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Spectral theory on the S -spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S -spectrum has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n -tuples of noncommuting operators. With this paper we show that the spectral theory on the S -spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. In fact, the S -spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S -functional calculus goes beyond quaternionic and Clifford analysis, indeed the S -functional calculus has a certain universality property. This fact makes the spectral theory on the S -spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case. [ABSTRACT FROM AUTHOR]

In this paper we extend the H ∞ functional calculus to quaternionic operators and to n -tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S -functional calculus. The S -functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S -functional calculus is based on the notion of S -spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ functional calculus based on the notion of S -spectrum for both quaternionic operators and for n -tuples of noncommuting operators. We remark that the H ∞ functional calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator. [ABSTRACT FROM AUTHOR]

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform. [ABSTRACT FROM AUTHOR]

Abstract: 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 [4]. 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. [Copyright &y& Elsevier]

Abstract: In this paper we prove the Poincaré Lemma for circulant matrices with entries and the Mittag–Leffler theorem for Hermitian monogenic matrix functions. We then prove algebraic and topological duality theorems for Hermitian monogenic matrix functions. Finally, we use one of these duality theorems to characterize the -module , where denotes the sheaf of Hermitian monogenic functions. These results extend and clarify those obtained in R. Abreu-Blaya et al. (2012) . [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]

Abstract: The new notion of slice monogenic functions introduced in the paper [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009) 385–403] led us to define a new functional calculus for an n-tuple of not necessarily commuting operators, see [F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008) 2255–2274]. In this paper we prove a Cauchy formula with slice monogenic kernel for the slice monogenic functions. This new Cauchy formula is the fundamental tool to prove that our functional calculus apply to a more general setting. Moreover, we deduce some fundamental properties of the functional calculus, for example: some algebraic properties, the Spectral Mapping Theorem and the Spectral Radius Theorem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point can be recovered by the values of f at the points and for any choice of imaginary units . This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions. [Copyright &y& Elsevier]

Abstract: In this paper we use the notion of slice monogenic functions [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., in press] to define a new functional calculus for an n-tuple T of not necessarily commuting operators. This calculus is different from the one discussed in [B. Jefferies, Spectral Properties of Noncommuting Operators, Lecture Notes in Math., vol. 1843, Springer-Verlag, Berlin, 2004] and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T. Our functional calculus is consistent with the Riesz–Dunford calculus in the case of a single operator. [Copyright &y& Elsevier]

Abstract: We consider a quasilinear integrodifferential system in non-normal form. Such a system is a generalization of a phase-field model with memory and includes, as a particular case, the system describing the combustion of a material with memory. In this paper, we study both the direct and the inverse problems. Our fundamental tools are: the theory of analytic semigroups, optimal regularity results and fixed point arguments. [Copyright &y& Elsevier]

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ ′ -potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function. [ABSTRACT FROM AUTHOR]

We first review realizations of Herglotz functions in the unit ball of C N and provide new insights. Then, we define the corresponding class and prove the extend the results in the case of several quaternionic variables. [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]