for the latter). The point evaluation at the origin, which we will denote by C, is also bounded since the space is contractively included in the Hardy space and its norm is larger than the Hardy space norm. We can thus rewrite (8.21) as Since the adjoint of a contraction between Hilbert space is still a contraction we have and we can factorize the quaternionic positive operator via a right quaternionic Hilbert space C as The operator matrix is co-isometric, and the L(C, H r )-valued function S defined by is a Schur multiplier. To conclude we show that H = H(S). From the definition of S we have for x 0 , y 0 ∈ (−1/3, 1/3) We have and the result follows from axially hyperholomorphic extension on the left with respect to x and on the right with respect to y.

Spaces L(Φ)
We now consider the counterpart of L(Φ) spaces, see (1.2), in the present setting, and first briefly review the classical case. Functions analytic in the open unit disk and with a positive real part there will be called here Herglotz functions (they are called Carathéodory functions in Akhiezer's book [5, p. 116]). An Herglotz function, say Φ, is characterized by an integral representation of the form where m ∈ R, σ is an increasing function, and (9.1) is a Stieltjes integral. They play an important role in the trigonometric moment problem, operators models for isometric and unitary operators in Hilbert spaces and in the theory of dissipative discrete systems, and have been extended to various more general frameworks; see e.g. [1,27,29,30,69,68,73].
In a way similar to Theorem 2.21, consider a function Φ defined in a real neighborhood (−ǫ, ǫ) of the origin and such that the kernel is positive definite there. Then it is the restriction to (−ǫ, ǫ) of a uniquely defined Herglotz function, and the corresponding kernel is positive definite in the open unit disk. The factor 2 in the kernels is to get nicer realization formulas (such as (9.2)) and follows basically from the Cayley transform z → 1 − z 1 + z , which maps Herglotz functions into Schur functions. The corresponding reproducing kernel Hilbert space and its applications to operator models was first characterized and studied by de Branges; see [34,38]. It is important to note that a Herglotz function need not be bounded, and hence need not be a multiplier of the Hardy space.
Using the reproducing kernel space L(Φ) associated with L Φ (or directly from (9.1)), one can characterize Herglotz functions in terms of a realization of the form where V is coisometric in some Hilbert space, and C * is a continuous map from the coefficient space (the complex numbers when the functions are scalar) into that Hilbert space. Note that (9.2) can be rewritten as In this section we study the counterpart of the kernel L Φ in our setting, and give the counterpart of the expansion (9.3), and study connections with Toeplitz operators. In the following definition (and also in Section 12 below) we use the term multiplier although the operator of CK-multiplication by the given function need not be bounded in the Hardy space. (P n ⊙ Φ)(x)P n (y) * + P n (x)((P n ⊙ Φ)(y)) * is positive definite in E.
When the operator of CK-multiplication by Φ is bounded in the Hardy space (H 2 (E)) r , we can replace (9.4) by the condition (see Remark 4.18) Proposition 9.2. We note the following property of Γ: Proof. Using the fundamental equality (4.10), we can write We make now some remarks on the above kernel. It seems difficult to find a direct counterpart of (9.2) (in view of (2.24)), let alone of (9.1). As for the case of Schur multiplier, we look for a characterization of the coefficients of Φ in its expansion along the P n . When x = x 0 and y = y 0 belong to (−1/3, 1/3) the kernel L Φ becomes As for the case of Schur multipliers, this restriction is enough to get back the kernel L Φ in view of the axial symmetry of the functions. where V is an isometry in a Hilbert space, say H, and C is a continuous map from H into H r .
Proof. We first prove the sufficiency. We have for a ∈ (−1, 1) Φ(a) = CC * + 2 ∞ n=1 a n CV * n C * = C(I + aV * )(I − aV * ) −1 C * . We now turn to the proof of the direct statement, and divide the proof in a number of steps. We follow [25, Proof of Theorem 5.2, p. 708]. We write P 1 (3y 0 ) rather than 3y 0 to emphasize the axially symmetric hypercomplex extension to be used. At the end of section alternative steps 1 and 2 are given when M Φ is bounded.
and on the other hand , and hence the result by axially hypercomplex extension. STEP 3: Let C denote the evaluation at the origin in L(Φ). It holds that We have, with p, q ∈ H r , and hence the result. STEP 4: We prove (9.8).
With q ∈ H r we write Corollary 9.5. Let Φ be a Herglotz multiplier. The space L(Φ) is R 0 invariant.
As for Schur multiplier one has: Theorem 9.6. A space L(Φ) is finite dimensional if and only if the operator V can be chosen to act in a finite dimensional space (and hence is unitary).
In terms of Toeplitz matrices we have: Proof. Let Φ be such that the kernel L Φ is positive definite in E (at this stage we do not assume yet that the associated operator M Φ is bounded). Then, setting x 1 = x 2 = x 3 = 0 and applying the map χ we see that the kernel Conversely, if (Φ n−m ) ∞ n,m=0 defines a bounded positive operator, the function . . .
We conclude with a computation of the linear relation associated to Φ when M Φ is a bounded operator. The computations are longer, but avoid axially symmetric extensions. The relation (9.9) becomes: where u 1 , v 1 . . . ∈ E and p 1 , q 1 , . . . ∈ H r .
NEW STEP 1: The linear relation spanned by the elements (9.10) is isometric and densely defined, and hence extends to the graph of an everywhere defined isometry.
To prove this claim, we let f = j k E (·, u j )q j and g = ℓ k E (·, v ℓ )p ℓ . Using (9.6) we can write: .
There is thus an everywhere defined isometric operator such that NEW STEP 2: We have Indeed, let V * F = √ Γf . On the one hand, On the other hand, Hence, using the formulas (7.4)-(7.5) for the operator range inner product, we have

