||B. Gramsch and D. Lay have studied spectral mapping theorems for the essential spectra of an operator acting in a complex Banach space. Firstly they consider operators belonging to the Banach algebra of all bounded linear operators on the space, and later they derive the theorems for unbounded closed linear operators with non-empty resolvent from the aboye case ; but bounded closed linear operators with domain a proper subspace are not included. In this note we introduce a notion of extended essential spectra for any closed linear operator with non-empty resolvent, which covers the above cases. Then, in this more general context, we are able to prove the spectral mapping theorems by means of a more unified approach based on a factorization of the operators provided by the Dunford-Taylor calculus and well-known properties of products of operators present in Fredholm theory.