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In this paper we prove a new version of Krein–Langer factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [9]. We treat both the case of functions with κ negative squares defined on subsets of the quaternionic unit ball or on subsets of the half-space of quaternions with positive real part. A crucial tool in the proof of our results is the Schauder–Tychonoff theorem and an invariant subspace theorem for contractions in a Pontryagin space.

An attempt to study the compatibility conditions, and the general free resolution, for the system associated with the Dirac operator in \(k\) vector variables appeared already in Sabadini et al. (Math Z 239: 293–320, 2002), from the point of view of Clifford analysis, and in Sabadini et al. (Exp Math 12: 351–364, 2003) using the tool of megaforms. Other studies have been carried out in other papers, like Krump (Adv Appl Clifford Alg 19: 365–374, 2009), Krump and Soucek (17: 537–548, 2007), Salac (The generalized Dolbeault complexes in Clifford analysis, Praha 2012), using methods of representation theory. In this paper, we restrict our attention to the case of three variables and we describe the free resolution associate to the module from various different angles. The comparison has several noteworthy consequences. In particular, it gives the explicit description of all the maps contained in the algebraic resolution and shows that they are invariant with respect to the action of \(SL(3)\times SO(m)\). We also discuss how the methods used in this paper can be generalized to the case of \(k>3\) Dirac operators.

In this paper we introduce the notion of quaternionic Weyl algebraA 1(ℍ) and we study its main properties. We give some examples ofA 1(ℍ)-modules useful in the resolution of systems of differential equations whose linear differential operators are inA 1(ℍ). We also introduce a notion of Fourier transform that is an isomorphism ofA 1(ℍ).

We introduce generalized Schur functions and generalized positive functions in setting of slice hyperholomorphic functions and study their realizations in terms of associated reproducing kernel Pontryagin spaces, Revised version,to appear in the Milan Journal of Mathematics

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball B into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space H 2 (B). In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

Attaching to a compact ball $\overline{\mathbb{B}_{r}}$ in the quaternion field $\mathbb{H}$ and to analytic functions in Weierstrass sense (slice regular functions on $\overline{\mathbb{B}_{r}}$) some convolution operators, the exact orders of approximation in $\overline{\mathbb{B}_{r}}$ for these operators are obtained. The results in this paper extend to quaternionic variables those in the case of approximation of analytic functions of a complex variable in disks by convolution operators of a complex variable, extensively studied in the Chapter 3 of the book [5]. More in general, the results extend also to the setting of analytic functions of paravector variable with coefficients in a Clifford algebra.

C. Bisi, C. Stoppato: Regular vs. classical Mobius transformations of the quaternionic unit ball.- F. Brackx, H. De Bie, Hennie De Schepper: Distributional Boundary Values of Harmonic Potentials in Euclidean Half-space as Fundamental Solutions of Convolution Operators in Clifford Analysis.- F. Colombo, J.O. Gonzalez-Cervantes, M.E. Luna-Elizarraras, I. Sabadini, M. Shapiro: On two approaches to the Bergman theory for slice regular functions.- C. Della Rocchetta, G. Gentili, G. Sarfatti: A Bloch- Landau Theorem for slice regular functions.- M. Ku, U. Kahler, P. Cerejeiras: Dirichlet-type problems for the iterated Dirac operator on the unit ball in Clifford analysis.- A. Perotti: Fueter regularity and slice regularity: meeting points for two function theories.- D.C. Struppa: A note on analytic functionals on the complex light cone.- M.B. Vajiac: The S-spectrum for some classes of matrices.- F. Vlacci: Regular Composition for SliceRegular Functions of Quaternionic Variable.

In this paper we provide an alternative method to construct the Fueter primitive of an axial monogenic function of degree $k$, which is complementary to the one used in [F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem in integral form using spherical monogenics, Israel Journal of Mathematics, 2012]. As a byproduct, we obtain an explicit description of the kernel of the Fueter mapping. We also apply our method to obtain the Fueter primitives of the Cauchy kernels with singularities on the unit sphere., Comment: 13 pages

