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Superstability for generalized module left derivations and generalized module derivations on a Banach module. I. (English) Zbl 1185.46033

Summary: We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let \(\mathcal A\) be a Banach algebra and X a Banach \(\mathcal A\)-module, \(f : {\text X}\to {\text X}\) and \(g :{\mathcal A}\to {\mathcal A}\). The mappings \(\Delta^1_{f,g}\), \(\Delta^2_{f,g}\), \(\Delta^3_{f,g}\), and \(\Delta^4_{f,g}\) are defined and it is proved that if \(\|\Delta^1_{f,g}(x,y,z,w)\|\) (resp., \(\|\Delta^3_{f,g}(x,y,z,w,\alpha,\beta)\|\)) is dominated by \((\varphi(x,y,z,w)\), then \(f\) is a generalized (resp., linear) module-\(\mathcal A\) left derivation and \(g\) is a (resp., linear) module-X left derivation. It is also shown that if \(\|\Delta^2_{f,g}(x,y,z,w)\|\) (resp., \(\|\Delta^4_{f,g}(x,y, z, w,\alpha,\beta)\|\)) is dominated by \(\varphi(x,y,z,w)\), then \(f\) is a generalized (resp., linear) module-\(\mathcal A\) derivation and \(g\) is a (resp., linear) module-X derivation.

MSC:

46H20 Structure, classification of topological algebras
39B82 Stability, separation, extension, and related topics for functional equations
47B47 Commutators, derivations, elementary operators, etc.
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