Operator ranges and spaceability
It is well known [\textit{A.~Plichko,} Funct. Anal. Appl, 15, 67--68 (1981; Zbl 0535.46008)] that, if for a continuous linear operator \(T\) from a Banach space \(X\) to a Banach space \(Y\) we have \(\dim Y/TX=\infty\), then \(Y\) contains a closed infinite-dimensional subspace \(Z\) such that \(Z\cap TX=\{0\}\). A similar fact is valid for Fréchet spaces [\textit{L.~Drewnowski}, Stud. Math. 77, 373--391 (1984; Zbl 0552.46003)]. In the paper under review, the authors prove some results of this kind. In particular, they prove that, if \(X_n\) are Banach spaces, \(Y\) is a Fréchet space, \(T_n:X_n\to Y\) are continuous linear operators and the subspace \(Y':=\text{lin}(T_nX_n)_1^{\infty}\subset Y\) is not closed, then \(Y\) contains a closed infinite-dimensional subspace \(Z\) with \(Z\cap Y'=\{0\}\). The authors ask if this result can be extended to allow the \(X_n\) to be Fréchet spaces.
scientific article; zbMATH DE number 5871224
scientific article