Banach Journal of Mathematical Analysis
- Banach J. Math. Anal.
- Volume 12, Number 2 (2018), 347-373.
Multiplicative operator functions and abstract Cauchy problems
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 -groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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. https://projecteuclid.org/euclid.bjma/1515402092