## 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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