90 research outputs found
On power series expansions of the S-resolvent operator and the Taylor formula
The -functional calculus is based on the theory of slice hyperholomorphic
functions and it defines functions of -tuples of not necessarily commuting
operators or of quaternionic operators. This calculus relays on the notion of
-spectrum and of -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
-resolvent operators associated to the sum of two -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
-resolvent operators. This paper is devoted to researchers working in
operators theory and hypercomplex analysis
An Application of the -Functional Calculus to Fractional Diffusion Processes
In this paper we show how the spectral theory based on the notion of
-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 functional calculus and we use it to define the
fractional powers of vector operators. The Fourier laws for the propagation of
the heat in non homogeneous materials is a vector operator of the form where , are orthogonal unit vectors, , ,
are suitable real valued function that depend on the space variables
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 so we can define the non local version
, for , of the Fourier law defined by . Our new
mathematical tools open the way to a large class of fractional evolution
problems that can be defined and studied using our theory based on the
-spectrum for vector operators. This paper is devoted to researchers in
different research fields such as: fractional diffusion and fractional
evolution problems, partial differential equations, non commutative operator
theory, and quaternionic analysis
Fractional powers of vector operators and fractional Fourier's law in a Hilbert space
In this paper we give a concrete application of the spectral theory based on
the notion of -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 ,
, , are given real valued functions that depend on the space
variables , and possibly also on time.
Using the -version of the -functional calculus we have recently
defined fractional powers of quaternionic operators, which contain, as a
particular case, the vector operator . Hence, we can define the non-local
version , for , of the Fourier law defined by .
We will see in this paper how we have to interpret , when we
introduce our new approach called: "The -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 -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
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