The half-space case
We first recall that the classical Hardy space of the open right half-plane C r is the reproducing kernel Hilbert space with reproducing kernel equal to 1 2π(z + w) (the factor 2π appears because of Cauchy's formula), and can be characterized as the space of power series of the form where the complex numbers b n satisfy ∞ n=0 |b n | 2 < ∞; see for instance [66]. We will denote this space by H 2 (C r ). The purpose of this section is to define and begin a study of the counterpart of the space H 2 (C r ) in the present setting. A more detailed analysis will be presented in a sequel to the present work. We give three equivalent characterizations of the new space, respectively in terms of a reproducing kernel, restriction to the positive real axis and series expansion analogous to (10.1). We first define what will be the counterpart of C r . To that purpose, consider the function w(x) defined in (1.3); it has for (unique) CK-extension This function is intrinsic hyperholomorphic of axial type by Remark 2.14, in fact in a neighborhood of the origin we can write it as We define W n (x) = W ⊙n 1 (x) and we set Note that W n (0) = 1 and also that W n (x) is hyperholomorphic of axial type, being a finite CK-product of intrinsic series in P 1 (x). On the other hand for x 1 = x 2 = x 3 = 0 and x 0 > 0, Using the arguments in Lemma 2.3 we can prove the following: By Lemma 2.3 we have that for any ρ > 0 there exists ǫ > 0 such that forx 2 0 +x 2 j < ǫ, j = 1, 2, 3, one has W 1 (x) ⊙n < ρ n , n = 1, 2, . . . .
SinceW n (x) = W n (1/3 + x) = (W 1 (1/3 + x)) ⊙n = (W 1 (x) ⊙n we deduce that |W n (x)| < ρ n for x 2 0 +x 2 j < ǫ, j = 1, 2, 3. Thus, for any 0 < ρ < 1 there exists ǫ > 0 such that for (x 0 − 1/3) 2 + x 2 j < ǫ, j = 1, 2, 3 and the statement follows. Definition 10.2. We denote by P the subset of H for which the series The set P is nonempty as the previous lemma shows, moreover we have: The P contains all the points of the positive real axis R + and for anỹ x 0 ∈ R + there exists a neighborhood ofx 0 in which the series converges.
As in Section 2, K W (x, y) solves the equation (8.1), with W 1 replacing P 1 , namely We set (note that we do not put a factor 2π): Theorem 10.4. The kernel K P (x, y) is positive definite in P and is the unique solution of the Lyapunov equation Proof. The first claim follows from the formula where we have used Proposition 2.20 relating the left and right CK-products.
We denote by H 2 (P) the reproducing kernel Hilbert space with reproducing kernel equal to K P (x, y). We also note that (10.7) and (10.11) are equivalent, but (10.11) is better adapted to use the lurking isometry method or the linear relation method, when one considers multipliers (see Section 11).
The counterpart of the expansion (10.1) is presented in the following theorem.
Theorem 10.5. The reproducing kernel Hilbert space associated with the kernel K P (x, y) consists of the power series where the coefficients f n are in H and satisfy ∞ n=0 |f n | 2 < ∞. The latter is then the square of the norm of f .
We now turn to the characterization of H 2 (P) in terms of restrictions to the real positive axis. Theorem 10.6. f ∈ H 2 (P) if and only if x 0 → χ(f (3x 0 )) is the restriction to x 0 > 0 of an element in (H 2 (C r )) 2×2 . The map which to f ∈ H 2 (P) associates the map x 0 → √ πχ(f (x 0 )) is then unitary.
Proof. Setting Applying the map χ and comparing with (10.1) we get the statement. The function is uniquely determined since (0, ∞) is a zero set. The converse follows from the fact that x 0 → (1 − 3x 0 ) n (1 + 3x 0 ) n+1 has as unique axially hyperholomorphic extension the function (1 + P 1 (x)) −⊙ ⊙ W n (x) (which is evidently of axial type), being the CK-product of series in P 1 (x).

