On the generators of Clifford semigroups: Polynomial resolvents and their integral transforms

This paper deals with generators A of strongly continuous right linear semigroups in Banach two-sided spaces whose set of scalars is an arbitrary Clifford algebra Cℓ(0,n). We study the invertibility of operators of the form P(A), where P(x)∈R[x] is any real polynomial, and we give an integral repres...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 521; no. 1; p. 126905
Main Authors: Ghiloni, Riccardo, Recupero, Vincenzo
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2023
Subjects:
Clifford algebras
Functional calculus
Functions of hypercomplex variables
Laplace transform
Quaternions
Slice regular semigroups
Clifford algebras
Functions of hypercomplex variables
Quaternions
Laplace transform
Slice regular semigroups
Functional calculus
ISSN:0022-247X, 1096-0813
Online Access:Get full text
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Summary:This paper deals with generators A of strongly continuous right linear semigroups in Banach two-sided spaces whose set of scalars is an arbitrary Clifford algebra Cℓ(0,n). We study the invertibility of operators of the form P(A), where P(x)∈R[x] is any real polynomial, and we give an integral representation for P(A)−1 by means of a Laplace-type transform of the semigroup T(t) generated by A. In particular, we deduce a new integral representation for the spherical quadratic resolvent of A (also called pseudoresolvent of A). As an immediate consequence, we also obtain a new proof of the well-known integral representation for the spherical resolvent of A.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126905