**Question 6.25.** *Do there exist $\ell_2$-saturated $d_2$-HI spaces?*

After this article was submitted, Argyros, Manoussakis and Motakis announced in [5] that they were able to build an $\ell_2$-saturated $d_2$-HI space, thus giving a positive answer to Question 6.25. The construction will be published in a forthcoming paper, and the preprint [5] exposes the construction of analogues of that space.

We now come back to Question 6.24. We conjecture that the answer to this question is negative, and we actually have the following stronger conjecture:

**Conjecture 6.26.** *A Banach space cannot be simultaneously $d_2$-minimal and $d_2$-HI.*

This conjecture is motivated by the fact that the $d_2$-HI property is a weakening of the HI property, and it is known that HI spaces have many different subspaces, up to isomorphism. For example, Gowers–Maurey’s Theorem 1.18 says that HI spaces cannot be isomorphic to any proper subspace of themselves. This implies, in particular, that they cannot be minimal. It would be tempting to adapt Gowers–Maurey’s approach to $d_2$-HI spaces. Note that, however, in the case of $d_2$-HI spaces, we cannot hope to have a result as strong as Gowers–Maurey’s one, since both spaces presented in Example 6.21 and in Example 6.22 are isomorphic to their hyperplanes and even, to their direct sum with $\ell_2$. However, we can hope that these spaces cannot be isomorphic to “too deep” subspaces of themselves. This is at least the case for our first example, as shown by the following lemma:

**Lemma 6.27.** Let $Y$ be an HI space and let $X = Y \oplus \ell_2$. Then every subspace of $X$ that is isomorphic to $X$ is complemented in $X$ by a (finite- or infinite-dimensional) Hilbertian subspace.

*Proof.* Denote by $P_Y: X \to Y$ and $P_{\ell_2}: X \to \ell_2$ the projections. Let $U \subseteq X$ be an isomorphic copy of $X$; we can write $U = V \oplus W$, where $V \cong Y$ and $W \cong \ell_2$. Suppose that $(P_Y)_{\upharpoonright V}$ is infinitely singular. Then by Proposition 6.20, we can find a subspace $V'' \subseteq V$ on which $P_Y$ has small norm. In particular, $P_{\ell_2}$ would induce an isomorphism between $V''$ and a subspace of $\ell_2$, a contradiction. Thus, $(P_Y)_{\upharpoonright V}$ is not infinitely singular: we can find a finite-codimensional subspace $V'$ of $V$ such that $P_Y$ induces an isomorphism between $V'$ and $P_Y(V')$.

Observe that $V' \cong P_Y(V')$, and that $V'$ and $P_Y(V')$ are respectively subspaces of $V$ and $Y$, that are HI and isomorphic. By Gowers–Maurey’s Theorem 1.18, we deduce that the codimension of $P_Y(V')$ in $Y$ is equal to the codimension of $V'$ in $V$, so is finite. So write $Y = P_Y(V') \oplus F$, where $F$ has finite dimension. We have $P_Y(V') \subseteq V + \ell_2$, so $Y \subseteq V + \ell_2 + F$, so $X = V + \ell_2 + F = U + \ell_2 + F$. Letting $Z$ be a complement of $U \cap (\ell_2 + F)$ in $\ell_2 + F$, we get that $Z$ is Hilbertian and that $X = U \oplus Z$.

□

**Question 6.28.** Let $X$ be $d_2$-HI and let $Y$ be a subspace of $X$ which is isomorphic to $X$. Does it follow that $Y$ is complemented by a (finite- or infinite-dimensional) Hilbertian subspace?
**Acknowledgments.** The authors would like to thank F. Le Maître and S. Todorcevic for useful suggestions, and Ch. Rosendal and O. Kurka for comments on drafts of the paper. They also thank the anonymous referees for comments leading to the improvement of this article.

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W. CUELLAR CARRERA, DEPARTAMENTO DE MATEMÁTICA, INSTITUTO DE MATEMÁTICA E ESTATÍSTICA, UNIVERSIDADE DE SÃO PAULO, RUA DO MATÃO 1010, 05508-090 SÃO PAULO SP, BRAZIL

*E-mail address: cuellar@ime.usp.br*

N. DE RANCOURT, UNIVERSITÄT WIEN, INSTITUT FÜR MATHEMATIK, KURT GÖDEL RESEARCH CENTER, AUGASSE 2-6, UZA 1 – BUILDING 2, 1090 WIEN, AUSTRIA

*Current adress: Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 49/83, 186 75 Praha 8, CZECH REPUBLIC*

*E-mail address: rancourt@karlin.mff.cuni.cz*

V. FERENCZI, DEPARTAMENTO DE MATEMÁTICA, INSTITUTO DE MATEMÁTICA E ESTATÍSTICA, UNIVERSIDADE DE SÃO PAULO, RUA DO MATÃO 1010, 05508-090 SÃO PAULO SP, BRAZIL, AND EQUIPE D'ANALYSE FONCTIONNELLE, INSTITUT DE MATHÉMATIQUES DE JUSSIEU, SORBONNE UNIVERSITÉ - UPMC, CASE 247, 4 PLACE JUSSIEU, 75252 PARIS CEDEX 05, FRANCE

*E-mail address: ferenczi@ime.usp.br*