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Abstract In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a Bargmann–Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane waves. We prove that this projection can be written in integral form in terms the so-called Bargmann–Radon kernel. Moreover, we have a characterization formula for the Bargmann–Radon transform of a function in the real Bargmann module in terms of its complex extension and then its restriction to the nullcone in C m . We also show that the formula holds for the Szegő–Radon transform that we introduced in Colombo et al. (2016) and we define the dual transform and we provide an inversion formula. Finally, in Theorem 5.6, we prove an integral formula for the monogenic part of an entire function. [ABSTRACT FROM AUTHOR]

The S -functional calculus is based on the theory of slice hyperholomorphic functions and it defines functions of n -tuples of not necessarily commuting operators or of quaternionic operators. This calculus relays on the notion of S -spectrum and of S -resolvent operator. Since most of the properties that hold for the Riesz–Dunford functional calculus extend to the S -functional calculus, it can be considered its non commutative version. In this paper we show that the Taylor formula of the Riesz–Dunford functional calculus can be generalized to the S -functional calculus. The proof is not a trivial extension of the classical case because there are several obstructions due to the non commutativity of the setting in which we work that have to be overcome. To prove the Taylor formula we need to introduce a new series expansion of the S -resolvent operators associated to the sum of two n -tuples of operators. This result is a crucial step in the proof of our main results, but it is also of independent interest because it gives a new series expansion for the S -resolvent operators. This paper is addressed to researchers working in operator theory and in hypercomplex analysis. [ABSTRACT FROM AUTHOR]

In the recent years the theory of slice hyperholomorphic functions has become an important tool to study two functional calculi for n -tuples of operators and also for its applications to Schur analysis. In particular, using the Cauchy formula for slice hyperholomorphic functions, it is possible to give the Fueter–Sce mapping theorem an integral representation. With this integral representation it has been defined a monogenic functional calculus for n -tuples of bounded commuting operators, the so called F -functional calculus. In this paper we show that it is possible to define this calculus also for n -tuples containing unbounded operators and we obtain an integral representation formula analogous to the one of the Riesz–Dunford functional calculus for unbounded operators acting on a complex Banach space. As we will see, it is not an easy task to provide the correct definition of the F -functional calculus in the unbounded case. This paper is addressed to a double audience, precisely to people with interests in hypercomplex analysis and also to people working in operator theory. [ABSTRACT FROM AUTHOR]

Abstract: In the paper [F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009) 601–627] the authors treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute. This functional calculus is the quaternionic version of the classical Riesz–Dunford functional calculus. When considering quaternionic operators it is natural to also consider the case of left linear operators. Furthermore, one can use left or right slice regular functions to construct a functional calculus for right (or left) linear operators. In this paper we discuss these possibilities, showing that, in all the cases, we can associate to an operator two so-called -resolvent operators but their interpretation depends on whether we are considering a left or a right linear operator. Also the -resolvent equations for right or left closed operators do not have the same interpretation. Moreover, we study the bounded perturbations of both the -resolvent operators. [Copyright &y& Elsevier]

Abstract: In a recent work, Colombo (in press) , we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from the above-mentioned work can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real. [Copyright &y& Elsevier]

Abstract: A proper generalization of complex function theory to higher dimension is Clifford analysis and an analogue of holomorphic functions of several complex variables were recently described as the space of solutions of several Dirac equations. The four-dimensional case has special features and is closely connected to functions of quaternionic variables. In this paper we present an approach to the Dolbeault sequence for several quaternionic variables based on symmetries and representation theory. In particular we prove that the resolution of the Cauchy–Fueter system obtained algebraically, via Gröbner bases techniques, is equivalent to the one obtained by R.J. Baston (J. Geom. Phys. 1992). [Copyright &y& Elsevier]

Recently the S -spectrum approach to fractional diffusion problems has been applied to vector operators with homogeneous Dirichlet boundary conditions. This method allows to determine the Fractional Fourier law under given boundary conditions. In this paper we consider first order linear boundary conditions and we study the case of vector operators with commuting components. Precisely, let Ω be an open bounded set in R 3 where its boundary ∂ Ω is considered suitably regular, we denote a point in Ω ¯ by x = (x 1 , x 2 , x 3) and a basis for the quaternions H will be indicated by e ℓ , for ℓ = 1 , 2 , 3. We prove that under suitable conditions on the coefficients a 1 , a 2 , a 3 : Ω ¯ ⊂ R 3 → R of the vector operator T = e 1 a 1 (x 1) ∂ x 1 + e 2 a 2 (x 2) ∂ x 2 + e 3 a 3 (x 3) ∂ x 3 , x ∈ Ω ¯ and on the coefficient a : ∂ Ω → R of the boundary operator B : B ≔ ∑ ℓ = 1 3 a ℓ 2 (x ℓ) n ℓ ∂ x ℓ + a (x) I , x ∈ ∂ Ω , where n = (n 1 , n 2 , n 3) is the outward unit normal vector to ∂ Ω , we can define the fractional powers T α , for α ∈ (0 , 1) , of T. In general the coefficients a 1 , a 2 , a 3 and a can depend on time. We omit the time dependence for the sake of simplicity but the proofs of our results can be easily extended to this more general setting considering the time as a parameter. [ABSTRACT FROM AUTHOR]