Paulsen and Projection Problems for Banach Spaces

K. Mahesh Krishna

uncertainhigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper explicitly formulates the Banach-space Paulsen problem using approximate Schauder frames (Problems 2.11–2.13) and leaves it open, while reviewing the Hilbert-space resolution and stating an n-independent bound f(ε,n,d)=20εd^2 (Theorem 1.24). There is no claim of a solution in Banach spaces, and the equivalence to a Banach-space projection problem is posed as open (Problem 2.16). These points appear verbatim in the PDF, so the status is open as of the paper’s date (Jan 5, 2022) .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

A concise, problem-formulating note: it clearly states Banach-space analogues of the Paulsen and projection problems, faithfully reviews the Hilbert-space resolution (including an n-independent bound), and collects natural variants and reductions. It does not claim results beyond the current state of knowledge, and it transparently leaves the Banach-space questions open, including the Paulsen–projection equivalence. Minor additions clarifying the choice of definitions (e.g., spectral inclusion vs. operator-norm control) and providing illustrative examples would improve readability for newcomers.