Splitting and parameter dependence in the category of \(\mathrm{PLH}\) spaces. (English) Zbl 1423.46100
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The central concept of the paper is to provide a splitting theory in the category of the so-called PLH-spaces, i.e., projective limits of inductive limits of Hilbert spaces. The main problem is to characterize those pairs \((F,E)\) of PLH-spaces such that, for any PLH-space \(G\), the short exact sequence \[ \begin{tikzcd} 0\arrow[r] & F\arrow[r, "f"] & G\arrow[r, "g"] & E\arrow[r] & 0 \end{tikzcd} \] splits. The results of the paper should be compared with their analogues obtained in the category of PLS-spaces by Bonet, Domański, Frerick, Langenbruch, Vogt, Wengenroth and others.
The main issue is a proper notion of exactness. The authors call the above short sequence exact if \(f\) is a topological embedding, \(g\) is surjective and open and \(\operatorname{im} f=\ker g\). This approach allows the authors to drop one of the crucial assumptions in the splitting business, namely, they do not need (in general) nuclearity of the spaces involved. This is an interesting novelty in the splitting theory.
The paper is divided into five parts. Sections 1 and 2 are ‘Introduction’ and ‘Preliminaries’. Section 3 is the most technical one. Here, the authors obtain their splitting results for specific pairs \((F,E)\). In the next section, their methods are applied to get parameter dependence results. This part, however, requires some nuclearity assumptions due to the fact that some interpolation techniques have to be used. The last section is devoted to applications in the theory of partial differential operators.
As for the last comment, the reviewer would like to point out that the paper is dedicated to the memory of Paweł Domański, a great Polish mathematician who passed away – far too early – in 2016. Paweł Domański was also a supervisor of the reviewer’s Ph.D. Thesis, and the reviewer wishes to thank the authors for keeping Paweł Domański in the memory of the mathematical community.
The central concept of the paper is to provide a splitting theory in the category of the so-called PLH-spaces, i.e., projective limits of inductive limits of Hilbert spaces. The main problem is to characterize those pairs \((F,E)\) of PLH-spaces such that, for any PLH-space \(G\), the short exact sequence \[ \begin{tikzcd} 0\arrow[r] & F\arrow[r, "f"] & G\arrow[r, "g"] & E\arrow[r] & 0 \end{tikzcd} \] splits. The results of the paper should be compared with their analogues obtained in the category of PLS-spaces by Bonet, Domański, Frerick, Langenbruch, Vogt, Wengenroth and others.
The main issue is a proper notion of exactness. The authors call the above short sequence exact if \(f\) is a topological embedding, \(g\) is surjective and open and \(\operatorname{im} f=\ker g\). This approach allows the authors to drop one of the crucial assumptions in the splitting business, namely, they do not need (in general) nuclearity of the spaces involved. This is an interesting novelty in the splitting theory.
The paper is divided into five parts. Sections 1 and 2 are ‘Introduction’ and ‘Preliminaries’. Section 3 is the most technical one. Here, the authors obtain their splitting results for specific pairs \((F,E)\). In the next section, their methods are applied to get parameter dependence results. This part, however, requires some nuclearity assumptions due to the fact that some interpolation techniques have to be used. The last section is devoted to applications in the theory of partial differential operators.
As for the last comment, the reviewer would like to point out that the paper is dedicated to the memory of Paweł Domański, a great Polish mathematician who passed away – far too early – in 2016. Paweł Domański was also a supervisor of the reviewer’s Ph.D. Thesis, and the reviewer wishes to thank the authors for keeping Paweł Domański in the memory of the mathematical community.
Reviewer: Krzysztof Piszczek (Poznan)
MSC:
| 46M18 | Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.) |
| 46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 35E20 | General theory of PDEs and systems of PDEs with constant coefficients |
| 35R20 | Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) |
| 46A63 | Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces |
Keywords:
splitting of short exact sequences; parameter dependence; functor \(\mathrm{Ext}^{1}\); hilbertizable locally convex spaces; Fréchet-Hilbert spaceReferences:
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