Graziano Gentili

Regular functions of a quaternionic variable

Introduction.- 1.Definitions and Basic Results.- 2.Regular Power Series.- 3.Zeros.- 4.Infinite Products.- 5.Singularities.- 6.Integral Representations.- 7.Maximum Modulus Theorem and Applications.- 8.Spherical Series and Differential.- 9.Fractional Transformations and the Unit Ball.- 10.Generalizations and Applications.- Bibliography.- Index.

The Open Mapping Theorem for Regular Quaternionic Functions

The basic results of a new theory of regular functions of a quaternionic variable have been recently stated, following an idea of Cullen. In this paper we prove the minimum modulus principle and the open mapping theorem for regular functions. The proofs involve some peculiar geometric properties of such functions which are of independent interest.

Twistor transforms of quaternionic functions and orthogonal complex structures

The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3.

Non Commutative Functional Calculus: Bounded Operators

In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the new notion of slice-regularity, see Gentili and Struppa (Acad Sci Paris 342:741–744, 2006) and the key tools are a new resolvent operator and a new eigenvalue problem.

A Bloch-Landau Theorem for Slice Regular Functions

The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc \(\mathbb{D}\) under a holomorphic function f (such that f(0)=0 and f′(0)=1) always contains an open disc with radius larger than a universal constant. In this paper we prove a Bloch-Landau type Theorem for slice regular functions over the skew field ℍ of quaternions. If f is a regular function on the open unit ball \(\mathbb{B}\subset\mathbb{H}\), then for every \(w \in \mathbb{B}\) we define the regular translation \(\tilde{f}_{w}\) of f. The peculiarities of the non commutative setting lead to the following statement: there exists a universal open set contained in the image of \(\mathbb{B}\) through some regular translation \(\tilde{f}_{w}\) of any slice regular function \(f: \mathbb{B}\to\mathbb{H}\) (such that f(0)=0 and ∂ C f(0)=1). For technical reasons, we introduce a new norm on the space of regular functions on open balls centred at the origin, equivalent to the uniform norm, and we investigate its properties.

The Bohr Theorem for slice regular functions

In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Caratheodory Theorem to the new setting.

Holomorphic Dynamical Systems

This chapter is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a large bibliography.

Recent Developments for Regular Functions of a Hypercomplex Variable

In this paper we survey a series of recent developments in the theory of functions of a hypercomplex variable. The central idea underlying these developments consists in requiring a function to be holomorphic on suitable slices of the space on which the function itself is defined. Specifically, we apply this approach to functions defined on the space ℍ of quaternions, on the space O of octonions, and finally on the Clifford algebra of type (0,3), denoted Cl (0,3). The properties of these functions resemble those of holomorphic functions, and yet the different nature of the three algebras on which we work introduces new and exciting phenomena.

Sheaves of Quaternionic Hyperfunctions and Microfunctions

The sheaf F - of quaternionic hyperfunctions is introduced as the sheaf of boundary values of quaternionic regular functions. A Kothe duality type theorem is established to prove the isomorphism between compactly supported quaternionic hyperfunctions and compactly supported regular functionals. Ordinary differential operators are studied on the sheaf F with the use of the C ? K product. Finally a sheaf of quaternionic microfunctions is introduced as the microlocalization of F , and its main properties are studied.

The Zero Sets of Slice Regular Functions and the Open Mapping Theorem

A new class of regular quaternionic functions, defined by power series in a natural fashion, has been introduced in [11, 12]. Several results of the theory recall the classical complex analysis, whereas other results reflect the peculiarity of the quaternionic structure. The recent work [1, 2] identified a larger class of domains, on which the study of regular functions is most natural and not limited to the study of quaternionic power series. In the present paper we extend some basic results concerning the algebraic and topological properties of the zero set to regular functions defined on these domains. We then use these results to prove the Maximum and Minimum Modulus Principles and a version of the Open Mapping Theorem in this new setting.