Non-analytic functional calculi in several variables

Generalizing a result of 1] and 2], we show that every-scalar system (see Def. 3) is decomposable (in the sense of 6], 7]). By means of this fact and by some results on decomposable n-tuples (contained in 6] and 7]) we prove the theorem of support and (in the case of an inverse closed admissible algebra) the spectral mapping theorem for-functional calculi in several variables (see Def. 3). In the case of a single operator we obtain a simplification of the definition of an-spectral function (in the sense of 5], Def. 3.1.3).