On some dense compact operator range subspaces in separable Fréchet spaces (Q809335)

Let E be a (Hausdorff) topological vector space and X, Y two closed subspaces of E with \(X\cap Y=0\). If \(X+Y\) is dense in E (and \(X+Y\neq E)\), then X, Y are called (proper) quasi-complements. With these notations, the author proves the following: If T: \(F\to E\) is a continuous not relatively open linear operator from a Fréchet space F into a separable Fréchet space E with dense range, then for every infinite dimensional separable Banach space Z, there exists a compact injective linear operator Q: \(Z\to E\) such that Q(Z) is dense and \(Q(Z)\cap T(F)=0.\) If X, Y are proper quasi-complements in a separable Fréchet space E, then E contains a dense subspace \(G\supset Y\) with \(G\cap X=0.\) This generalizes a theorem of \textit{V. V. Shevchik} [Dokl. Akad. Nauk SSSR 263, 817-819 (1982; Zbl 0514.46009)].