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Published in:
2020 | OriginalPaper | Chapter
10. Infinitesimal Generators
Authors : Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal
Published in: Continuous Semigroups of Holomorphic Self-maps of the Unit Disc
Publisher: Springer International Publishing
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Abstract
After having defined the Koenigs function of a semigroup in the previous chapter, now we turn our attention to the second characteristic feature of a semigroup: the infinitesimal generator. We see how to relate semigroups to Cauchy problems, showing that every semigroup is completely determined by a holomorphic vector field in the unit disc, its infinitesimal generator. Once shown the existence of such a vector field, we will focus on different descriptions and characterizations of infinitesimal generators and discuss several of their properties and examples.