Structural theory of Banach spaces and its interplay with analysis and probability · 1983-ana-214

Remaining questions recorded by the source

These are retained within the parent record and are not counted as additional index entries.

Question 1

Let $\ell_2=\ell_2(\mathbb R)$, and define $\operatorname{KP}\colon\ell_2\to\mathbb R^{\mathbb N}$ by $\operatorname{KP}(0)=0$ and

$$\operatorname{KP}(x)_n=2x_n\log\!\left(\frac{|x_n|}{\lVert x\rVert_2}\right)$$

for $x\ne0$, with $0\log 0=0$. Let

$$Z_2=\bigl\{(\omega,x)\in\mathbb R^{\mathbb N}\times\ell_2:\omega-\operatorname{KP}(x)\in\ell_2\bigr\}, \qquad \lVert(\omega,x)\rVert_{Z_2}=\lVert\omega-\operatorname{KP}(x)\rVert_2+\lVert x\rVert_2,$$

where $Z_2$ is equipped with any equivalent Banach norm. Determine whether there exists a closed linear subspace $H\subset Z_2$ of codimension one that admits a bounded real-linear operator $J\colon H\to H$ such that $J^2=-I_H$.

Source reference: Jesús M. F. Castillo, Wilson Cuellar, Valentin Ferenczi, and Yolanda Moreno, “Complex structures on twisted Hilbert spaces,” Israel Journal of Mathematics 222(2) (2017), 787–814, Problem 1. DOI: https://doi.org/10.1007/s11856-017-1605-9; arXiv:1511.05867.

Question 2

Let $\ell_2=\ell_2(\mathbb R)$, and define $\operatorname{KP}\colon\ell_2\to\mathbb R^{\mathbb N}$ by $\operatorname{KP}(0)=0$ and

$$\operatorname{KP}(x)_n=2x_n\log\!\left(\frac{|x_n|}{\lVert x\rVert_2}\right)$$

for $x\ne0$, with $0\log 0=0$. Let

$$Z_2=\bigl\{(\omega,x)\in\mathbb R^{\mathbb N}\times\ell_2:\omega-\operatorname{KP}(x)\in\ell_2\bigr\}, \qquad \lVert(\omega,x)\rVert_{Z_2}=\lVert\omega-\operatorname{KP}(x)\rVert_2+\lVert x\rVert_2,$$

where $Z_2$ is equipped with any equivalent Banach norm, and let $i\colon\ell_2\to Z_2$ be the canonical embedding $i(y)=(y,0)$. Suppose that $H\subset Z_2$ is a closed linear subspace of codimension one containing $i(\ell_2)$, that $j\colon i(\ell_2)\to i(\ell_2)$ and $J\colon H\to H$ are bounded real-linear operators satisfying $j^2=-I_{i(\ell_2)}$ and $J^2=-I_H$, and that $j$ is viewed as an operator from $i(\ell_2)$ into $H$. Determine whether $J|_{i(\ell_2)}-j\colon i(\ell_2)\to H$ must fail to be strictly singular, where an operator is strictly singular if its restriction to every infinite-dimensional subspace of its domain fails to be an isomorphism onto its image.

Source reference: W. Cuellar Carrera and V. Ferenczi, “On complex structures and uniqueness of algebra norms in Banach spaces,” arXiv:2602.05045 [math.FA] (2026), Question 7.4. https://arxiv.org/abs/2602.05045.