Abstract For an operator having a uniformly bounded resolvent on a strip around the imaginary axis, the existence of—possibly unbounded—spectral projections corresponding to the left and right half‐plane is proved. The operator is dichotomous if these projections are bounded, and an abstract perturbation theorem for dichotomy is derived. All results apply, with certain simplifications, to bisectorial operators. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)