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Dariusz Idczak - One of the best experts on this subject based on the ideXlab platform.

Irene Sabadini - One of the best experts on this subject based on the ideXlab platform.

  • Differential Forms and Clifford Analysis
    Trends in Mathematics, 2016
    Co-Authors: Irene Sabadini, Franciscus Sommen
    Abstract:

    In this paper we use a calculus of differential forms which is defined using an axiomatic approach. We then define integration of differential forms over chains in a new way and we present a short proof of Stokes’ Formula using distributional techniques. We also consider differential forms in Clifford analysis, vector differentials and their powers. This framework enables an easy proof for a Cauchy Formula on a k-surface. Finally, we discuss how to compute winding numbers in terms of the monogenic Cauchy kernel and the vector differentials with a new approach which does not involve cohomology of differential forms.

  • perturbation of the generator of a quaternionic evolution operator
    Analysis and Applications, 2015
    Co-Authors: Daniel Alpay, F. Colombo, Irene Sabadini
    Abstract:

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

  • Perturbation of the generator of a quaternionic evolution operator
    Analysis and Applications, 2015
    Co-Authors: Daniel Alpay, F. Colombo, Irene Sabadini
    Abstract:

    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.

  • Monogenic plane waves and the W-functional calculus
    Mathematical Methods in the Applied Sciences, 2015
    Co-Authors: Fabrizio Colombo, Irene Sabadini, Roman Lávička, Vladimír Souček
    Abstract:

    In this paper, we introduce some integral transforms that map slice monogenic functions to monogenic functions. We then show that one of these integral transforms, which is based on the Cauchy Formula of slice monogenic functions, is useful to define a functional calculus depending on a parameter for n-tuples of bounded operators. Copyright © 2015 John Wiley & Sons, Ltd.

  • An Invitation to the S-functional Calculus
    Spectral Theory Mathematical System Theory Evolution Equations Differential and Difference Equations, 2012
    Co-Authors: Fabrizio Colombo, Irene Sabadini
    Abstract:

    In this paper we give an overview of the S-functional calculus which is based on the Cauchy Formula for slice monogenic functions.S uch a functional calculus works for n-tuples of noncommuting operators and it is based on the notion of S-spectrum.Th ere is a commutative version of the S-functional calculus, due to the fact that the Cauchy Formula for slice monogenic functions admits two representations of the Cauchy kernel.W e will call SC-functional calculus the commutative version of the S-functional calculus. This version has the advantage that it is based on the notion of ℱ-spectrum, which turns out to be more simple to compute with respect to the S-spectrum. For commuting operators the two spectra are equal, but when the operators do not commute among themselves the ℱ-spectrum is not well defined.W e finally briefly introduce the main ideas on which the ℱ-functional calculus is inspired.T his functional calculus is based on the integral version of the Fueter-Sce mapping theorem and on the ℱ-spectrum.

Rafal Kamocki - One of the best experts on this subject based on the ideXlab platform.

Setsuo Taniguchi - One of the best experts on this subject based on the ideXlab platform.

  • Analytic Functions on Abstract Wiener Spaces
    Journal of Functional Analysis, 2001
    Co-Authors: Setsuo Taniguchi
    Abstract:

    Abstract Let ( X ,  H ,  μ ) be a real abstract Wiener space. A new definition of analytic functions on X is introduced, and it is shown that stochastic line integrals of real analytic 1-forms along Brownian motion and solutions to stochastic differential equations with real analytic coefficients are analytic Wiener functionals. An L p -theoretical sufficient condition for Wiener functionals to be analytic and an associated Cauchy Formula are also established.

  • Analytic Functions, Cauchy Formula, and Stationary Phase on a Real Abstract Wiener Space
    Journal of Functional Analysis, 1997
    Co-Authors: P. Malliavin, Setsuo Taniguchi
    Abstract:

    Abstract A new complexification of a real abstract Wiener space will be introduced, and some analogs of the algebra of analytic functions on finite dimensional Euclidean space will be considered; analytic functions on the original space, their holomorphic prolongation to the complexified space, and holomorphic functions and a Cauchy Formula on the complexified space. The Cauchy Formula is a key tool to study probabilistic quantities via “deformation of the contour integration.” Namely, it will be applied to establish (i) an explicit representation of stochastic oscillatory integrals with quadratic phase function and (ii) a stationary phase estimation of the integrals. Further, the later estimation is applicable to study Gevrey type smoothness of density functions. An integration by parts Formula on a totally real submanifold in the complexified space is also studied.

Fabrizio Colombo - One of the best experts on this subject based on the ideXlab platform.

  • An introduction to hyperholomorphic spectral theories and fractional powers of vector operators
    arXiv: Spectral Theory, 2020
    Co-Authors: Fabrizio Colombo, Jonathan Gantner, Stefano Pinton
    Abstract:

    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.

  • The Cauchy transform in the slice hyperholomorphic setting and related topics
    Journal of Geometry and Physics, 2019
    Co-Authors: Fabrizio Colombo, Samuele Mongodi
    Abstract:

    Abstract In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy Formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs.

  • Monogenic plane waves and the W-functional calculus
    Mathematical Methods in the Applied Sciences, 2015
    Co-Authors: Fabrizio Colombo, Irene Sabadini, Roman Lávička, Vladimír Souček
    Abstract:

    In this paper, we introduce some integral transforms that map slice monogenic functions to monogenic functions. We then show that one of these integral transforms, which is based on the Cauchy Formula of slice monogenic functions, is useful to define a functional calculus depending on a parameter for n-tuples of bounded operators. Copyright © 2015 John Wiley & Sons, Ltd.

  • An Invitation to the S-functional Calculus
    Spectral Theory Mathematical System Theory Evolution Equations Differential and Difference Equations, 2012
    Co-Authors: Fabrizio Colombo, Irene Sabadini
    Abstract:

    In this paper we give an overview of the S-functional calculus which is based on the Cauchy Formula for slice monogenic functions.S uch a functional calculus works for n-tuples of noncommuting operators and it is based on the notion of S-spectrum.Th ere is a commutative version of the S-functional calculus, due to the fact that the Cauchy Formula for slice monogenic functions admits two representations of the Cauchy kernel.W e will call SC-functional calculus the commutative version of the S-functional calculus. This version has the advantage that it is based on the notion of ℱ-spectrum, which turns out to be more simple to compute with respect to the S-spectrum. For commuting operators the two spectra are equal, but when the operators do not commute among themselves the ℱ-spectrum is not well defined.W e finally briefly introduce the main ideas on which the ℱ-functional calculus is inspired.T his functional calculus is based on the integral version of the Fueter-Sce mapping theorem and on the ℱ-spectrum.

  • The Cauchy Formula with s-monogenic kernel and a functional calculus for noncommuting operators
    Journal of Mathematical Analysis and Applications, 2011
    Co-Authors: Fabrizio Colombo, Irene Sabadini
    Abstract:

    Abstract The new notion of slice monogenic functions introduced in the paper [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009) 385–403] led us to define a new functional calculus for an n -tuple of not necessarily commuting operators, see [F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008) 2255–2274]. In this paper we prove a Cauchy Formula with slice monogenic kernel for the slice monogenic functions. This new Cauchy Formula is the fundamental tool to prove that our functional calculus apply to a more general setting. Moreover, we deduce some fundamental properties of the functional calculus, for example: some algebraic properties, the Spectral Mapping Theorem and the Spectral Radius Theorem.