Stability of evolution equations with small commutators in a Banach space

Abstract

Let $A$ be a generator of a $C_0$-semigroup in a Banach space $\mathcal{X}$, and $B(t)$ $(t\ge 0)$ be a variable bounded piece-wise strongly continuous operator in $\mathcal{X}$. We consider the equation $dy(t)/dt=(A+B(t))y(t)$ $(t\ge 0)$. 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.