SPECTRAL MAPPING THEOREMS FOR HOLOMORPHIC FUNCTIONAL CALCULI

A spectral inclusion theorem and a spectral mapping theorem are proved for the functional calculus for sectorial operators. Most proofs are generic, so that similar results can be obtained for other functional calculi. Applications are a new proof for the spectral mapping theorem for fractional powers and the identity σ(logA)= log(σ(A) \ {0}) for any injective sectorial operator A.