Maximal Factorization of Operators Acting in Köthe–Bochner Spaces


Using some representation results for Köthe–Bochner spaces of vector valued functions by means of vector measures, we analyze the maximal extension for some classes of linear operators acting in these spaces. A factorization result is provided, and a specific representation of the biggest vector valued function space to which the operator can be extended is given. Thus, we present a generalization of the optimal domain theorem for some types of operators on Banach function spaces involving domination inequalities and compactness. In particular, we show that an operator acting in Bochner spaces of p-integrable functions for any \(1<p<\infty \) having a specific compactness property can always be factored through the corresponding Bochner space of 1-integrable functions. Some applications in the context of the Fourier type of Banach spaces are also given.

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  1. 1.

    Abasov, N., Pliev, M.: On two definitions of a narrow operator on Köthe–Bochner spaces. Arch. Math. 111, 167–176 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bartle, R.G., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Can. J. Math. 7, 289–305 (1955)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bochner, S.: Integration von Funktionen, deren Werte die Elemente eines Vectorraumes sind. Fundam. Math. 20, 262–276 (1933)

    Article  MATH  Google Scholar 

  4. 4.

    Calabuig, J.M., Fernández Unzueta, M., Galaz Fontes, F., Sánchez Pérez, E.A.: Extending and factorizing bounded bilinear maps defined on order continuous Banach function spaces. RACSAM 108, 353–367 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Calabuig, J.M., Jiménez-Fernández, E., Juan, M.A., Sánchez-Pérez, E.A.: Optimal extensions of compactness properties for operators on Banach function spaces. Topol. Appl. 203, 57–66 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cembranos, P., Mendoza, J.: Banach spaces of vector-valued functions, Lecture Notes in Mathematics, vol. 1676. Springer, Berlin (1997)

  7. 7.

    Cerdà, J., Hudzik, H., Mastyło, M.: Geometric properties of Köthe-Bochner spaces. Math. Proc. Camb. Philos. Soc. 120(3), 521–533 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Choi, C., Lee, H.H.: Operators of Fourier type p with respect to some subgroups of a locally compact abelian group. Arch. Math. 81(4), 457–466 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)

    Google Scholar 

  10. 10.

    Defant, A., López Molina, J.A., Rivera, M.J.: On Pitt’s theorem for operators between scalar and vector-valued quasi-Banach sequence spaces. Monatshefte für Mathematik 130(1), 7–18 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)

    Google Scholar 

  12. 12.

    Duru, H., Kitover, A., Orhon, M.: Multiplication operators on vector-valued function spaces. Proc. Am. Math. Soc. 141, 3501–3513 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Feledziak, K.: Absolutely continuous linear operators on Köthe–Bochner spaces. Banach Center Publ. 92, 85–89 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Feledziak, K., Nowak, M.: Integral representation of linear operators on Orlicz-Bochner spaces. Collect. Math. 61, 277–290 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Huerta, P.G.: Espacios de medidas vectoriales. Thesis, Universidad de Valencia, ISBN: 8437060591 (2005)

  16. 16.

    Kusraev, A.G.: Dominated Operators. Springer, Dordrecht (2000)

    Google Scholar 

  17. 17.

    Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Lin, P.-K.: Köthe-Bochner Function Spaces. Birkhauser, Boston (2004)

    Google Scholar 

  19. 19.

    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)

    Google Scholar 

  20. 20.

    Nowak, M.: Bochner representable operators on Köthe–Bochner spaces. Comment. Math. 48, 113–119 (2008)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Okada, S.: Does a compact operator admit a maximal domain for its compact linear extension? In: Curbera, G., Mockenhaupt, G., Ricker, W.J. (eds.) Vector Measures, Integration and Related Topics, pp. 313–322. Basel, Birkhäuser (2009)

    Google Scholar 

  22. 22.

    Okada, S., Ricker, W.J., Pérez, E.A.S.: Optimal Domains and Integral Extensions of Operators acting in Function Spaces, Operator Theory Advances and Applications, vol. 180. Birkhäuser, Basel (2008)

    Google Scholar 

  23. 23.

    Sánchez Pérez, E.A., Szwedek, R.: Vector measures with values in \(\ell ^\infty (\Gamma )\) and interpolation of Banach lattices. J. Convex Anal. 25, 75–92 (2018)

    MathSciNet  Google Scholar 

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First author is supported by Grant MTM2011-23164 of the Ministerio de Economía y Competitividad (Spain). Second author is supported by Grant 284110 of CONACyT (Mexico). Fourth author is supported by Grant MTM2016-77054-C2-1-P of the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigaciones (Spain) and FEDER.

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Correspondence to E. A. Sánchez-Pérez.

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Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F. et al. Maximal Factorization of Operators Acting in Köthe–Bochner Spaces. J Geom Anal 31, 560–578 (2021).

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  • Köthe Bochner function space
  • Mixed norm space
  • Tensor product
  • Vector measure
  • Operator
  • Bilinear
  • Compactness
  • Fourier type

Mathematics Subject Classification

  • Primary 46E40
  • Secondary 46G10
  • 46E30