In this paper we show that the classical Bergman theory admits two possible settings for the class of slice regular functions. Let Ω be a suitable open subset of the space of quaternions ℍ that intersects the real line and let \(\mathbb{S}^{2}\) be the unit sphere of purely imaginary quaternions. Slice regular functions are those functions f:Ω→ℍ whose restriction to the complex planes ℂ(i), for every \(\mathbf{i}\in \mathbb{S}^{2}\), are holomorphic maps. One of their crucial properties is that from the knowledge of the values of f on Ω∩ℂ(i) for some \(\mathbf{i}\in \mathbb{S}^{2}\), one can reconstruct f on the whole Ω by the so called Representation Formula. We will define the so-called slice regular Bergman theory of the first kind. By the Riesz representation theorem we provide a Bergman kernel which is defined on Ω and is a reproducing kernel. In the slice regular Bergman theory of the second kind we use the Representation Formula to define another Bergman kernel; this time the kernel is still defined on Ω but the integral representation of f requires the calculation of the integral only on Ω∩ℂ(i) and the integral does not depend on \(\mathbf{i}\in \mathbb{S}^{2}\).

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions. These are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper, we focus on the case of Hilbert spaces and introduce, in particular, a version of the Hardy space. Then we define Blaschke factors and Blaschke products and consider an interpolation problem. In the second part of the paper, we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Caratheodory functions.

Superoscillatory functions were introduced in Aharonov and Vaidman (1990) [5] , and recently studied in detail in Aharonov et al. (2011) [2] , Berry (1994) [7] and Berry and Popescu (2006) [9] . In this paper we study the time evolution of a superoscillating function, by taking it as initial value for the Cauchy problem for the Schrodinger equation. By using convolution operators on spaces of entire functions with suitable growth conditions, we prove the surprising fact that the superoscillatory phenomenon persists for all values of t .

In this paper we prove an integral representation formula for the inverse Fueter mapping theorem for monogenic functions defined on axially symmetric open sets U ⊆ ℝ n+1, i.e. on open sets U invariant under the action of SO(n), where n is an odd number. Every monogenic function on such an open set U can be written as a series of axially monogenic functions of degree k, i.e. functions of type $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x) = \left[ {A\left( {x_{0,\rho } } \right) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } {\rm B}\left( {x_{0,\rho } } \right)} \right]\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ , where A(x 0, ρ) and B(x 0, ρ) satisfy a suitable Vekua-type system and $$\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a homogeneous monogenic polynomial of degree k. The Fueter mapping theorem says that given a holomorphic function f of a paravector variable defined on U, then the function $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ given by $$\Delta ^{k + \tfrac{{n - 1}} {2}} \left( {f(x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )} \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ is a monogenic function. The aim of this paper is to invert the Fueter mapping theorem by determining a holomorphic function f of a paravector variable in terms of $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$ . This result allows one to invert the Fueter mapping theorem for any monogenic function defined on an axially symmetric open set.

We prove that regular functions of one quaternionic variahle which satisfy a large class of differential equations cannot have compact singulanties. This result is equivalent to the fact that a large family of 8 × 4 matrices has torsion-free cokernel. The result (obvrons in inc complex case) eaisly extends to Clifford algebray.

The purpose of this paper is to establish a topological isomorphism between the space FK of H-hyperfunctions supported by a compact K and the space (G(K))′ of left H-linear functionals on G(K) space of germs of regular functions. The space FK is also topologically isomorphic to the space R ∞ 1(HP1\K) of left regular functions vanishing at infinity. We also introduce two different families of infinite order differential operators: one of them is useful to characterize the convergence in G(K) while we need the other one to write an H-hyperfunction with support at the origin as infinite sum of erivatives of Dirac's delta.

The aim of this note is to present a brief state of the art of Schur analysis in the slice hyperholomorphic setting.

Slice regular functions have been introduced in 20 as solutions of a special partial differential operator with variable coefficients. As such they do not naturally form a sheaf. In this paper we use a modified definition of slice regularity, see 21, to introduce the sheaf of slice regular functions with values in in the algebra of quaternions and, more in general, in a Clifford algebra and we study its cohomological properties. We show that the first cohomology group with coefficients in the sheaf of slice regular functions vanishes for any open set in the space of quaternions (resp. the space of paravectors in ). However, we prove that not all the open sets are domains of slice regularity but only those special sets which are axially symmetric, i.e., invariant with respect to rotations that fix the real axis.

In this paper we expand on some ideas originally put forward by Ehrenpreis in his monograph (Fourier Analysis in Several Complex Variables, Wiley Interscience, New York, 1970), and we show how to extend approximate solutions to the Cauchy–Fueter system in n variables.

