Nuclear operators on the Banach space of totally measurable functions.

Let \Sigma be a \sigma-algebra of subsets of a set \Omega and X be a Banach space, and let B(\Sigma,X) stand for the Banach space of all X-valued totally \Sigma-measurable functions on \Omega, equipped with the sup-norm. We study nuclear operators T:B(\Sigma,X)\rightarrow Y between Banach spaces B(\Sigma,X) and Y in terms of their representing measures m:\Sigma \rightarrow \mathcal {N}(X,Y), where \mathcal {N}(X,Y) stands for the Banach space of all nuclear operators U:X\rightarrow Y, equipped with the nuclear norm. We establish the relationship between nuclearity of a bounded linear operator T:B(\Sigma,X)\rightarrow Y and nuclearity of its conjugate operator. [ABSTRACT FROM AUTHOR]