The functional equation of multiplicative derivation is superstable on standard operator algebras

Abstract

LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let ϕ: ℝ+ → ℝ+ be a function with the property limt→∞ φ(t)t−1=0. Assume that a mappingD:A →B(X) satisfies ‖D(AB)−AD(B)−D(A)B‖<φ(‖A‖ ‖B‖) for all operatorsA, B ∈D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT−TA for someT∈B(X).