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.

我们研究了Weyl量化\（a ^ w \）的幂级数的全局正则性和谱性质，其中\（a（x，\ xi）\）是经典的椭圆舒宾多项式。对于合适的整个函数P，我们将两个自然无限阶运算符与\（a ^ {w} \），\（P（a ^ w）\）和\（（Pcirca）^ {w}，\ ），并证明这些算子及其低阶摄动在全球范围内是Gelfand–Shilov的正则。它们具有由分离为\（\ infty \）的实际孤立特征值组成的频谱，为此我们找到了它们的特征值计数函数的渐近行为。在本文的第二部分，我们通过以下方式介绍了Shubin-Sobolev型空间f - \（\ Gamma ^ {*，\ infty} _ {A_p，\ rho} \）-椭圆符号，其中f是超多项式增长的函数，\（\ Gamma ^ {*，\ infty} _ {A_p， \ rho} \）是在本文档和我们之前的论文中研究的一类无穷阶符号。我们研究了这些空间的正则性质，并表明所考虑的伪微分算子是它们上的Fredholm算子。它们的索引与Shubin-Sobolev空间的顺序无关。最后，我们表明该指数可以通过Fedosov-Hörmander积分公式表示。