1,720,969 research outputs found
Formulations of the F-functional calculus and some consequences
No full textIn this paper we introduce the two possible formulations of the-functional calculus that are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo-resolvent equation. In the case of dimension 3 we prove the-resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the-functional calculus
An Application of the S-Functional Calculus to Fractional Diffusion Processes
No full textIn this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the H∞ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the formT= e1a(x) ∂x1+ e2b(x) ∂x2+ e3c(x) ∂x3where el, l= 1 , 2 , 3 are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables x= (x1, x2, x3) and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version Tα, for α∈ (0 , 1) , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis
Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity
Full text linkIn 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
On power series expansions of the S-resolvent operator and the Taylor formula
No full textThe 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
Fractional powers of vector operators and fractional Fourier's law in a Hilbert space
No full textIn this paper we give a concrete application of the spectral theory based on the notion of S-spectrum to fractional diffusion process. Precisely, we consider the Fourier law for the propagation of the heat in non homogeneous materials, that is the heat flow is given by the vector operator: where , are orthogonal unit vectors in , a, b, c are given real valued functions that depend on the space variables , and possibly also on time. Using the -version of the S-functional calculus we have recently defined fractional powers of quaternionic operators, which contain, as a particular case, the vector operator T. Hence, we can define the non-local version for of the Fourier law defined by T. We will see in this paper how we have to interpret when we introduce our new approach called: 'The S-spectrum approach to fractional diffusion processes'. This new method allows us to enlarge the class of fractional diffusion and fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations and non commutative operator theory. Our theory applies not only to the heat diffusion process but also to Fick's law and more in general it allows to compute the fractional powers of vector operators that arise in different fields of science and technology
Functions of the infinitesimal generator of a strongly continuous quaternionic group
Full text linkThe 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 T is the quaternionic infinitesimal generator of a strongly continuous group of operators (ZT(t)tâR and we show how we can define bounded operators f(T), where f 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 S-spectrum of T but not necessarily at infinity. Moreover, we establish the relation between f(T) and the quaternionic functional calculus and we study the problem of finding the inverse of f(T)
Schatten class and Berezin transform of quaternionic linear operators
No full textIn 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
A New Resolvent Equation for the S-Functional Calculus
Full text linkThe S-functional calculus is a functional calculus for (n + 1)-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S−1 L (s, T ) and the right one S−1 R (s, T ), where s = (s0, s1, . . . , sn) ∈ Rn+1 and T = (T0, T1, . . . , Tn) is an (n + 1)-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations S−1 L (s, T )s − TS−1 L (s, T ) = I, and sS−1 R (s, T )−S−1 R (s, T )T = I, respectively, where I denotes the identity operator. These equations allows to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation for the S-functional calculus which is the analogue of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously
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