Quaternionic inner and outer functions

We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.


Introduction
An essential role in the function theory of the unit disk of the complex plane is played by the property that any function in the Hardy space factorizes (uniquely) as the product of an inner and an outer function. The connections of inner and outer functions are ubiquitous in mathematical analysis, ranging from operator theory to dynamical systems and PDEs (see [4] and [7], for instance). One of the main reasons for this is the fact that the closed invariant subspaces (for the shift operator in the Hardy space) can be described via an identi cation with inner functions, whereas outer functions contain information about approximation properties, and, in fact, coincide with cyclic functions. In the recent paper [11], the rst two authors proved an inner-outer factorization theorem for the Hardy space of slice regular functions on the quaternionic unit ball H (B). Thus, it seems natural to investigate the properties of inner and outer functions in the quaternionic setting more deeply, and the present paper is a rst step in this direction. We will see that some properties of holomorphic inner and outer functions are straightforwardly generalized to the quaternionic setting, whereas some other properties are more peculiar of slice regular functions.
The paper is organized as follows. In Section 2 we x the notation and we recall some basic de nitions and properties of slice regular functions and the quaternionic Hardy space H (B). We devote Section 3 to properties of inner functions, whereas in Section 4 we focus on outer ones. Then, in Section 5 we investigate cyclicity and properties of optimal approximant polynomials in the quaternionic setting. We conclude formulating some open problems in Section 6.

Notation and basic de nitions
In this section we recall a few de nitions and properties of slice regular functions and the quaternionic Hardy space H (B). We do not include any proofs; we refer the reader to the monograph [8] for the basics on slice regular functions and to [5] for results concerning H (B).
Let H denote the skew eld of quaternions, let B = {q ∈ H : |q| < } be the quaternionic unit ball and let ∂B be its boundary, containing elements of the form q = e tI = cos t + sin tI, I ∈ S, t ∈ R, where S = {q ∈ H : q = − } is the two dimensional sphere of imaginary units in H. Then, where the slice L I := R + RI can be identi ed with the complex plane C for any I ∈ S.
A function f : B → H is a slice regular function if the restriction f I of f to B I := B ∩ L I is holomorphic, i.e., it has continuous partial derivatives and it is such that for all x+yI ∈ B I . The relationship between slice regular functions and holomorphic functions of one complex variable is made clear in the following lemma.
It is a well-known fact that every slice regular function on the unit ball B admits a power series expansion of the form The conjugate of f , which we denote by f c , is the function de ned by Morevover, we denote by f the function f (q) := f (q).
The function f is not slice regular but it is a slice function. The class of slice functions was introduced in [9] in a more general setting than the present one. In this paper we say that a function f : B → H is a slice function if for any I, J ∈ S, Slice regular functions are examples of slice functions. Moreover, Formula (2) furnishes a tool to uniquely extend a holomorphic function de ned on the complex disk B J to a slice regular function de ned on the whole unit ball B (see [8]). Given f J : B J → H, holomorphic function with respect to the complex variable x + yJ, the function ext(f J ) : B → H de ned for any x + yI ∈ B as is slice regular on B. Formula (2) can also be used to prove the following result concerning the zeros of a slice regular function.

Proposition 2.2. Let f be a slice regular function on
The structure of the zero set of a slice regular function is completely understood. A -dimensional sphere x + yS ⊆ B of zeros of f is called a spherical zero of f . Any point x + yI of such a sphere is called a generator of the spherical zero x + yS. Any zero of f that is not a generator of a spherical zero is called an isolated zero of f . Moreover, on each sphere x + yS contained in B, the zeros of f are in one-to-one correspondence with the zeros of f c , see [8,Proposition 3.9]. In general, the pointwise product of two slice regular functions is not a slice regular function and a suitable product must be considered, namely, the so-called slice or *-product. If f (q) = n∈N q n an and g(q) = n∈N q n bn are two slice regular functions on B, then This product is related to the pointwise product by the formula where T f c (q) := f (q) − qf (q). By means of the *-product, we can associate to a function f its symmetrization f s , that is, We remark here that the symmetrization f s is a slice preserving function, namely f s (B I ) ⊆ L I for all I ∈ S. In particular, this is equivalent to the fact that the coe cients in the power series expansion of f s are all real numbers. Finally, we denote by f −* the inverse of f with respect to the *-product, which is given by .
This inner product on H (B) admits also an integral representation. Let us endow ∂B with the measure dΣ e tI = dσ(I)dt, where dt is the Lebesgue measure on [ , π) and dσ is the standard area element of S, normalized in such a way that σ(S) = Σ(∂B) = . Then, The measure Σ, and not the induced Lebesgue measure on ∂B, is naturally associated to the Hardy space H (B), as reasoned in [2,5]. An important feature of this inner product is that it can be actually computed by restricting it to any slice L I . In more detail, given any I ∈ S we set f , g I = π π g(e θI )f (e θI )dθ, (9) where dθ is the Lebesgue measure on [ , π). For any I ∈ S, it holds that We denote by H ∞ (B) the space of bounded slice regular functions on the unit ball. Notice that De nitions of inner and outer functions in the quaternionic setting are similar to the classical ones for holomorphic functions and rst appeared in [5].
We point out that the de nition of inner and outer functions were given in terms of the induced Lebesgue measure m on ∂B. It is not di cult to show that Σ and m are mutually absolutely continuous.
The following theorem was proved in [11] by the rst two authors.
Then, f has a factorization f = φ * g where φ is inner and g is outer. Moreover, this factorization is unique up to a unitary constant in the following sense: if f = φ * g = φ * g , then φ = φ * λ and g = λ * g for some λ ∈ H such that |λ| = .
The proof of this theorem makes use of the concept of cyclicity.

