Norm Estimates for a Semigroup Generated by the Sum of Two Operators with an Unbounded Commutator

Let A be the generator of an analytic semigroup (eAt)t≥0 on a Banach space X, B be a bounded operator in X and K= AB- BA be the commutator. Assuming that there is a linear operator S having a bounded left-inverse operator Sl-1, such that ∫0∞‖SeAt‖‖eBt‖dt<∞, and the operator KSl-1 is bounded and has a sufficiently small norm, we show that ∫0∞‖e(A+B)t‖dt<∞, where (e(A+B)t)t≥0 is the semigroup generated by A+ B. In addition, estimates for the supremum- and L¹-norms of the difference e^{( A + B ) t}- e^Ate^Bt are derived.