Superstability for Generalized Module Left Derivations and Generalized Module Derivations on a Banach Module (I)

Abstract

We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let 𝒜 be a Banach algebra and X a Banach 𝒜-module, f:X→X and g:𝒜→𝒜. The mappings Δf,g1, Δf,g2, Δf,g3, and Δf,g4 are defined and it is proved that if ∥Δf,g1(x,y,z,w)∥ (resp., ∥Δf,g3(x,y,z,w,α,β)∥) is dominated by φ(x,y,z,w), then f is a generalized (resp., linear) module-𝒜 left derivation and g is a (resp., linear) module-X left derivation. It is also shown that if ∥Δf,g2(x,y,z,w)∥ (resp., ∥Δf,g4(x,y,z,w,α,β)∥) is dominated by φ(x,y,z,w), then f is a generalized (resp., linear) module-𝒜 derivation and g is a (resp., linear) module-X derivation.