Nuclear operators on the Banach space of totally measurable functions

Nowak, Marian

doi:10.1090/proc/16886

Nuclear operators on the Banach space of totally measurable functions
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by Marian Nowak;

Proc. Amer. Math. Soc. 152 (2024), 3977-3990

DOI: https://doi.org/10.1090/proc/16886

Published electronically: July 29, 2024

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Abstract:

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.

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