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

2021/04/14 by Riccardo Ghiloni, Ghiloni, Riccardo, Vincenzo Recupero +1
Mathematics · #30G35 #47A10 #47A60 #47D03 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2104.07110

openalex publication_date 2021/04/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

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 operator (A² - 2Re(q) A + |q|²)^-1. As an immediate consequence, we also obtain a new proof of the well-known integral representation for the S-resolvent operator of A (also called spherical resolvent operator of A).

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