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On traces of Bochner representable operators on the space of bounded measurable functions
Published online by Cambridge University Press: 11 January 2024
Abstract
Let Σ be a σ-algebra of subsets of a set Ω and $B(\Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let $\tau(B(\Sigma),ca(\Sigma))$ denote the natural Mackey topology on $B(\Sigma)$. It is shown that a linear operator T from $B(\Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space $(B(\Sigma),\tau(B(\Sigma),ca(\Sigma)))$ and the Banach space E. We derive a formula for the trace of a Bochner representable operator $T:B({\cal B} o)\rightarrow B({\cal B} o)$ generated by a function $f\in L^1({\cal B} o, C(\Omega))$, where Ω is a compact Hausdorff space.
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- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 224 - 235
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.