The Fueter mapping theorem can be stated at several levels of generality and the literature on this very active field is quite rich. In the recent years, the authors have started the study of the inversion of the Fueter mapping in various settings. In this note we discuss the state of the art.

In this paper we give an overview of the S-functional calculus which is based on the Cauchy formula for slice monogenic functions.S uch a functional calculus works for n-tuples of noncommuting operators and it is based on the notion of S-spectrum.Th ere is a commutative version of the S-functional calculus, due to the fact that the Cauchy formula for slice monogenic functions admits two representations of the Cauchy kernel.W e will call SC-functional calculus the commutative version of the S-functional calculus. This version has the advantage that it is based on the notion of ℱ-spectrum, which turns out to be more simple to compute with respect to the S-spectrum. For commuting operators the two spectra are equal, but when the operators do not commute among themselves the ℱ-spectrum is not well defined.W e finally briefly introduce the main ideas on which the ℱ-functional calculus is inspired.T his functional calculus is based on the integral version of the Fueter-Sce mapping theorem and on the ℱ-spectrum.

We study various aspects of Schur analysis in the slice hyperholomorphic setting. We present two sets of results: first, we give new results on the functional calculus for slice hyperholomorphic functions. In particular, we introduce and study some properties of the Riesz projectors. Then we prove a Beurling-Lax type theorem, the so-called structure theorem. A crucial fact which allows to prove our results, is the fact that the right spectrum of a quaternionic linear operator and the S-spectrum coincide. Finally, we study the Krein-Langer factorization for slice hyperholomorphic generalized Schur functions. Both the Beurling-Lax type theorem and the Krein-Langer factorization are far reaching results which have not been proved in the quaternionic setting using notions of hyperholomorphy other than slice hyperholomorphy, Comment: Version to appear in the Journal of Geometric Analysis

In [14], Pincherle studies the surjectivity of a difference operator with constant coefficients in the space of holomorphic functions. In this paper, we discuss how this work can be rephrased in the context of modern functional analysis and we conclude by extending his results and we show that difference equations act surjectively on the space of quaternionic regular functions.

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allow to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels and positive definite functions in this setting and we show how they can be obtained using the extension operator and the slice hyperholomorphic product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges–Rovnyak space.

In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price (An Introduction to Multicomplex Spaces and Functions, Dekker, New York, 1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields. The intended audience includes researchers, PhD students, postgraduate students who are interested in the field and in possible connection between hypercomplex analysis and other disciplines, including mathematical analysis, mathematical physics, algebra.

In this paper we show that the systems introduced in [Eelbode, Adv. Appl. Clifford Algebr. 17: 635–649, 2007] and [Peña-Peña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97–113, 2007] are equivalent, both giving the notion of quaternionic Hermitian monogenic functions. This makes it possible to prove that the free resolution associated to the system is linear in any dimension, and that the first cohomology module is nontrivial, thus generalizing the results in [Peña-Peña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97–113, 2007]. Furthermore, exploiting the decomposition of the spinor space into 𝔰𝔭(m)-irreducibles, we find a certain number of “algebraic” compatibility conditions for the system, suggesting that the usual spinor reduction is not applicable.

The goal of this chapter is to construct a functional calculus for n-tuples of not necessarily commuting operators on a Banach space V over the real numbers. We start by introducing the basic notions which will allow us, given an n-tuple of linear operators acting on V , to construct a new operator acting on a suitable module over a real Clifford algebra. The idea to use a Clifford algebra approach is not new and goes back to Coifman and Murray, see [80] and also to the works of McIntosh, Pryde and Jefferies (see [62] and the references therein).

We discuss the notion of slice monogenic function as well as its variation and we compare the various definitions among them.

In this paper, we consider bicomplex holomorphic functions of several variables in \({{\mathbb B}{\mathbb C}^n}\) .We use the sheaf of these functions to define and study hyperfunctions as their relative 3n-cohomology classes. We show that such hyperfunctions are supported by the Euclidean space \({{\mathbb R}^n}\) within the bicomplex space \({{\mathbb B}{\mathbb C}^n}\), and we construct an abstract Dolbeault complex that provides a fine resolution for the sheaves of bicomplex holomorphic functions. As a corollary, we show how that the bicomplex hyperfunctions can be represented as classes of differential forms of degree 3n − 1.