Schur multipliers in the half-plane setting
In the classical case of the complex numbers, a C r×r -valued function is contractive in C r if and only if the kernel (11.1) I r − s(z)s(w) * z + w is positive definite in C r . More generally, if a function s is defined in a zero set, say Z, of the open right half-plane and the kernel (11.1) is positive definite on Z, then s is the restriction to Z of a uniquely defined function analytic and contractive in C r . This can be seen from the disk case (see Theorem 2.21) using a Cayley transform. In the present section we study the counterpart of the Schur multipliers for the space H 2 (P), and characterize them in three equivalent ways: (1) In terms of a positive definite kernel.
(2) In terms of an appropriately defined multiplication operator.
(3) In terms of a realization.
Definition 11.1. A H r×r -valued S function is called a Schur multiplier if there is a kernel K S (x, y) positive definite in P, left-hyperholomophic in x and right-hyperholomorphic in y, and such that (11.2) 2(P 1 (x) ⊙ K S (x, y) + K S (x, y) ⊙ r P 1 (y)) = I r − S(x)S(y) * , x, y ∈ P.
In the complex setting case, a function, say s, analytic and contractive in C r does not need belong to H 2 (C r ), but z → s(z)/(1 + z) does belong to H 2 (C r ). Here, at least in the present analysis, we need a supplementary condition to get the counterpart of this result. We have (the notion of spectral radius is defined for a quaternionic operator A as in the classical case by the formula ρ(A) = lim sup n→∞ A n 1/n ): Corollary 11.3. In the notation of Theorem 11.2, assume ρ(A) < 1. Then, the entries of the function (1 + P 1 (x)) −⊙ ⊙ S(x) belong to H 2 (P).
Proof. This follows from the fact that CA n B ≤ C · B · A n , and the series ∞ n=1 A n converges since ρ(A) < 1. (1 + P 1 (x)) −⊙ (W n ⊙ S)(x)f n is a contraction from H 2 (P) into itself, and K S is given by (W n ⊙ S)(x)(W n ⊙ S)(y) ⊙ r (1 + P 1 (y)) −⊙r . (11.9) Proof. We consider the scalar case to ease the notation and first remark that, if S is a Schur multiplier we have K S (3x 0 , 3y 0 ) = 1 − S(3x 0 )S(3y 0 ) 3(x 0 + y 0 ) with axially hyperholomorphic extension (11.9), and the positivity of (11.9) expresses that the operator M S is a contraction. The converse is proved by defining a contractive relation from the positivity of the kernel, and show that the relation extends to the graph of M * S . As in the E setting, the case where the isometry in the above realizations is unitary in a finite dimensional space corresponds to finite dimensional H(S) spaces isometrically included in the Hardy space (H 2 (P)) r . When the space has dimension 1 the function S is the counterpart of a Blaschke factor of the half-plane. (1 + P 1 (x)) −⊙ ⊙ W m+n (x)CA n Bh.
The same computations as in the proof of Proposition 8.6 show that M S 1 + P 1 (x)) −⊙ ⊙ W n 1 h(x) , M S 1 + P 1 (x)) −⊙ ⊙ W n 2 (x)k = = ∞ n,m=0 n 1 +n=n 2 +m k * B * A * n AC * C m Bh = δ n,m k * h, h, k ∈ H s , and this allows to end the proof.

Carathéodory multipliers in the half-plane setting
A function Φ analytic and with a positive real part in the open right-half plane is called a Carathéodory function, and is characterized by the positivity of the kernel Φ(z) + Φ(w) z + w in C r . As for Herglotz functions, a Carathéodory function need not be a multiplier of the Hardy space H 2 (C r ). We now introduce the counterpart of this class of functions in the present setting. As in Section 9.1 we use the term multiplier (rather than, for instance pseudo-multiplier) although the CK-multiplication by the function Φ is not assumed bounded.
A first example is given by Φ(x) = aP 1 (x) with a > 0 and K Φ (x, y) = a/2. It follows from the definition that a sum of Carathéodory multipliers is a Carathéodory multiplier, and so is Φ −⊙ and aΦ with a > 0. Therefore, any sum of the form Φ(x) = a 0 P 1 (x) + N n=1 b n (a n + P 1 (x)) −⊙ is a Carathéodory multiplier for every choice of a 0 ≥ 0, a 1 , . . . , a N > 0 and b 1 , . . . , b N ≥ 0.
Here too, the space L(Φ) will be finite dimensional if and only if the space H can be chosen finite dimensional.

A table
We conclude this paper with a table comparing the slice hyperholomorphic case, the case of Fueter variables and the present setting.