Completability and optimal factorization norms in tensor products of Banach function spaces
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.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas