Maximal Factorization of Operators Acting in Köthe–Bochner Spaces

Abstract

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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Acknowledgements

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). https://doi.org/10.1007/s12220-019-00290-4

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Keywords

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

Mathematics Subject Classification

  • Primary 46E40
  • Secondary 46G10
  • 46E30