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.