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This paper is a continuation of the research of our previous work [5] and considers quaternionic generalized Carathéodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Carathéodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S-functional calculus. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space R N = { ( x 1 , x 2 , … ) : x d ∈ R } endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner–Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner–Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties. [ABSTRACT FROM AUTHOR]

In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded antiself adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics. [ABSTRACT FROM AUTHOR]

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]

Abstract: The S-spectrum has been introduced for the definition of the S-functional calculus that includes both the quaternionic functional calculus and a calculus for n-tuples of nonnecessarily commuting operators. The notion of right spectrum for right linear quaternionic operators has been widely used in the literature, especially in the context of quaternionic quantum mechanics. Moreover, several results in linear algebra, like the spectral theorem for quaternionic matrices, involve the right spectrum. In this Note we prove that the two notions of S-spectrum and of right spectrum coincide. [Copyright &y& Elsevier]

Abstract: 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) , we study the quaternionic semigroups and groups generated by a quaternionic (bounded or unbounded) linear operator . It is crucial to note that we consider operators with components 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 is the S-resolvent operator , 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. [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]

We define and study the counterparts of Gleason's problem, of the Drury–Arveson's space and of Schur multipliers when the unit ball of C N is replaced by the unit ball of H N , where H denotes the algebra of real quaternions. Schur multipliers are characterized in terms of coisometric operator matrices in quaternionic spaces. Finally, we define the counterpart of Blaschke factors in this setting and study a one point interpolation problem. [ABSTRACT FROM AUTHOR]