arXiv subjects

Jonathan Gantner

Publications and source records attributed to Jonathan Gantner.

14 recordsLinked to original sources

arXiv2021

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory 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. More precisely, we show that 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 we show that the $S$-functional calculus has a certain {\em 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.

math.FA↗

arXiv2020

An introduction to hyperholomorphic spectral theories and fractional powers of vector operators

The 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 $(A_1,...,A_n)$. 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. Here we also discuss how to define the fractional Fourier's law for nonhomogeneous materials, such definition is based on the spectral theory on the $S$-spectrum.

math.SP↗

arXiv2018

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 $S$-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: $$ T=e_1\,a(x)\partial_{x_1} + e_2\,b(x)\partial_{x_2} + e_3\,c(x)\partial_{x_3} $$ where $e_\ell$, $\ell=1,2,3$ are orthogonal unit vectors in $\mathbb{R}^3$, $a$, $b$, $c$ are given real valued functions that depend on the space variables $x=(x_1,x_2,x_3)$, and possibly also on time. Using the $H^\infty$-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 $T^\alpha$, for $\alpha\in (0,1)$, of the Fourier law defined by $T$. We will see in this paper how we have to interpret $T^\alpha$, 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.

math.FA↗

arXiv2018

Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators

Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defined a priori but must be chosen randomly. Spectral properties of an operator should hence be independent of this left multiplication. We show that results derived from functional calculi for intrinsic slice functions can be formulated without the assumption of a left multiplication. We develop the S-functional calculus in this setting and a new approach to spectral integration. This approach has a clear interpretation in terms of the right linear structure on the space and allows to formulate the spectral theorem without using any randomly chosen structure. Our techniques only apply to intrinsic slice functions, but only these functions are compatible with the basic intuition of a functional calculus that $f(T)$ should be defined by letting $f$ act on the spectral values of $T$. Using these tools, we develop a theory of quaternionic spectral operators. In particular, we show the existence of a canonical decomposition of such operator and discuss its behavior under the S-functional calculus. Finally, we show a relation with complex operator theory: if we embed the complex numbers into the quaternions, then complex and quaternionic operator theory are consistent. The symmetry of intrinsic slice functions guarantees that this compatibility is true for any imbedding of the complex numbers.

math.SP↗

arXiv2018

An Application of the $S$-Functional Calculus to Fractional Diffusion Processes

In 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 $H^\infty$ 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 \[ T=e_1\,a(x)\partial_{x_1} + e_2\,b(x)\partial_{x_2} + e_3\,c(x)\partial_{x_3}, \] where $e_\ell$, $e_\ell=1,2,3$ are orthogonal unit vectors, $a$, $b$, $c$ are suitable real valued function that depend on the space variables $x=(x_1,x_2,x_3)$ 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 $T^\alpha$, for $\alpha\in (0,1)$, 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 theory based on the $S$-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.

math.SP↗

arXiv2017

On the Equivalence of Complex and Quaternionic Quantum Mechanics

Due to the existence of incompatible observables, the propositional calculus of a quantum system does not form a Boolean algebra but an orthomodular lattice. Such lattice can be realised as a lattice of subspaces on a real, complex or quaternionic Hilbert space, which motivated the formulation of real and quaternionic quantum mechanics in addition to the usual complex formulation. It was argued that any real quantum system admits a complex structure that turns it into a complex quantum system and hence real quantum mechanics was soon discarded. Several authors however developed a quaternionic version of quantum mechanics and this version did not seem to be equivalent its standard formulation on a complex Hilbert space. Motivated by some recently developed techniques from quaternionic operator theory, we conjecture in this article that this not correct and that any quaternionic quantum system is actually simply the quaternionification of a complex quantum system. We then show that this conjecture holds true for quaternionic relativistic elementary systems by applying some recent arguments that were used to show the equivalence of real and complex elementary relativistic systems. Finally, we conclude by discussing how the misconception that complex and quaternionic quantum mechanics are inequivalent arose from assuming the existence of a left multiplication on the Hilbert space, which is physically not justified.

