Operator Ranges and Spaceability: Extending a Result of Kitson and Timoney

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.

6, \keywords{Lineability; spaceability; operator range; Fréchet space; cone method; uncountable families.}