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 \(a(x,\xi ) \) 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–Hörmander integral formula.