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

  • Bernhard DierolfEmail author
  • Dennis Sieg
Original Paper
  • 48 Downloads

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 

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.

References

  1. 1.
    Bierstedt, K.D.: An introduction to locally convex inductive limits. In: Functional Analysis and Its Applications (Nice, 1986). ICPAM Lecture Notes, pp. 35–133. World Scientific Publishing, Singapore (1988)Google Scholar
  2. 2.
    Bonet, J., Domański, P.: Real analytic curves in Fréchet spaces and their duals. Monatsh. Math. 126(1), 13–36 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bonet, J., Domański, P.: Parameter dependence of solutions of partial differential equations in spaces of real analytic functions. Proc. Am. Math. Soc. 129(2), 495–503 (2001) (electronic) Google Scholar
  4. 4.
    Bonet, J., Domański, P.: Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences. J. Funct. Anal. 230(2), 329–381 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bonet, J., Domański, P.: The structure of spaces of quasianalytic functions of Roumieu type. Arch. Math. (Basel) 89(5), 430–441 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bonet, J., Domański, P.: The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations. Adv. Math. 217(2), 561–585 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bonet, J., Domański, P., Vogt, D.: Interpolation of vector-valued real analytic functions. J. Lond. Math. Soc. 66(2), 407–420 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Browder, F.E.: Analyticity and partial differential equations. I. Am. J. Math. 84, 666–710 (1962)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam (1993)zbMATHGoogle Scholar
  11. 11.
    Dierolf, B.: Splitting theory for PLH-spaces. Ph.D. thesis, University of Trier. http://ubt.opus.hbz-nrw.de/volltexte/2014/869/pdf/DissertationBernhardDierolf.pdf (2014)
  12. 12.
    Dierolf, B.: The Hilbert tensor product and inductive limits. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM (2015). doi: 10.1007/s13398-015-0229-3
  13. 13.
    Dierolf, B., Sieg, D.: A homological approach to the splitting theory for PLS\(_w\)-spaces. J. Math. Anal. Appl. 433(2), 1305–1328 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Domański, P.: Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives. In: Orlicz Centenary Volume, Banach Center Publ., vol. 64, pp. 51–70. Polish Academy of Sciences, Warsaw (2004)Google Scholar
  15. 15.
    Domański, P.: Real analytic parameter dependence of solutions of differential equations. Rev. Mat. Iberoam. 26(1), 175–238 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Domański, P.: Real analytic parameter dependence of solutions of differential equations over Roumieu classes. Funct. Approx. Comment. Math. 44(part 1), 79–109 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Domański, P., Frerick, L., Vogt, D.: Fréchet quotients of spaces of real-analytic functions. Studia Math. 159(2), 229–245 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Domański, P., Mastyło, M.: Characterization of splitting for Fréchet–Hilbert spaces via interpolation. Math. Ann. 339(2), 317–340 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Domański, P., Vogt, D.: Distributional complexes split for positive dimensions. J. Reine Angew. Math. 522, 63–79 (2000)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Domański, P., Vogt, D.: The space of real-analytic functions has no basis. Studia Math. 142(2), 187–200 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Domański, P., Vogt, D.: A splitting theory for the space of distributions. Studia Math. 140(1), 57–77 (2000)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Frerick, L.: On vector valued sequence spaces. Ph.D. thesis, University of Trier (1994)Google Scholar
  23. 23.
