  For an unbounded operator $S$ on a Banach space the existence of invariant
subspaces corresponding to its spectrum in the left and right half-plane is
proved. The general assumption on $S$ is the uniform boundedness of the
resolvent along the imaginary axis. The projections associated with the
invariant subspaces are bounded if $S$ is strictly dichotomous, but may be
unbounded in general. Explicit formulas for these projections in terms of
resolvent integrals are derived and used to obtain perturbation theorems for
dichotomy. All results apply, with certain simplifications, to bisectorial
operators.
