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

  • J. M. CalabuigEmail author
  • M. Fernández-Unzueta
  • F. Galaz-Fontes
  • E. A. Sánchez-Pérez
Original Paper


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.


Product measure Banach function space Bilinear operator Tensor product Factorization 

Mathematics Subject Classification

46E30 28A35 47H60 



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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Centro de Investigación em Matemáticas, A.C.GuanajuatoMexico

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