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In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S-functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra Rn. The methodology can be generalized, for example, to handle the case of noncommuting matrix variables. [ABSTRACT FROM AUTHOR]

In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let eℓ, eℓ=1,2,3 be orthogonal unit vectors in R3 and let Ω⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation is obtained replacing the Fourier law given by T=ax_∂xe1+bx_∂ye2+cx_∂ze3into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, c:Ω→R such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure. [ABSTRACT FROM AUTHOR]

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two S-resolvent operators and the S-spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the S-spectrum of a quaternionic linear operator T is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S-spectrum and of slice hyperholomorphicity of the S-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator. [ABSTRACT FROM AUTHOR]

Let eℓ, for ℓ = 1,2,3, be orthogonal unit vectors in R3 and let Ω⊂R3 be a bounded open set with smooth boundary ∂Ω. Denoting by x_ a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T=a(x_)∂xe1+b(x_)∂ye2+c(x_)∂ze3,into the conservation of energy law, here a, b, c:Ω→R are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c:Ω→R the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory. [ABSTRACT FROM AUTHOR]

In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice hyperholomorphic functions and on the S-resolvent operators of the quaternionic functional calculus. The integral representation formulas of the fractional powers and the quaternionic version of Kato’s formula are based on the notion of S-spectrum of a quaternionic operator. The proofs of several properties of the fractional powers of quaternionic operators rely on the S-resolvent equation. This equation, which is very important and of independent interest, has already been introduced in the case of bounded quaternionic operators, but for the case of unbounded operators some additional considerations have to be taken into account. Moreover, we introduce a new series expansion for the pseudo-resolvent, which is of independent interest and allows to investigate the behavior of the S-resolvents close to the S-spectrum. The paper is addressed to researchers working in operator theory and in complex analysis. [ABSTRACT FROM AUTHOR]

The quaternionic analogue of the Riesz-Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that is the quaternionic infinitesimal generator of a strongly continuous group of operators and we show how we can define bounded operators , where belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace-Stieltjes transform. This class includes functions that are slice regular on the -spectrum of but not necessarily at infinity. Moreover, we establish the relation between and the quaternionic functional calculus and we study the problem of finding the inverse of . [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]

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]

The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz-Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille-Phillips-Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille-Phillips-Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator we study under which condition a closed operator P is such that T + P generates the evolution operator . This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory. [ABSTRACT FROM AUTHOR]

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]