Fully nuclear and completely nuclear operators with applications to ℒ₁- and ℒ_{∞}-spaces

Stegall, C. P.; Retherford, J. R.

doi:10.1090/S0002-9947-1972-0415277-3

Fully nuclear and completely nuclear operators with applications to $\mathcal {L}_1-$ and $\mathcal {L}_\infty$-spaces
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by C. P. Stegall and J. R. Retherford

Trans. Amer. Math. Soc. 163 (1972), 457-492

DOI: https://doi.org/10.1090/S0002-9947-1972-0415277-3

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Abstract:

This paper is devoted to a study of the conjecture of A. Grothendieck that if $E$ and $F$ are Banach spaces and all operators from $E$ to $F$ are nuclear, then $E$ or $F$ must be finite dimensional. Two partial solutions are given to this conjecture (Chapters II and IV). In these chapters, operators we call fully nuclear and completely nuclear are introduced and studied. The principal result of these two chapters is that if $\mathcal {L}(E,F) = \operatorname {FN} (E,F)$ or $\mathcal {L}(E,F) = \operatorname {CN} (E,F)$ (and $E$ is isomorphic to a conjugate space or $E’$ contains a reflexive subspace in the latter case) then one of $E$, $F$ is finite dimensional. Two new properties of Banach spaces are introduced in Chapter I. We call these properties “sufficiently Euclidean” and “the two-series property". Chapter I provides the machinery for all the subsequent chapters. The principal part of the paper (Chapters II and V) is devoted to internal characterizations of the ${\mathcal {L}_\infty }$ - and ${\mathcal {L}_1}$-spaces of Lindenstrauss and Pełlczyhski. These characterizations are in terms of the behavior of various classes of operators from or into these spaces. As a by-product an apparently new characterization of Hilbert spaces is obtained. Finally, Chapter VI is a summary of the known characterizations of ${\mathcal {L}_1}$ - and ${\mathcal {L}_\infty }$ -spaces.

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