    Frerick, L., Kalmes, T.: Some results on surjectivity of augmented semi-elliptic differential operators. Math. Ann. 347(1), 81–94 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Frerick, L., Kunkle, D., Wengenroth, J.: The projective limit functor for spectra of webbed spaces. Studia Math. 158(2), 117–129 (2003)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Frerick, L., Wengenroth, J.: A sufficient condition for vanishing of the derived projective limit functor. Arch. Math. (Basel) 67(4), 296–301 (1996)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires, vol. 16, p. 140. Memoirs of the American Mathematical Society, Providence, Rhode Island (1955)Google Scholar
  27. 27.
    Hermanns, V.: Zur Existenz von Rechtsinversen linearer partieller Differentialoperatoren mit konstanten Koeffizienten auf \({B}_{p,\kappa }^{loc}({\Omega })\)-Räumen. Ph.D. thesis, Bergische Universität Wuppertal. http://d-nb.info/975806661/34 (2005)
  28. 28.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients. Classics in Mathematics. Springer, Berlin (2005) (Reprint of the 1983 original)Google Scholar
  29. 29.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Classics in Mathematics. Springer, Berlin (2003)Google Scholar
  30. 30.
    Jarchow, H.: Locally Convex Spaces. Mathematische Leitfäden [Mathematical Textbooks]. B. G. Teubner, Stuttgart (1981)zbMATHGoogle Scholar
  31. 31.
    Kalmes, T.: The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective. Bull. Lond. Math. Soc. 44(3), 610–614 (2012)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Köthe, G.: Topological Vector Spaces. II. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 237. Springer, New York (1979)zbMATHGoogle Scholar
  33. 33.
    Kunkle, D.: Splitting of power series spaces of (PLS)-type. Ph.D. thesis, Bergische Universität Wuppertal. http://elpub.bib.uni-wuppertal.de/servlets/DerivateServlet/Derivate-333/d070106.pdf (2001)
  34. 34.
    Langenbruch, M.: Characterization of surjective partial differential operators on spaces of real analytic functions. Studia Math. 162(1), 53–96 (2004)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mantlik, F.: Linear equations depending differentiably on a parameter. Integral Equ. Oper. Theory 13(2), 231–250 (1990)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Mantlik, F.: Partial differential operators depending analytically on a parameter. Ann. Inst. Fourier (Grenoble) 41(3), 577–599 (1991)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Mantlik, F.: Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter. Trans. Am. Math. Soc. 334(1), 245–257 (1992)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press, Oxford University Press, New York (Translated from the German by M. S. Ramanujan and revised by the authors) (1997)Google Scholar
  39. 39.
    Mitchell, B.: Theory of Categories. Pure and Applied Mathematics, vol. XVII. Academic, New York (1965)zbMATHGoogle Scholar
  40. 40.
    Murphy, G.J.: \(C^*\)-Algebras and Operator Theory. Academic, Boston (1990)zbMATHGoogle Scholar
  41. 41.
    Palamodov, V.P.: Homological methods in the theory of locally convex spaces. Uspekhi Mat. Nauk 26 (1), 3–66 (in Russian) (English transl., Russian Math. Surveys 26(1), 1–64 (1971)) (1971)Google Scholar
  42. 42.
    Palamodov, V.P.: The projective limit functor in the category of topological linear spaces. Mat. Sb. 75, 567–603 (in Russian) (English transl., Math. USSR. Sbornik 17, 189–315 (1972)) (1968)Google Scholar
  43. 43.
    Pietsch, A.: Nuclear Locally Convex Spaces. Springer, New York (Translated from the second German edition by William H. Ruckle, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 66) (1972)Google Scholar
  44. 44.
    Piszczek, K.: On a property of PLS-spaces inherited by their tensor products. Bull. Belg. Math. Soc. Simon Stevin 17(1), 155–170 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Retakh, V.S.: The subspaces of a countable inductive limit. Dokl. Akad. Nauk SSSR 194, 1277–1279 (1970)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Schwartz, L.: Théorie des distributions à valeurs vectorielles. I. Ann. Inst. Fourier Grenoble 7, 1–141 (1957)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Schwartz, L.: Théorie des distributions à valeurs vectorielles. II. Ann. Inst. Fourier Grenoble 8, 1–209 (1958)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Sieg, D.: A homological approach to the splitting theory of PLS-spaces. Ph.D. thesis, University of Trier. http://ubt.opus.hbz-nrw.de/volltexte/2010/572/pdf/SiegDiss.pdf (2010)
