Multiplicative operator functions and abstract Cauchy problems

Felix Früchtl

doi:10.1215/17358787-2017-0042

April 2018 Multiplicative operator functions and abstract Cauchy problems

Felix Früchtl

Banach J. Math. Anal. 12(2): 347-373 (April 2018). DOI: 10.1215/17358787-2017-0042

Abstract

We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including $C_{0}$ -groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.

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Felix Früchtl "Multiplicative operator functions and abstract Cauchy problems," Banach Journal of Mathematical Analysis 12(2), 347-373, (April 2018). https://doi.org/10.1215/17358787-2017-0042

Received: 22 February 2017; Accepted: 12 May 2017; Published: April 2018

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