Entropy Numbers of Vector-Valued Sequence Operators

Definitions and mathematical statement

Let $X,Y$ be Banach spaces, $T:X\to Y$ bounded and linear, and $N\geq1$. For $0<p<\infty$, write \[ \|(x_i)_{i=1}^N\|_{\ell_p^N(X)} =\left(\sum_{i=1}^N\|x_i\|_X^p\right)^{1/p}, \] and use the maximum for $p=\infty$. These are quasi-norms for $p<1$; the source does not specify whether that range or quasi-Banach underlying spaces is intended. In the coordinatewise-lift interpretation supplied by the curation, \[ \widetilde T:\ell_p^N(X)\to\ell_q^N(Y),\qquad \widetilde T(x_1,\ldots,x_N)=(Tx_1,\ldots,Tx_N). \]

Use dyadic entropy numbers: for an operator $S:E\to F$, \[ e_k(S)=\inf\left\{\varepsilon>0:S(B_E)\subseteq \bigcup_{j=1}^{2^{k-1}}(y_j+\varepsilon B_F) \text{ for some }y_j\in F\right\},\qquad k\geq1. \] The unit balls refer to the declared norms or quasi-norms. Other entropy indexing conventions require an explicit adjustment.

The question is to relate $e_k(\widetilde T)$ to the entropy profile $(e_m(T))_{m\geq1}$, with the dependence on $N,p,q$ specified. A precise result might give sharp upper/lower estimates or asymptotic equivalences, including whether constants are uniform in $N,k$ and whether extra assumptions on $X,Y,T$ are needed. The source does not choose one such strength, a range of $p,q$, compactness assumptions, or geometric hypotheses. It should not be interpreted as asserting that one exact universal equality has already been identified.

The supplied triage dated 21 August 2026 reports broad two-sided estimates for vector-valued and sequence-space lifts, but no complete exact relation for the full extracted formulation. Their hypotheses and formulas are not reproduced. The record is therefore retained as a partially resolved comparison program, without importing a particular estimate or silently restricting it to one exponent regime.

Short English statement

Compare dyadic entropy numbers of the coordinatewise lift with the entropy profile of $T$, keeping track of $N,p,q$ and any geometry assumptions. The source leaves the exponent range and desired sharpness unspecified; its cited partial estimates are not given as a universal formula.

Sources

• Local UnsolvedMath registry, permanent ID $30000118$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; supplied verification reports a self-contained notation edit against the official report.
• "Compactness Problems in Interpolation Theory and Function Spaces (2005)," Oberwolfach Reports, pp. 2103--2132, DOI 10.4171/owr/2004/40: https://doi.org/10.4171/owr/2004/40. Proposed year and DOI refer to 2004, while the citation label says 2005.
• Supplied "Edmunds--Netrusov": https://onlinelibrary.wiley.com/doi/10.1002/mana.201100195.
• Supplied "Vector-valued entropy estimates": https://arxiv.org/abs/1309.7885. The other source is the original report; complete theorem statements are not supplied.
• Literature checked 21 August 2026; supplied status partially_solved. Registry created 25 August 2026 and updated 16 September 2026. This dated curation triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.