  We study global regularity and spectral properties of power series of the
Weyl quantisation $a^w$, where $a(x,\xi) $ is a classical elliptic Shubin
polynomial. For a suitable entire function $P$, we associate two natural
infinite order operators to $a^{w}$, $P(a^w)$ and $(P\circ a)^{w},$ and prove
that these operators and their lower order perturbations are globally
Gelfand-Shilov regular. They have spectra consisting of real isolated
eigenvalues diverging to $\infty$ for which we find the asymptotic behaviour of
their eigenvalue counting function. In the second part of the article, we
introduce Shubin-Sobolev type spaces by means of
$f$-$\Gamma^{*,\infty}_{A_p,\rho}$-elliptic symbols, where $f $ is a function
of ultrapolynomial growth and $\Gamma^{*,\infty}_{A_p,\rho}$ is a class of
symbols of infinite order studied in this and our previous papers. We study the
regularity properties of these spaces, and show that the pseudo-differential
operators under consideration are Fredholm operators on them. Their indices are
independent on the order of the Shubin-Sobolev spaces; finally, we show that
the index can be expressed via a Fedosov-H\"{o}rmander integral formula.
