Riesz Dunford Functional Calculus Research Articles

Double exponential quadrature for fractional diffusion

We introduce a novel discretization technique for both elliptic and parabolic fractional diffusion problems based on double exponential quadrature formulas and the Riesz–Dunford functional calculus. Compared to related schemes, the new method provides faster convergence with fewer parameters that need to be adjusted to the problem. The scheme takes advantage of any additional smoothness in the problem without requiring a-priori knowledge to tune parameters appropriately. We prove rigorous convergence results for both, the case of finite regularity data as well as for data in certain Gevrey-type classes. We confirm our findings with numerical tests.

Functions of the infinitesimal generator of a strongly continuous quaternionic group

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 [Formula: see text] is the quaternionic infinitesimal generator of a strongly continuous group of operators [Formula: see text] and we show how we can define bounded operators [Formula: see text], where [Formula: see text] 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 [Formula: see text]-spectrum of [Formula: see text] but not necessarily at infinity. Moreover, we establish the relation between [Formula: see text] and the quaternionic functional calculus and we study the problem of finding the inverse of [Formula: see text].

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.

The H∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators

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.

The Spectral Theorem for Unitary Operators Based on the S-Spectrum

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

Perturbation of the generator of a quaternionic evolution operator

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 [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.

A New Resolvent Equation for the $$S$$ S -Functional Calculus

The $$S$$ -functional calculus is a functional calculus for $$(n+1)$$ -tuples of not 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 noncommuting operators. The 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 allow us to prove some properties of the $$S$$ -functional calculus. In this paper we prove a new resolvent equation which is the analog of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right $$S$$ -resolvent operators simultaneously.

The -spectrum and the -functional calculus

In some recent papers (called $\mathcal{S}$-functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The $\mathcal{S}$-functional calculus is based on the notion of $\mathcal{S}$-spectrum, which naturally arises from the definition of the $\mathcal{S}$-resolvent operator for n-tuples of operators. The $\mathcal{S}$-resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the $\mathcal{S}$-spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the $\mathcal{S}$-spectrum. In the case of commuting operators, our functional calculus is based on the $\mathcal{F}$-spectrum and will be called $\mathcal{SC}$-functional calculus. We point out that for a correct definition of the $\mathcal{S}$-resolvent operator and of the $\mathcal{SC}$-resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the $\mathcal{F}$-spectrum is that it is related to the $\mathcal{F}$-functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.

On the formulations of the quaternionic functional calculus

In the paper [F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009) 601–627] the authors treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute. This functional calculus is the quaternionic version of the classical Riesz–Dunford functional calculus. When considering quaternionic operators it is natural to also consider the case of left linear operators. Furthermore, one can use left or right slice regular functions to construct a functional calculus for right (or left) linear operators. In this paper we discuss these possibilities, showing that, in all the cases, we can associate to an operator two so-called S -resolvent operators but their interpretation depends on whether we are considering a left or a right linear operator. Also the S -resolvent equations for right or left closed operators do not have the same interpretation. Moreover, we study the bounded perturbations of both the S -resolvent operators.

On stationary Schrödinger-Poisson equations modelling an electron gas with reduced dimension

We regard the Schrodinger–Poisson system arising from the modelling of an electron gas with reduced dimension in a bounded up to three-dimensional domain and establish the method of steepest descent. The electrostatic potentials of the iteration scheme will converge uniformly on the spatial domain. To get this result we investigate the Schrodinger operator, the Fermi level and the quantum mechanical electron density operator for square integrable electrostatic potentials. On bounded sets of potentials the Fermi level is continuous and bounded, and the electron density operator is monotone and Lipschitz continuous. As a tool we develop a Riesz–Dunford functional calculus for semibounded self-adjoint operators using paths of integration which enclose a real half-axis. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Solving algebraic and differential Riccati operator equations

By application of the Riesz-Dunford functional calculus a method for solving generalized algebraic Riccati operator equations in Hilbert space is given. Results are applied to the problem of finding explicit solutions for Cauchy problems and two-point boundary-value problems for generalized Riccati operator differential equations.