Infinite order $\Psi$DOs: Composition with entire functions, new Shubin-Sobolev spaces, and index theorem

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.