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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 C0-groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.

Citation

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

Information

Received: 22 February 2017; Accepted: 12 May 2017; Published: April 2018
First available in Project Euclid: 8 January 2018

zbMATH: 06873505
MathSciNet: MR3779718
Digital Object Identifier: 10.1215/17358787-2017-0042

Subjects:
Primary: 47D99
Secondary: 34G10, 39B42, 43A62, ‎43A65, 45N05

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.12 • No. 2 • April 2018
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