math.FA↗

arXiv2016

BMO- and VMO-spaces of slice hyperholomorphic functions

In this paper we continue the study of important Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball by investigating the BMO- and VMO-spaces of slice hyperholomorphic functions. We discuss in particular conformal invariance and a refined characterization of these spaces in terms of Carleson measures. Finally we show the relations with the Bloch and Dirichlet space and the duality relation with the Hardy space $\mathcal{H}^1(\mathbb{D})$. The importance of these spaces in the classical theory is well known. It is therefore worthwhile to study their slice hyperholomorphic counterparts, in particular because slice hyperholomorphic functions were found to have several applications in operator theory and Schur analysis.

math.CV↗

arXiv2016

A direct approach to the S-functional calculus for closed operators

The $S$-functional calculus for slice hyperholomorphic functions generalizes the Riesz-Dunford-functional calculus for holomorphic functions to quaternionic linear operators and to $n$-tuples of noncommuting operators. For an unbounded closed operator, it is defined, as in the classical case, using an appropriate transformation and the $S$-functional calculus for bounded operators. This is however only possible if the $S$-resolvent set of the operator contains a real point. In this paper, we define the $S$-functional calculus directly via a Cauchy integral, which allows us to consider also operators whose resolvent sets do not contain real points. We show that the product rule and the spectral mapping theorem also hold true with this definition and that the $S$-functional calculus is compatible with polynomials, although polynomials are not included in the set of admissible functions if the operator is unbounded. We also prove that the $S$-functional calculus is able to create spectral projections. In order to do this, we remove another assumption usually made: we do not assume that admissible functions are defined on slice domains. Instead, we also consider functions that are defined on not necessarily connected sets. This leads to an unexpected phenomenon: the $S$-functional calculi for left and right slice hyperholomorphic functions become inconsistent and give different operators for functions that are both left and right slice hyperholomorphic. We show that any such function is the sum of a locally constant and an intrinsic function. For intrinsic functions both functional calculi agree, but for locally constant functions they must give different operators unless any spectral projection commutes with arbitrary scalars.

math.SP↗

arXiv2015

Schatten class and Berezin transform of quaternionic linear operators

In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is 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 for 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.

math.FA↗

arXiv2015

Fractional powers of quaternionic operators and Kato's formula using slice hyperholomorphicity

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.

math.FA↗

arXiv2015

Functions of the infinitesimal generator of a strongly continuous quaternionic group

The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator of a strongly continuous group of operators $(\mathcal{Z}_T(t))_{t \in \mathbb{R}}$ and we show how we can define bounded operators $f(T)$, where $f$ belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the $S$-spectrum of $T$ but not necessarily at infinity. Moreover, we establish the relation of $f(T)$ with the quaternionic functional calculus and we study the problem of finding the inverse of $f(T)$.

math.SP↗

arXiv2015

On power series expansions of the S-resolvent operator and the Taylor formula

The $S$-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 devoted to researchers working in operators theory and hypercomplex analysis.

math.FA↗

arXiv2014

Formulations of the F-functional calculus and some consequences

In this paper we introduce the two possible formulations of the F-functional calculus which are based on the Fueter-Sce mapping theorem in integral form and we introduce the pseudo F-resolvent equation. In the case of dimension 3 we prove the F-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 F-functional calculus.

math.FA↗

arXiv2013

A new resolvent equation for the S-functional calculus

The 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 $S_L^{-1}(s,T)$ and the right one $S_R^{-1}(s,T)$, where $s=(s_0,s_1,\ldots,s_n)\in \mathbb{R}^{n+1}$ and $T=(T_0,T_1,\ldots,T_n)$ is an $(n+1)$-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations $S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal{I}$, and $sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal{I}$, respectively, where $\mathcal{I}$ 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.

math.FA↗