Banach Spaces and Ramsey Theory — Problem 10

Definitions and mathematical statement

Let $\mathbb K$ denote the underlying scalar field. The supplied record does not specify whether $\mathbb K={\mathbb{R}}$ or $\mathbb K={\mathbb{C}}$ is intended; the formulation below is relative to a fixed common scalar field. Write $C[0,1]=C([0,1];\mathbb K)$ for the Banach space of continuous functions with norm $\|f\|_\infty=\sup_{t\in[0,1]}|f(t)|$, and write \[ c_0=\{(a_n)_{n\geq1}\in\mathbb K^{{\mathbb{N}}}:a_n\longrightarrow0\},\qquad \|(a_n)\|_\infty=\sup_{n\geq1}|a_n|. \] A Banach space contains a copy of another Banach space if it has a closed linear subspace isomorphic to that space by a bounded linear map with bounded inverse. A bounded linear operator $T:C[0,1]\to C[0,1]$ fixes a copy of a Banach space $Y$ if there exist a closed subspace $E\subseteq C[0,1]$, isomorphic to $Y$, and a constant $c>0$ such that \[ \|Tf\|_\infty\geq c\|f\|_\infty\qquad(f\in E). \] Thus $T|_E$ is an isomorphism onto its image.

Question as supplied (H. P. Rosenthal, source label [Ro2]). For every bounded linear operator $T:C[0,1]\to C[0,1]$ and every separable Banach space $X$ containing no copy of $c_0$, does \[ T\text{ fixes a copy of }X \quad\Longrightarrow\quad T\text{ fixes a copy of }C[0,1] \] hold?

For the continuous dual $C[0,1]^*$, define the adjoint by \[ T^*:C[0,1]^*\longrightarrow C[0,1]^*,\qquad T^*\varphi=\varphi\circ T. \] The source presents the following as an equivalent question under the same hypotheses: must $\operatorname{ran}(T^*)$ be nonseparable in its inherited dual norm?

Unresolved qualifications. The supplied statement imposes no infinite-dimensionality assumption on $X$, so its literal quantifiers include finite-dimensional spaces and the zero space. Whether a dimensional restriction was intended is a material formulation issue; none has been inserted here. The scalar field is also unspecified. The equivalence of the two targets is reported from the supplied source and is not independently established in this draft.

Short English statement

The source asks whether an operator on $C[0,1]$ that is an isomorphism on a copy of a separable Banach space containing no $c_0$ must also be an isomorphism on a copy of all of $C[0,1]$. It presents nonseparability of the adjoint range as an equivalent target. The absent dimensional restriction on $X$ and the unspecified scalar field require review.

Sources

• Local UnsolvedMath registry: permanent ID $800747$, exact source problem code catalogue entry, title Banach Spaces and Ramsey Theory --- Problem 10. The supplied registry labels the problem open and lists Dodos, Lopez-Abad as proposers, with proposed year 2010. These are supplied metadata, not an independently verified status assessment.
• Dodos, Lopez-Abad, Banach Spaces and Ramsey Theory: Some Open Problems (2010), Problem 10. Supplied source URL: https://arxiv.org/abs/1006.2668. The record reports access on 2026-10-16 and extraction by arxiv_latex; this draft does not claim independent access or verification.
• The supplied problem heading attributes Fixing properties of operators on $C[0,1]$ to H. P. Rosenthal, with reference label [Ro2]. No bibliographic entry resolving [Ro2] is supplied.