Operator Ranges and Spaceability: Extending a Result of Kitson and Timoney
Geivison Ribeiro
TL;DR
The paper addresses spaceability of the complement of operator-range spans in Fréchet spaces, extending Kitson–Timoney’s results from Banach domains to Fréchet domains and to uncountable operator families. The authors encode a (countable or uncountable) family of operators into a single Fréchet-space operator $S:E\to X$ defined on a suitable $E$ so that $S(D)=Y$, and apply a Drewnowski–type geometric argument to the range of $S$. They prove that if $Y=\operatorname{span}(\bigcup_i T_i(Z_i))$ is not closed in $X$, then $X\setminus Y$ is spaceable; the argument handles both when $S(E)$ is closed or not, using a separable reduction in the closed case. The uncountable extension follows by restricting to a countable obstruction subset inside a closed separable subspace, applying the countable case, and transferring the result back to the original space. Overall, the work broadens large linear-structure results for nonlinear sets to general Fréchet-space operator ranges with potentially uncountable indexing.
Abstract
We revisit the results of Kitson and Timoney \emph{[J.~Math.~Anal.~Appl.~\textbf{378} (2011), 680--686]} on the spaceability of complements of operator ranges, extending one of their main theorems to the general Fréchet setting. In particular, we provide an affirmative answer to the question posed in \emph{Remark~3.4} of that paper, showing that the conclusion remains valid when the operators act between Fréchet spaces. Moreover, we show that the same phenomenon occurs for arbitrary (possibly uncountable) families of operators. The arguments presented here follow the spirit of the original work.