pp 1–26 | Cite as

Interpolation of nonlinear positive or order preserving operators on Banach lattices

  • Ralph ChillEmail author
  • Alberto Fiorenza
  • Sebastian Król


We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo–Peetre operators, and exact partially K-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple \((L^1,L^\infty )\) due to Bénilan and Crandall.


Nonlinear interpolation Exact interpolation spaces Order preserving operators Positive Gagliardo–Peetre operators Normal spaces 

Mathematics Subject Classification

Primary 46B42 46B70 47H07 Secondary 47H20 



  1. 1.
    Astashkin, S.V.: Exact \({\cal{K}}\)-monotonicity of a class of Banach couples. Sibirsk. Mat. Zh. 43, 14–32 (2002)MathSciNetGoogle Scholar
  2. 2.
    Astashkin, S.V., Maligranda, L., Tikhomirov, K.E.: New examples of \(K\)-monotone weighted Banach couples. Studia Math. 218, 55–88 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics. Springer, New York (2010)CrossRefGoogle Scholar
  4. 4.
    Barthélemy, L.: Invariance d’un convex fermé par un semi-groupe associé à une forme non-linéaire. Abstr. Appl. Anal. 1, 237–262 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bénilan, Ph., Crandall, M.G.: Completely accretive operators. In: Semigroup Theory and Evolution Equations (Delft, 1989), Lecture Notes in Pure and Appl. Math., vol. 135, pp. 41–75. Dekker, New York (1991)Google Scholar
  6. 6.
    Bénilan, Ph., Picard, C.: Quelques aspects non linéaires du principe du maximum. In: Séminaire de Théorie du Potentiel, No. 4 (Paris, 1977/1978), Lecture Notes in Math., vol. 713, pp. 1–37. Springer, Berlin (1979)Google Scholar
  7. 7.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston (1988)zbMATHGoogle Scholar
  8. 8.
    Berezhnoĭ, E.I.: Interpolation of positive operators in the spaces \(\phi (X_0,X_1)\). In: Qualitative and Approximate Methods for the Investigation of Operator Equations (Russian), Yaroslav. Gos. Univ., Yaroslavl’, pp. 3–12, 160 (1981)Google Scholar
  9. 9.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  10. 10.
    Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland Mathematics Studies, vol. 5. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  11. 11.
    Brézis, H., Strauss, W.A.: Semi-linear second-order elliptic equations in \(L^{1}\). J. Math. Soc. Jpn. 25, 565–590 (1973)CrossRefzbMATHGoogle Scholar
  12. 12.
    Browder, F.E.: Remarks on nonlinear interpolation in Banach spaces. J. Funct. Anal. 4, 390–403 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brudnyĭ, Y.A., Krugljak, N.Y.: Interpolation Functors and Interpolation Spaces, vol. I. North-Holland Mathematical Library, Amsterdam (1991)zbMATHGoogle Scholar
  14. 14.
    Cipriani, F., Grillo, G.: Nonlinear Markov semigroups, nonlinear Dirichlet forms and application to minimal surfaces. J. Reine Angew. Math. 562, 201–235 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Coulhon, T., Hauer, D.: Regularisation Effects of Nonlinear Semigroups, Springer Briefs in Mathematics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2018, Classical Methods and Recent Advances, BCAM SpringerBriefs (2018)Google Scholar
  16. 16.
    Crandall, M.G., Liggett, T.M.: Generation of semi-groups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cwikel, M.: Monotonicity properties of interpolation spaces. Ark. Mat. 14, 213–236 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cwikel, M., Keich, U.: Optimal decompositions for the \(K\)-functional for a couple of Banach lattices. Ark. Mat. 39, 27–64 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cwikel, M., Nilsson, P.G., Schechtman, G.: Interpolation of Weighted Banach Lattices. A Characterization of Relatively Decomposable Banach Lattices, vol. 165. Mem. Amer. Math. Soc. (2003)Google Scholar
  20. 20.
    Karlovich, A., Maligranda, L.: On the interpolation constant for subadditive operators in Orlicz spaces. In: Banach and Function Spaces II, Yokohama (2008)Google Scholar
  21. 21.
    Kreĭn, S.G., Petunīn, Yu., Semënov, E.M.: Interpolation of Linear Operators, Translations of Mathematical Monographs, vol. 54 (1982)Google Scholar
  22. 22.
    Lorentz, G.G., Shimogaki, T.: Interpolation theorems for operators in function spaces. J. Funct. Anal. 2, 31–51 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lozanovskiĭ, G.J.: A remark on a certain interpolation theorem of Calderón. Funkcional Anal. Priložen 6, 89–90 (1972)Google Scholar
  24. 24.
    Maligranda, L.: Calderón-Lozanovskii spaces and interpolation of operators. Semesterbericht Funktionalanalysis Tübingen 8, 83–92 (1985)Google Scholar
  25. 25.
    Maligranda, L.: Some remarks on Orlicz’s interpolation theorem. Studia Math. 95, 43–58 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Maligranda, L.: Orlicz spaces and interpolation. In: Seminários de Matemática [Seminars in Mathematics], Universidade Estadual de Campinas (1989)Google Scholar
  27. 27.
    Maligranda, L.: On interpolation of nonlinear operators. Comment. Math. Prace Mat. 28, 253–275 (1989)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Maligranda, L.: The \(K\)-functional for \(p\)-convexifications. Positivity 17, 707–710 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mastyło, M.: Lattice structures on some Banach spaces. Proc. Am. Math. Soc. 140, 1413–1422 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  31. 31.
    Orlicz, W.: On a class of operations over the space of integrable functions. Studia Math. 14(1954), 302–209 (1955)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Sedaev, A.A.: A description of the interpolation spaces of the couple \((L^{p}_{a_{0}},\, L^{p}_{a_{1}})\), and certain related questions. Dokl. Akad. Nauk SSSR 209, 798–800 (1973)MathSciNetGoogle Scholar
  33. 33.
    Sedaev, A.A., Semenov, E.M.: The possibility of describing interpolation spaces in terms of Peetre’s \(K\)-method. Optimizacija 4, 98–114 (1971)MathSciNetGoogle Scholar
  34. 34.
    Shestakov, V.A.: Transformations of Banach ideal spaces and interpolation of linear operators. Bull. Acad. Polon. Sci. Sér. Sci. Math. 29(1981), 569–577 (1982)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Sparr, G.: Interpolation of weighted \(L_{p}\)-spaces. Studia Math. 62, 229–271 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Veselova, L.V., Sukochev, F.A., Tikhonov, O.E.: Interpolation of positive operators. Mat. Zametki 81, 43–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Analysis, Fakultät für MathematikTU DresdenDresdenGermany
  2. 2.Dipartimento di ArchitetturaUniversitá di Napoli Federico IINaplesItaly
  3. 3.Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di NapoliConsiglio Nazionale delle RicercheNaplesItaly
  4. 4.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

Personalised recommendations