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We derive new results on the characterization of Gelfand--Shilov spaces$\mathcal{S}^\mu_\nu (\R^n)$, $\mu,\nu >0$, $\mu+\nu \geq 1$ by Gevreyestimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globallyelliptic Shubin (or $\Gamma$) type operators, $(-\Delta)^{m/2} +| x |^k$ with$m,k\in 2\N$ being a model operator, and on the decay of the Fouriercoefficients in the related eigenfunction expansions. Similar results areobtained for the spaces $\Sigma^\mu_\nu (\R^n)$, $\mu,\nu >0$, $\mu+\nu > 1$,cf. \eqref{GSdef}. In contrast to the symmetric case $\mu = \nu$ and $k=m$(classical Shubin operators) we encounter resonance type phenomena involvingthe ratio $\kappa:=\mu/\nu$; namely we obtain a characterization of$\mathcal{S}^\mu_\nu(\R^n)$ and $\Sigma^\mu_\nu(\R^n)$ in the case$\mu=kt/(k+m), \nu= mt/(k+m), t \geq 1$, that is, when $\kappa=k/m \in \Q$.

Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces