Delay-dependent stability conditions for differential-difference equations with small commutators in a banach space

Let <FOR VERIFICATION>A and <FOR VERIFICATION>B be bounded operators in a Banach space. For the equation <FOR VERIFICATION>du(t)/dt = Au(t) + Bu(t - h)(h > 0) we derive sharp explicit delay-dependent exponential stability conditions via the commutator <FOR VERIFICATION>AB - BA. The stability conditions in terms of the commutator enable us to construct a feedback with delay, which stabilizes a system. Gives us a possibility construct Stabilization of Systems with Retarded Feedback in a Banach Space. Our results are new even in the finite-dimensional case.