Infinite order $\mathrm{\Psi }$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 $\mathrm{\infty }$ 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 $f$-${\mathrm{\Gamma }}_{A_p,\rho }^{\ast ,\mathrm{\infty }}$-elliptic symbols, where $f$ is a function of ultrapolynomial growth and ${\mathrm{\Gamma }}_{A_p,\rho }^{\ast ,\mathrm{\infty }}$ 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.