Completability and optimal factorization norms in tensor products of Banach function spaces

Abstract

Given \(\sigma \)-finite measure spaces \((\Omega _1,\Sigma _1, \mu _1)\) and \((\Omega _2,\Sigma _2,\mu _2)\), we consider Banach spaces \(X_1(\mu _1)\) and \(X_2(\mu _2)\), consisting of \(L^0 (\mu _1)\) and \(L^0 (\mu _2)\) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product \(X_1(\mu _1) \otimes _\pi X_2(\mu _2)\) is continuously included in the metric space of measurable functions \(L^0(\mu _1 \otimes \mu _2)\). In particular, we prove that the elements of the completion of the projective tensor product of \(L^p\)-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally show that given a bounded linear operator \(T:X_1(\mu _1) \otimes _\pi X_2(\mu _2) \rightarrow E\) (where E is a Banach space), a norm can be found for T to be bounded, which is ‘minimal’ with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.

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Correspondence to J. M. Calabuig.

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J. M. Calabuig and M. Fernández-Unzueta were supported by Ministerio de Economía, Industria y Competitividad (Spain) under project MTM2014-53009-P. M. Fernández-Unzueta was also suported by CONACyT 284110. F. Galaz-Fontes was supported by Ministerio de Ciencia e Innovación (Spain) and FEDER under project MTM2009-14483-C02-01. E. A. Sánchez Pérez was supported by Ministerio de Economía, Industria y Competitividad (Spain) and FEDER under project MTM2016-77054-C2-1-P.

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Calabuig, J.M., Fernández-Unzueta, M., Galaz-Fontes, F. et al. Completability and optimal factorization norms in tensor products of Banach function spaces. RACSAM 113, 3513–3530 (2019). https://doi.org/10.1007/s13398-019-00711-7

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Keywords

  • Product measure
  • Banach function space
  • Bilinear operator
  • Tensor product
  • Factorization

Mathematics Subject Classification

  • 46E30
  • 28A35
  • 47H60