Stability of evolution equations with small commutators in a Banach space (Q1730283)

Summary: Let \(A\) be a generator of a \(C_0\)-semigroup in a Banach space \(\mathcal X\), and \(B(t)\) \((t\geq 0)\) be a variable bounded piece-wise strongly continuous operator in \(\mathcal X\). We consider the equation \(dy(t)/dt = (A+B(t))y(t)\) \((t\geq 0)\). It is assumed that the commutator \(K(t)=AB(t)-B(t)A\) is a bounded operator. Under that condition, exponential stability conditions are derived in terms of the commutator.