De nition 2.7. A function g is cyclic in H
We stress out that [g] is the smallest closed invariant subspace of H (B) containing g. Thus, g is cyclic if the smallest closed subspace containing g is the space H (B) itself. In [11] it is rstly proved that each function f ∈ H (B) admits a factorization f = φ * g with φ inner and g cyclic. Afterwards, being cyclic is proved equivalent to being outer in the sense of De nition 2.5. We remark that we work with right quaternionic Hilbert spaces, therefore the left-hand side of (11) denotes the closure in H (B) of elements of the form where pm is a quaternionic polynomial with scalar coe cients {an} m n= ⊆ H.

Inner functions
Let us now focus on inner functions. To start, we would like to better understand any connection between f being an inner function in H (B) and the properties of f I (the restriction of f to the slice L I = R + RI), or of the splitting components of f (see Lemma 2.1). Some of the results we include in this section are implicit in [11].
Here we state them explicitly and we make some remarks.
We rst prove a characterization of inner functions in H (B). In the following statement the *-product is the extension of (3) to the more general setting of slice L functions on ∂B, that is, the space L s (∂B) of functions of the form q → k∈Z q k a k with q ∈ ∂B and {a k } k∈Z ∈ , see [11]. If f (q) = n∈Z q n an and g(q) = n∈Z q n bn belong to L s (∂B), then Clearly, the boundary value function of every f ∈ H (B) is a slice L function.
Then, the following are equivalent: Then, if f is inner, we immediately get the rst implication of the statement.
Suppose now that f * f c = f c * f = Σ-almost everywhere on ∂B, so that |f | = Σ-almost everywhere on ∂B. Let g ∈ H (B), g ≢ , and denote by Z g s = {q ∈ B : g s (q) = }. Proposition 2.3 in [11] guarantees that Σ(Z g s ) = , whereas Proposition 5.32 in [8] guarantees that the map Tg : ∂B \ Z g s → ∂B \ Z g s is a bijection. Therefore, we have where we used that |f | = Σ-almost everywhere on ∂B if and only if the same holds true for |f c | (see [6,Proposition 5]). This implies that f is a multiplier for H (B). Thanks to [1,Corollary 3.5] we conclude that f ∈ H ∞ (B) and, in particular, that f is an inner function. Hence, conditions (i) and (ii) are equivalent. Clearly (ii) implies (iii). Suppose now that condition (iii) holds. Then, for almost every t ∈ [ , π), we have both = f * f c (e tI ) and = f * f c (e (t+π)I ) = f * f c (e −tI ).

