Pseudodifferential operators on $\mathbf R^n$ and limit operators

The Fredholm property and spectral properties are considered for pseudodifferential operators on $\mathbf R^n$ with symbol satisfying the estimates
\begin{equation} |\partial^\beta_x\partial^\alpha_\xi a(x,\xi)|\leqslant C_{\alpha\beta}\lambda(x,\xi)\qquad\forall\,\alpha,\beta,C_{\alpha\beta}>0, \end{equation}
where $\lambda(x,\xi)$ is a basic weight function.
As follows from (1), differentiation of the symbol does not improve its behavior at infinity.
The family of limit operators is introduced for a pseudodifferential operator. A theorem is proved giving necessary and sufficient conditions for the Fredholm property in terms of invertibility of the family of limit operators. Some properties of the spectrum are formulated in the same terms. Examples are given which illustrate the main results.
Bibliography: 14 titles.