  49. 49.
    Trèves, F.: Fundamental solutions of linear partial differential equations with constant coefficients depending on parameters. Am. J. Math. 84, 561–577 (1962)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Trèves, F.: Un théorème sur les équations aux dérivées partielles à coefficients constants dépendant de paramètres. Bull. Soc. Math. Fr. 90, 473–486 (1962)zbMATHGoogle Scholar
  51. 51.
    Valdivia, M.: Representations of the spaces \({\fancyscript {D}}(\Omega )\) and \({\fancyscript {D}}^{\prime } (\Omega )\). Rev. Real Acad. Cienc. Exact. Fís. Natur. Madr. 72(3), 385–414 (1978)Google Scholar
  52. 52.
    Varol, O.: Kriterien für Tor\(^1_{\alpha }\)(E,F)= 0 für (DF)- und Frécheträume. Ph.D. thesis, Bergische Universtät Wuppertal. http://elpub.bib.uni-wuppertal.de/edocs/dokumente/fb07/diss2002/varol/d070205.pdf (2002)
  53. 53.
    Varol, O.: On the derived tensor product functors for (DF)- and Fréchet spaces. Studia Math. 180(1), 41–71 (2007)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Vogt, D.: Vektorwertige Distributionen als Randverteilungen holomorpher Funktionen. Manuscr. Math. 17(3), 267–290 (1975)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Vogt, D.: On the solvability of \({P}({D})f = g\) for vector valued functions. RIMS Kokyoroku 508, 168–182 (1983)Google Scholar
  56. 56.
    Vogt, D.: Sequence space representations of spaces of test functions and distributions. In: Functional Analysis, Holomorphy, and Approximation Theory (Rio de Janeiro, 1979). Lecture Notes in Pure and Applied Mathematics, vol. 83, pp. 405–443. Dekker, New York (1983)Google Scholar
  57. 57.
    Vogt, D.: On the functors \({\text{ Ext }}^{1}(E, F)\) for Fréchet spaces. Studia Math. 85(2), 163–197 (1987)MathSciNetGoogle Scholar
  58. 58.
    Vogt, D.: Topics on projective spectra of (LB)-spaces. In: Advances in the Theory of Fréchet Spaces (Istanbul, 1988). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 287, pp. 11–27. Kluwer Academic Publishers, Dordrecht (1989)Google Scholar
  59. 59.
    Vogt, D.: Regularity properties of (LF)-spaces. In: Progress in Functional Analysis (Peníscola, 1990). North-Holland Mathematics Studies, vol. 170, pp. 57–84. North-Holland, Amsterdam (1992)Google Scholar
  60. 60.
    Vogt, D.: Fréchet valued real analytic functions. Bull. Soc. Roy. Sci. Liège 73(2–3), 155–170 (2004)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Vogt, D.: Invariants and spaces of zero solutions of linear partial differential operators. Arch. Math. (Basel) 87(2), 163–171 (2006)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Vogt, D.: On the splitting relation for Fréchet–Hilbert spaces. Funct. Approx. Comment. Math. 44(part 2), 215–225 (2011)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Vogt, D., Wagner, M.J.: Charakterisierung der Quotientenräume von \(s\) und eine Vermutung von Martineau. Studia Math. 67(3), 225–240 (1980)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  65. 65.
    Wengenroth, J.: Retractive (LF)-spaces. Ph.D. thesis, University of Trier (1995)Google Scholar
  66. 66.
    Wengenroth, J.: Acyclic inductive spectra of Fréchet spaces. Studia Math. 120(3), 247–258 (1996)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Wengenroth, J.: A splitting theorem for subspaces and quotients of \(\fancyscript {D}^{\prime }\). Bull. Pol. Acad. Sci. Math. 49(4), 349–354 (2001)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Wengenroth, J.: Derived Functors in Functional Analysis. Lecture Notes in Mathematics, vol. 1810. Springer, Berlin (2003)zbMATHGoogle Scholar

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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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