TY - JOUR

T1 - Functions of the infinitesimal generator of a strongly continuous quaternionic group

AU - Alpay, Daniel

AU - Colombo, Fabrizio

AU - Gantner, Jonathan

AU - Kimsey, David P.

N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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 T is the quaternionic infinitesimal generator of a strongly continuous group of operators (ZT(t)t∈R and we show how we can define bounded operators f(T), where f 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 S-spectrum of T but not necessarily at infinity. Moreover, we establish the relation between f(T) and the quaternionic functional calculus and we study the problem of finding the inverse of f(T).

AB - 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 T is the quaternionic infinitesimal generator of a strongly continuous group of operators (ZT(t)t∈R and we show how we can define bounded operators f(T), where f 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 S-spectrum of T but not necessarily at infinity. Moreover, we establish the relation between f(T) and the quaternionic functional calculus and we study the problem of finding the inverse of f(T).

KW - Quaternionic infinitesimal generators

KW - S-resolvent operator

KW - S-spectrum

KW - functions of the infinitesimal generator

KW - quaternionic Laplace-Stieltjes transform

KW - quaternionic functional calculus

UR - http://www.scopus.com/inward/record.url?scp=84970021432&partnerID=8YFLogxK

U2 - 10.1142/S021953051650007X

DO - 10.1142/S021953051650007X

M3 - Article

AN - SCOPUS:84970021432

VL - 15

SP - 279

EP - 311

JO - Analysis and Applications

JF - Analysis and Applications

SN - 0219-5305

IS - 2

ER -