Banach Journal of Mathematical Analysis

Multiplicative operator functions and abstract Cauchy problems

Felix Früchtl

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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.

Article information

Banach J. Math. Anal., Volume 12, Number 2 (2018), 347-373.

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

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Zentralblatt MATH identifier

Primary: 47D99: None of the above, but in this section
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 39B42: Matrix and operator equations [See also 47Jxx] 43A62: Hypergroups 43A65: Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45] 45N05: Abstract integral equations, integral equations in abstract spaces

abstract Cauchy problem special functions functional equation hypergroup representation theory homogeneous Banach space weakly stationary process


Früchtl, Felix. Multiplicative operator functions and abstract Cauchy problems. Banach J. Math. Anal. 12 (2018), no. 2, 347--373. doi:10.1215/17358787-2017-0042.

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