For certain Sheffer sequences (sn)n=0∞ on C, Grabiner (1988) proved that, for each α∈0,1, the corresponding Sheffer operator zn↦sn(z) extends to a linear self-homeomorphism of Eminα(C), the Fréchet topological space of entire functions of order at most α and minimal type (when the order is equal to α>0). In particular, every function f∈Eminα(C) admits a unique decomposition f(z)=∑n=0∞cnsn(z), and the series converges in the topology of Eminα(C). Within the context of a complex nuclear space Φ and its dual space Φ′, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on Φ′. In particular, for Φ=Φ′=Cn with n≥2, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space Φ′, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of Eminα(Φ′) when α>1. The latter result is new even in the one-dimensional case.