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

Abstract

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 $$0\to X\stackrel{\iota }{\to }Y\to Z\to 0$$ the map $$L(Y,E)\to L(X,E),T\mapsto T\circ \iota$$ is surjective, meaning that each continuous linear map $X\to E$ can be extended to a continuous linear map $Y\to E$ via $\iota$, 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].

30 pages

References in corpus (1)

• Linear topological invariants for kernels of convolution and differential operators

Cited by in corpus (1)

• The vector-valued Stieltjes moment problem with general exponents