In the recent years, the notion of slice regular functions has allowed the introduction of a quaternionic functional calculus. In this paper, motivated also by the applications in quaternionic quantum mechanics, see Adler (1995) [1], we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator T=T0+iT1+jT2+kT3. It is crucial to note that we consider operators with components Tℓ (ℓ=0,1,2,3) that do not necessarily commute. Among other results, we prove the quaternionic version of the classical Hille–Phillips–Yosida theorem. This result is based on the fact that the Laplace transform of the quaternionic semigroup etT is the S-resolvent operator (T2−2Re[s]T+|s|2I)−1(s¯I−T), the quaternionic analogue of the classical resolvent operator. The noncommutative setting entails that the results we obtain are somewhat different from their analogues in the complex setting. In particular, we have four possible formulations according to the use of left or right slice regular functions for left or right linear operators.

In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function $f$ of the form $f=\alpha+\underline{\omega}\beta$ (where $\alpha$, $\beta$ satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function $\bar{f}=A+\underline{\omega}B$ (where $A,B$ satisfy the Vekua's system) given by $\bar{f}(x)=\Delta^{\frac{n-1}{2}}f(x)$ where $\Delta$ is the Laplace operator in dimension $n+1$. In this paper we solve the inverse problem: given an axially monogenic function $\bar{f}$ determine a slice monogenic function $f$ (called Fueter's primitive of $\bar{f}$ such that $\bar{f}=\Delta^{\frac{n-1}{2}}f(x)$. We prove an integral representation theorem for $f$ in terms of $\bar{f}$ which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution $f$ of the equation $\Delta^{\frac{n-1}{2}}f(x)=\bar{f} (x)$ in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.

In this Appendix we collect some basic material on the Riesz–Dunford functional calculus useful for the readers who are not familiar with this subject. This background, with all the details, can be found in [35] and [91].

Recently, we have introduced the F-functional calculus and the SC-functional calculus. Our theory can be developed for operators of the form T = T 0 + e 1 T 1 +...+ e n T n where (T 0, T 1,...,T n) is an (n + 1)-tuple of linear commuting operators. The SC-functional calculus, which is defined for bounded but also for unbounded operators, associates to a suitable slice monogenic function f with values in the Clifford algebra ℝn the operator f(T). The F-functional calculus has been defined, for bounded operators T, by an integral transform. Such an integral transform comes from the Fueter’s mapping theorem and it associates to a suitable slice monogenic function f the operator \(\breve{f} (T)\), where \(\breve{f}(x)=\Delta^{\frac{n-1}{2}}f(x)\) and Δ is the Laplace operator. Both functional calculi are based on the notion of F-spectrum that plays the role that the classical spectrum plays for the Riesz-Dunford functional calculus. The aim of this paper is to study the bounded perturbations of the SC-resolvent operator and of the F-resolvent operator. Moreover we will show some examples of equations that lead to the F-spectrum.

In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem.

This paper is devoted to the algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary group. In the one variable case, we show that it is possible to give explicit formulae for all the maps of the resolution associated to the system. Moreover, we compute the minimal generators for the first syzygies also in the case of the Hermitian system in several vector variables. Finally, we study the removability of compact singularities. We also show a major difference with the orthogonal case: in the odd dimensional case it is possible to perform a reduction of the system which does not affect the behavior of the free resolution, while this is not always true for the case of even dimension.

In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup e tA when A is a linear quaternionic operator.

In this paper we give an overview of some different versions of functional calculus in a noncommutative setting. In particular, we will focus on a recent functional calculus based on the notion of slice-hyperholomorphy. This notion, in suitable versions, will allow us to study the case of linear quaternionic operators, as well as the case of n-tuples of linear (real or complex) operators.

In this paper we show a structure formula for slice monogenic functions (see Lemma 2.2 and [1] for further details): we will show that this formula is a key tool to prove several results, among which we mention the Cauchy integral formula with slice monogenic kernel. This Cauchy formula allows us to extend the validity of the functional calculus for n-tuples of noncommuting operators introduced in [6]. In this wider setting, most of the properties which hold for the Riesz-Dunford functional calculus of a single operator, such as the Spectral Mapping Theorem and the Spectral Radius Theorem, still hold.

This paper presents some algorithmic techniques for computing explicitly the Noetherian operators associated with a class of ideals and modules over a polynomial ring. The procedures we include in this work can be easily encoded in computer algebra packages such as CoCoA [CoCoATeam 05].