Splitting and parameter dependence in the category of PLH spaces

We extend the splitting theory for PLS spaces and the corresponding parameter dependence problem to the context of hilbertizable spaces. In particular, we characterize for fixed PLH spaces E and X , i.e. strongly reduced projective limits of inductive limits of Hilbert spaces, the splitting of each short exact sequence 0 → X → f G → g E → 0 of PLH spaces, i.e. g has a continuous linear right inverse or f has a continuous linear left inverse, if E is either a Fréchet–Hilbert space or the strong dual of a Fréchet–Hilbert space by Bonet and Domański’s conditions ( T ) and ( T ε ) . Thus we extend the splitting relation for Fréchet–Hilbert spaces due to Domański and Mastyło and the ( D N ) - ( Ω ) splitting theorem of Vogt and Wagner. Due to the lack of nuclearity significantly different methods have to be applied. Through the connection to the vanishing of proj 1 of a spectrum of spaces of operators the above methods are also linked to the parameter dependence problem, albeit under some nuclearity assumptions as we need interpolation. These theoretical results are applied to several non- PLS (non-nuclear) spaces, as the space D L 2 , its strong dual, Hörmander’s B 2 , k loc ( Ω ) spaces and the Köthe PLH spaces.