# On hereditarily reflexive Banach spaces

It was proved by W.B. Johnson and H.P. Rosenthal [Studia Math. 43 (1972), 77–92] that every Banach space $$X$$ with $$X^{**}$$ separable is hereditarily reflexive: every infinite dimensional closed subspace of $$X$$ contains an infinite dimensional reflexive subspace.

Suppose that $$X$$ is separable and $$X^{**}/X$$ reflexive. Is $$X$$ hereditarily reflexive?

Of course, we would have a positive answer if each infinite dimensional closed subspace of such a space $$X$$ contains an infinite dimensional subspace $$Y$$ with $$Y^{**}$$ separable.

• Gowers' dichotomy theorem reduces to the case that $𝑋$ is HI. Have you asked Argyros whether there exists a hereditarily non reflexive HI space $X$ with $X^{**}/X$ reflexive? Jul 14, 2020 at 18:11
• I have looked at Spiros A. Argyros, Alexander D. Arvanitakis, Andreas G. Tolias. Saturated extensions, the attractors method and Hereditarily James Tree Spaces. pp. 1-90 in "Methods in Banach space theory". J.M.F. Castillo and W.B. Johnson eds. Cambridge University Press, 2006. Jul 14, 2020 at 19:04

One of these spaces $$X$$ has the additional property that $$X^{**}/X$$ is isomorphic to $$\ell_2(\Gamma)$$ with $$\Gamma$$ uncountable: see Proposition 5.4 in the paper.