Towards a general \(\mathcal{F}\)-resolvent equation and Riesz projectors
type: Item
instance of: scholarly article
title: Towards a general \(\mathcal{F}\)-resolvent equation and Riesz projectors
author: Fabrizio Colombo
author: Antonino De Martino
author: Irene Sabadini
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-09-30T00:00:00Z
review text: In Section 1, Introduction, the authors start explaining the steps in which the Fueter-Sce-Qian extension (or construction) consists (step A consists in the extension of holomorphic functions to the class of slice hyperholomorphic functions and step B consists in the extension of slice hyperholomorphic functions to monogenic functions). Step A has been helpful in the foundation of the quaternionic spectral theory on the \(S\)-spectrum. Step B generates the monogenic functional calculus based on the monogenic spectrum. The main goal of this paper is to study some properties of the \(\mathcal F\)-functional calculus, which is a bridge between the spectral theory on the \(S\)-spectrum and the monogenic spectral theory. Also in the Introduction there were made some considerations on the resolvent equations in various settings. It is recalled that in the slice hyperholomorphic setting there exist two different resolvent operators (according to left of right slice hyperholomorphicity) and they both are used to obtain the \(S\)-resolvent equation. Some of their properties are recalled. There are considered paravector operators \(T=T_0+e_1 T_1 + \cdots + e_n T_n\) with commuting components and the \(\mathcal F\)-spectrum of \(T\) is defined. It turns out that the \(\mathcal F\)-spectrum is the commutative version of the \(S\)-spectrum. At the end of the Introduction there are described the main results of the paper. ``Preliminary material'' \, is the he title of Section 2. The authors start this section recalling the story of how to extend a holomorphic function to another one that belongs to a wider class. Fueter started with holomorphic functions defined on open sets of the complex upper half plane and that satisfy few more conditions and gave a method to generate quaternionic Cauchy-Fueter regular functions. For his part, Sce started with the same kind of functions and gave a method to generate monogenic functions defined on a subset of \(\mathbb R^{n+1}\) and with value…
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32A36
zbMATH DE Number: 7595154
zbMATH Keywords: slice regular weighted composition operator
zbMATH Keywords: quaternionic Fock space
zbMATH Keywords: essential norm
zbMATH Keywords: S-spectrum
zbMATH Keywords: C-symmetry
zbMATH Keywords: path connected component
reviewed by: M. E. Luna-Elizarrarás
MaRDI profile type: Publication
full work available at URL: j.jmaa.2022.126652
OpenAlex ID: W4293571916
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DOI: 10.1016/J.JMAA.2022.126652
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description: scientific article; zbMATH DE number 7595154