A tame splitting theorem for Köthe spaces
Let E and F be Fréchet spaces with fixed gradings \((\mu _ p)\) and \((\nu _ p)\). A linear mapping T:E\(\to F\) is linearly (or multiplicatively) tame if there is a positive integer a and positive numbers \(C_ p\), \(p\in {\mathbb{N}}\), such that \(\nu _ p(Tx)\leq C_ p\mu _{p+a}(x)\) (or \(\nu _ p(Tx)\leq C_ p\mu _{ap}(x))\), for every \(x\in E\) and \(p\in {\mathbb{N}}\). A short exact sequence of Fréchet spaces \(0\to E\to ^{i}X\to ^{q}F\to 0\) is tame if the isomorphisms i:E\(\to i(E)\subset X\) and q:X/i(E)\(\to F\) and their inverses are tame. By definition, a tame short exact sequence splits tamely if q has a tame right inverse or, equivalently, if i has a tame left inverse. Using the method of twisted sums [\textit{T. Ketonen} and \textit{K. Nyberg}: Ann. Acad. Sci. Fenn., Ser. AI 7, 323-335 (1982; Zbl 0531.46003)] a necessary and sufficient condition is given in terms of the gradings of nuclear Köthe spaces E and F under which every tame short exact sequence splits tamely. Concerning the question whether and when tameness implies splitting, this condition is applied to nuclear power series spaces. It is shown that if E is of infinite type and F is of finite type then tameness considerations are irrelevant, while in the other three cases the tameness with respect to certain gradings gives a tame splitting.
scientific article; zbMATH DE number 4042532