Infinite order ΨDOs: 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 , ξ ) 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 ∘ 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 ∞ 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 - Γ A p , ρ ∗ , ∞ -elliptic symbols, where f is a function of ultrapolynomial growth and Γ A p , ρ ∗ , ∞ 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–Hörmander integral formula.