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For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner(1988) proved that, for each $\alpha\in[0,1]$, the corresponding Shefferoperator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of$\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fr\'echet topologicalspace of entire functions of order at most $\alpha$ and minimal type (when theorder is equal to $\alpha>0$). In particular, every function $f\in \mathcalE^{\alpha}_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition$f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topologyof $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$. Within the context of acomplex nuclear space $\Phi$ and its dual space $\Phi'$, in this work wegeneralize Grabiner's result to the case of Sheffer operators corresponding toSheffer sequences on $\Phi'$. In particular, for $\Phi=\Phi'=\mathbb C^n$ with$n\ge2$, we obtain the multivariate extension of Grabiner's theorem.Furthermore, for an Appell sequence on a general co-nuclear space $\Phi'$, wefind a sufficient condition for the corresponding Sheffer operator to extend toa linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\Phi')$ when$\alpha>1$. The latter result is new even in the one-dimensional case.

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis