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This article has a twofold purpose: on one hand, we deepen the study of slice regular functions by studying their behaviour with respect to the so-called C-property and anti-C-property. We show that, for any fixed basis of the algebra of quaternions ℍ any slice regular function decomposes into the sum of four slice regular components, each of them satisfying the C-property. Then, we will use these results to show a reproducing property of the Bergman kernels of the second kind. [ABSTRACT FROM AUTHOR]

In this article, we prove two main results related to the 𝒮-spectrum which hold for an important subclass of the slice monogenic functions. The first result consists of an alternative proof of the 𝒮-spectral mapping theorem which follows from the invertibility of functions of bounded operators. The second result assures that for a subclass of slice monogenic polynomialsP, if (T0, T1, … , Tn) is ann-tuple of closed operators thenP(T0, T1, … , Tn) is closed. [ABSTRACT FROM PUBLISHER]

In this article we deepen the study of the Bergman theory in the slice regular setting by providing a description of the Bergman projection and by considering a subclass of slice regular functions which generate isometric isomorphisms of Bergman spaces. As a tool to show our result, we further study the multiplication of slice regular functions and their composition. [ABSTRACT FROM AUTHOR]

In this article we show that it is possible to construct a Koszul-type complex for maps given by suitable pairwise commuting matrices of polynomials. This result has applications to surjectivity theorems for constant coefficients differential operators of finite and infinite order. In particular, we construct a large class of constant coefficients differential operators which are surjective on the space of regular (or monogenic) functions on open convex sets. [ABSTRACT FROM AUTHOR]