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Let be a generator of a -semigroup in a Banach space , and be a variable bounded piece-wise strongly continuous operator in . We consider the equation . It is assumed that the commutator is a bounded operator. Under that condition, exponential stability conditions are derived in terms of the commutator.
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Michael Gil', Stability of evolution equations with small commutators in a Banach space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 4, pp. 589–596DOI 10.4171/RLM/822