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Splitting and parameter dependence in the category of \(\hbox {PLH}\) spaces

  • Bernhard DierolfEmail author
  • Dennis Sieg
Original Paper
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Abstract

We extend the splitting theory for \(\hbox {PLS}\) spaces and the corresponding parameter dependence problem to the context of hilbertizable spaces. In particular, we characterize for fixed \(\hbox {PLH}\) spaces E and X, i.e. strongly reduced projective limits of inductive limits of Hilbert spaces, the splitting of each short exact sequence
$$\begin{aligned} 0 \rightarrow X \xrightarrow {f} G \xrightarrow {g} E \rightarrow 0 \end{aligned}$$
of \(\hbox {PLH}\) 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 \((T_{\varepsilon })\). Thus we extend the splitting relation for Fréchet–Hilbert spaces due to Domański and Mastyło and the \((DN)-(\Omega )\) 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 \(\hbox {proj}^{1}\) 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-\(\hbox {PLS}\) (non-nuclear) spaces, as the space \(\mathscr {D}_{{\text {L}}_2}\), its strong dual, Hörmander’s \(\hbox {B}_{2,k}^{\mathrm{loc}}(\Omega )\) spaces and the Köthe \(\hbox {PLH}\) spaces.

Keywords

Splitting of short exact sequences Parameter dependence Functor \(\hbox {Ext}^{1}\) Hilbertizable locally convex spaces Fréchet–Hilbert space 

Mathematics Subject Classification

Primary 46M18 46F05 35E20 35R20 Secondary 46A63 46A13 
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Notes

Acknowledgements

The authors thank L. Frerick and J. Wengenroth for many fruitful discussions on the subjects of this article during the supervision of the PhD theses. Furthermore the financial support of the “Stipendienstiftung Rheinland-Pfalz” for both PhD projects is acknowledged.

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Authors and Affiliations

  1. 1.Math.-Geogr. FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany
  2. 2.Faculty IV MathematicsUniversity of TrierTrierGermany

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