Topological \(p\)-adic vector spaces and index theory
Suppose \(u\) is a linear transformation of a vector space \(E\) over a complete ultrametric field \(k\). This \(u\) is said to have an index \(\chi (u)\) if both kernel and cokernel of \(u\) are finite-dimensional. We set \(\chi (u)= \chi(u, E)= \dim(\ker u)- \dim(\text{coker } u)\). The basic question is to prove the existence of the index when \(u\) is an ordinary differential operator on certain power series rings. The main theorem of this paper says that if \(r_n\) is an increasing sequence of positive reals tending to \(r\) and \(P\) is a differential operator in the ring \({\mathcal H}^+ (r_n)\) of Laurent series of \({1\over x} k[[{1\over x}]]\) converging in a disk \(s\leq |x|\) for some \(s< r_n\) having an index in \({\mathcal H}^+ (r_n)\) for all \(n\), then it has an index in \({\mathcal H}^+ (r)\) and \(\chi (P,{\mathcal H}^+ (r))\) is equal to the limit of \(\chi (P,{\mathcal H}^+ (r_n))\) as \(n\) tends to \(\infty\).
scientific article; zbMATH DE number 800318