Remark 3.2.
We point out that the previous proof actually showed that condition (iii) implies ( f * f c ) I = (f c * f ) I = almost everywhere on ∂B I for any I ∈ S.
By means of the previous result we obtain another characterization of inner functions in H (B) which is often used as the de nition of inner functions in more abstract Hilbert spaces (see, for instance, [12]). Recalling the notations from (6) and (9) and denoting by δ k (j) the Kronecker delta, we have the following theorem. (i) f is inner; (ii) for all k ∈ N, we have q k * f , f = δ k ( ); (iii) there exists I ∈ S such that, for all k ∈ N, we have Proof. Let f (q) = n∈N q n an ∈ H (B). Then, for q ∈ ∂B, we get an a n−k .
We point out that q k * f has to be interpreted as a *-product in the setting of slice L functions as de ned in (12). Moreover, for Σ-almost any q ∈ ∂B, it holds that f c (q) = n≥ q n an and f (q) = n≤ q n a−n .
Hence, f c * f (q) = k∈Z q k n∈Z an a n−k = k∈Z q k n≥max{ ,k} an a n−k .
Therefore, for any k ∈ N, we obtain that q k * f , f is the k-th coe cient in the power series expansion of f c * f . From this fact and Proposition 3.1 is now easy to deduce that (ii) follows from (i). The reverse implication can be proved using the natural extension of the H inner product to the bigger space of slice L functions, namely, In fact, suppose (ii) holds. Then, the k-th coe cient in the power series expansion of f c * f vanishes for k > and equals for k = . To show that all the coe cients with k = −n < equal zero consider thus the rst part of the theorem is proved. From (10), we obtain that (ii) is equivalent to (iii). In fact, the inner product of H (B) can be computed on a single slice and it does not depend on the choice of the slice.
In general, the restriction f I of f to the slice L I is a function of one complex variable, but still quaternionvalued. If f I were a complex-valued function, then f I would truly be an inner function of H (D). Therefore, Theorem 3.3 guarantees that the restriction to any slice L I of a slice regular inner function of H (B) is "almost" an inner function of H (D). At this point it is natural to question about a simple converse. We wonder whether any inner function F ∈ H (D) admits a slice regular extension f := ext(F) to the unit ball B such that f is inner for H (B). Here we are identifying D with B i . We obtain an answer in the form of the following corollary.

Proof. Since the inner product of H (B) can be computed on any slice, it is clear that f = ext(F) ∈ H (B) whenever F ∈ H (D). The conclusion now follows from Theorem 3.3 since condition (iii) is satis ed for I = i.
We explicitly point out that not all the inner functions of H (B) trivially arise as the regular extension of some inner function of H (D). In fact, if a slice regular function f is the extension of a complex inner function F, then it necessarily preserves the slice L i , i.e. f (B i ) ⊆ L i . Recalling Proposition 2.2, we have that if a function preserves a slice, then all its isolated, non-spherical, zeros are contained in that slice. It is enough to take a slice regular Blaschke product which has at least two zeros that are not on the same slice (and neither on the same sphere). See [5] for an explicit construction of such a function.
So far we investigated how f being an inner function in H (B) a ects the restriction of f to any slice. We also want to understand how being an inner function a ects the splitting components of the function (see Lemma 2.1). The following is our best result in this direction.

Then, f is inner if and only if for Lebesgue-almost every x ∈ ∂B I , the following conditions hold:
Proof. If F, G are the splitting components of f I as in (13) Hence, for almost every z ∈ ∂B I , Combining (16) with Remark 3.2 we get for almost every z ∈ ∂B I . This holds if and only if (14) is satis ed.
We conclude this section showing that the characterization of inner functions in [12] involving their H and H ∞ norms works in the quaternionic setting as well.
Theorem 3.6. Let f ∈ H (B). The following are equivalent: (iii) f H = and for all k ∈ N and λ ∈ H we have Proof. Clearly, if f is inner, then its H ∞ norm equals and that is, (i) implies (ii). We deduce that (ii) implies (iii) from the fact that the multiplier space of H (B) can be isometrically identi ed with H ∞ (B) and the fact that f H ∞ = implies g*f H ≤ g H for any g ∈ H (B). To see that (iii) implies (i) we exploit Theorem 3.3. If f H = and f is not inner, then there must be k ∈ N\{ } such that q k * f , f ≠ .
Choose such k and notice that, for all λ ∈ H, we have Let us now compute the left-hand side in the condition in (iii) with a λ to be chosen later. It holds that Since the shift is an isometry over its image on H (B), we see that the rst two addends of the right-hand side sum to q k + λ H . Choosing λ = q k * f , f we contradict the hypothesis in (iii).

Outer functions
In [11] it was shown that the De nition 2.
for any z ∈ D. In this section we prove some preliminary results that go in the direction of nding an analogous characterization for quaternionic outer functions. We provide some necessary conditions as well as su cient ones for a function to be outer. Some of these conditions are in terms of the symmetrization f s of the function f . We will see that, in some cases, f being outer is equivalent to f s being outer. Since f s is slice preserving, a logarithm characterization for f s to be outer is available.
Most of our proofs rely on cyclicity, thus similar proofs could work for function spaces in which outer and cyclic functions do not necessarily coincide.
Our rst nding does not come as a surprise, but it will be used in what follows.

