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We consider the adiabatic regime of two parameters evolution semigroupsgenerated by linear operators that are analytic in time and satisfy thefollowing gap condition for all times: the spectrum of the generator consistsin finitely many isolated eigenvalues of finite algebraic multiplicity, awayfrom the rest of the spectrum. The restriction of the generator to the spectralsubspace corresponding to the distinguished eigenvalues is not assumed to bediagonalizable. The presence of eigenilpotents in the spectral decomposition ofthe generator forbids the evolution to follow the instantaneous eigenprojectorsof the generator in the adiabatic limit. Making use of superadiabaticrenormalization, we construct a different set of time-dependent projectors,close to the instantaneous eigeprojectors of the generator in the adiabaticlimit, and an approximation of the evolution semigroup which intertwinesexactly between the values of these projectors at the initial and final times.Hence, the evolution semigroup follows the constructed set of projectors in theadiabatic regime, modulo error terms we control

General Adiabatic Evolution with a Gap Condition