674 research outputs found
Quaternionic closed operators, fractional powers and fractional diffusion processes
No full textThis book presents a new theory for evolution operators and a new method for defining fractional powers of vector operators. This new approach allows to define new classes of fractional diffusion and evolution problems. These innovative methods and techniques, based on the concept of S-spectrum, can inspire researchers from various areas of operator theory and PDEs to explore new research directions in their fields. This monograph is the natural continuation of the book: Spectral Theory on the S-Spectrum for Quaternionic Operators by Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey (Operator Theory: Advances and Applications, Vol. 270)
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
An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators
No full textThe aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators (A1, ... , An). A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum
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 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
Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity
No full textIn 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
Universality property of the S-functional calculus, noncommuting matrix variables and Clifford operators
No full text\ua9 2022 Elsevier Inc.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
A New Resolvent Equation for the S-Functional Calculus
No full textThe 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
Adaptive isogeometric methods with optimal convergence rates
No full textThe CAD standard for spline representation in 2D or 3D relies on tensor-product splines.
To allow for adaptive refinement, several extensions have emerged, e.g., analysis-suitable T-splines, hierarchical splines, or LR-splines.
All these concepts have been studied via numerical experiments.
However, so far there exists only little literature concerning the thorough analysis of adaptive isogeometric finite element methods (IGAFEMs).
[Buffa, Giannelli, Math. Mod. Meth. Appl. S. 26 (2016)] investigates linear convergence of an IGAFEM with truncated hierarchical B-splines, where optimal convergence of the proposed algorithm was only recently proved in [Buffa, Giannelli, Math. Mod. Meth. Appl. S. 27 (2017)] .
This talk is based on our recent work [Gantner, Haberlik, Praetorius, Math. Mod. Meth. Appl. S. 27 (2017)] .
We consider an adaptive IGAFEM for second-order linear elliptic PDEs.
We employ hierarchical B-splines.
We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions, where adaptivity is driven by some weighted-residual a posteriori error estimator.
The adaptive algorithm guarantees linear convergence of the error estimator (or equivalently: energy error plus data oscillations) with optimal algebraic rates.
Unlike our strategy, the algorithm of [Buffa, Giannelli, Math. Mod. Meth. Appl. S. 26 (2016)] was designed for truncated hierarchical B-splines only and the use of hierarchical B-splines may lead to non-sparse Galerkin matrices.
Further, the analysis of [Buffa, Giannelli, Math. Mod. Meth. Appl. S. 27 (2017)] which appeared independently of [Gantner, Haberlik, Praetorius, Math. Mod. Meth. Appl. S. 27 (2017)] is restricted to symmetric PDEs.
Similar results were also obtained for an adaptive 3D boundary element method in [Gantner, PhD thesis, TU Wien (2017)]
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