Lemma 4.1. Let f ∈ H (B). Then, f is outer if and only if f c is outer.
Proof. Let f = f i * fo be the inner-outer factorization of f where f i denotes the inner part of f and fo denotes the outer part. Assume that f is outer, that is, f = fo, so that f c = (fo) c . A priori, we do not know if the conjugate of an outer factor is still an outer factor. Thus, assume for the moment that f c is not outer, that is, Then, on the one hand f = fo, on the other hand In particular, thanks to [11,Proposition 2.3], for Σ-almost every q ∈ ∂B, Since f is outer, we get |f (q)| = |fo(q)| ≥ |((f c )o) c (q)| for any q ∈ B. However, recall that a function is inner if and only if its conjugate function is inner ([11, Proposition 2.1]), hence |((f c ) i ) c (q)| ≤ inside the ball and we also get . We remark here that (f c )o is never zero in B since it is the outer factor of f c , hence ((f c )o) c never vanishes as well. As a consequence, T f c is a bijection of B to itself; see [8,Proposition 5.32]. Therefore, we obtain that |((f c ) i ) c | = in B, hence, for the maximum modulus principle in the quaternionic setting, we conclude that ((f c ) i ) c ≡ α where α is a quaternion of modulus . Thus, that is, f c is outer and the proof is concluded.
Let us now introduce the concept of optimal approximants which will be needed in what follows.

De nition 4.2.
Let n ∈ N, f ∈ H (B) and Pn := {p(q) = n k= q k a k : a k ∈ H}. A polynomial pn ∈ Pn is an optimal approximant of degree n of f −* if pn is such that f * pn − H = min{ f * p − H : p ∈ Pn}.
The existence and uniqueness of such a minimizer is guaranteed by the projection theorem for quaternionic Hilbert spaces, see [10]. In particular, the minimizer f *pn is given by the orthogonal projection of the constant function on the closed subspace f * Pn H (B).
The constant function plays a special role because it is cyclic. Then, to show the cyclicity of a function f is equivalent to show that its optimal approximants satisfy In fact, if the constant function satis es equation (18)  Proof. Let {pn} n∈N be a sequence of polynomials such that fo * pn − H → as n tends to ∞. Since f i is bounded, it is a multiplier, so that This shows that f i ∈  Before proving the lemma, notice that even in the complex case the assumption f , g ∈ H ∞ (D) cannot be discarded since there are functions in H (D) whose square is not an element of H (D).

Proof.
To show that f * g cyclic implies g cyclic, thanks to Lemma 4.1, it is enough to show that f * g cyclic implies f cyclic and then apply the result to g c * f c . Suppose now that f * g is cyclic and consider the innerouter factorization of f = f i * fo with f i inner and fo outer. Then, [f ] = f i * H (B). Hence, f * g = f i * (fo * g) is an element of [f ], since fo ∈ H (B) and g ∈ H ∞ (B) guarantee that fo * g ∈ H (B) (see [5]). Then, H (B). Hence, f is cyclic.
Suppose now that f and g are cyclic, and let {pn} n∈N and {rm} m∈N be sequences of polynomials with the property that both the norms f * pn − H and g * rm − H tend to zero as n goes to in nity. Then, for each n, m ∈ N, from the triangle inequality we get The last term on the right-hand side will be arbitrarily small whenever n is large enough. The other one may be estimated using the fact that f and pn are both multipliers, that is, Now, whatever the value of f H ∞ pn H ∞ is, it does not depend on m. Hence, if we x ε > and n ∈ N, taking m large enough we obtain g * rm − H ≤ ε( f H ∞ pn H ∞ ) − .
Therefore, f * g is cyclic.
The previous result will prove particularly useful when applied to the symmetrization f s = f * f c = f c * f of a function f ∈ H (B). For each p ∈ [ , ∞], the function f s is in H p (B) provided that f is in H p (B). In particular, if f ∈ H ∞ (B), then f s ∈ H ∞ (B), see [5]. From this equality, the equivalence (i) − (ii) is easily deduced. Also, from (20) we see that either (i) or (ii) is equivalent to − Kn( , ) → as n → ∞. However, Kn( , ) tending to is equivalent to (iii) and this concludes the proof.

Some open problems
We consider that the topic needs more development. We propose a few questions that seem natural from where we stand, beyond the obvious elimination of the boundedness hypothesis in Theorem 4.6. (