An extension result for (LB)-spaces and the surjectivity of tensorized mappings

We study an extension problem for continuous linear maps in the setting of (LB)-spaces. More precisely, we characterize the pairs (E, Z), where E is a locally complete space with a fundamental sequence of bounded sets and Z is an (LB)-space, such that for every exact sequence of (LB)-spaces the map \(\begin{aligned} L(Y,E) \rightarrow L(X, E), ~ T \mapsto T \circ \iota \end{aligned}\) $$\begin{array}{c}\hfill L(Y,E)\to L(X,E),T\mapsto T\circ \iota \end{array}$$ is surjective, meaning that each continuous linear map \(X \rightarrow E\) $X\to E$ can be extended to a continuous linear map \(Y \rightarrow E\) $Y\to E$ via \(\iota \) $\iota$ , under some mild conditions on E or Z (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24].