On Characterizations of Nuclear Spaces and Quasi-Integral Operators.

H. Millington

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Operators whose adjoints are quasi p-nuclear

J. M. Delgado, C. Piñeiro, E. Serrano (2010)

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For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xₙ) in X with $K \subseteq \sum ₙ α ₙ x ₙ : (α ₙ) \in B_{ℓ_{p^{'}}}$ . We prove that an operator T: X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T* is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

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