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 lim t→∞ 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).
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This work was supported by the Research Council of Slovenia
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Šemrl, P. The functional equation of multiplicative derivation is superstable on standard operator algebras. Integr equ oper theory 18, 118–122 (1994). https://doi.org/10.1007/BF01225216
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DOI: https://doi.org/10.1007/BF01225216