Necessary Conditions for Fredholmness of Partial Differential Operators of Irregular Singular Type

The convergence of formal power series solutions of ordinary differential equations are extensively studied by many authors in connection with irregularities, (cf. [6] and [10].) In case of partial differential equations of regular singular type, several sufficient conditions are known, (cf. [4] and [1].) In the preceeding paper [9], we gave sufficient conditions for the Fredholmness of partial differential operators of irregular singular type of two independent variables in analytic and Gevrey spaces. Then we deduced the convergence of formal power series solutions from the Fredholmness of the operators. These conditions are expressed in terms of Toeplitz symbols, and they are equivalent to a Riemann-Hilbert factorization condition, (cf. [3] and (2.5), (2.6) , (2.7) which follow.) In this paper, we shall show the necessity of these sufficient conditions. More precisely, we will prove that the Riemann-Hilbert factorization conditions (2.6) and (2.7) are necessary and sufficient for the partial differential operators of irregular singular type to be Fredholm operators on certain analytic and Gevrey spaces. The proof of our theorem is based on the analysis of the main (principal) part of irregular singular type operators via Toeplitz operators on the torus T • = R/27rZ. In fact, we will show that the essential parts of these operators in studying Fredholm properties are precisely Toeplitz operators. This