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. 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 infinity 8 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(Ap,rho)*(infinity)-elliptic symbols, where f is a function of ultrapolynomial growth and Gamma(Ap,rho)*(infinity) 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-Hormander integral formula.

Jaar van publicatie:2021

Toegankelijkheid:Open