Isomorphisms and derivations in proper JCQ\(^*\)-triples. (English) Zbl 1253.39031
Rassias, Themistocles M. (ed.) et al., Functional equations in mathematical analysis. Dedicated to the memory of Stanisław Marcin Ulam on the occasion of the 100th anniversary of his birth. Berlin: Springer (ISBN 978-1-4614-0054-7/hbk; 978-1-4614-0055-4/ebook). Springer Optimization and Its Applications 52, 229-245 (2011).
Summary: We investigate homomorphisms in proper JCQ\(^{\ast}\)-triples and derivations on proper JCQ\(^{\ast}\)-triples associated with the 3-variable Jensen functional equation
\[
2f\left(\frac{x+y+z}{2}\right)=f(x)+f(y)+f(z),
\]
which was introduced and investigated by C. Park, Y. Cho and M. Han [J. Inequal. Appl. 2007, Article ID 41820, 13 p. (2007; Zbl 1133.39024)]. We moreover prove the Hyers-Ulam-Rassias stability of homomorphisms in proper JCQ\(^{\ast}\)-triples and of derivations on proper JCQ\(^{\ast}\)-triples. This is applied to investigate isomorphisms between proper JCQ\(^{\ast}\)-triples.
For the entire collection see [Zbl 1225.39001].
For the entire collection see [Zbl 1225.39001].
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 17C65 | Jordan structures on Banach spaces and algebras |
| 47B48 | Linear operators on Banach algebras |
| 47L60 | Algebras of unbounded operators; partial algebras of operators |
