Semigroups of sums of two operators with small commutators

Let A be the generator of a C-semigroup (eAt)t≥0 on a Banach space X and B be a bounded operator in X. Assuming that ∫0∞‖eAt‖‖eBt‖dt<∞ and the commutator AB- BA 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 L ¹ -norms of the difference e ^{( A + B ) t} - e ^At e ^Bt are derived.