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

Abstract

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 * ,∞ Ap,ρ -elliptic symbols, where f is a function of ultrapolynomial growth and Γ * ,∞ Ap,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.