Discrepancies in Euclidean Operator Radii in Hilbert C∗-Modules
Bibliographic record
Abstract
In this research, we establish precise limits for the Euclidean operator radius of two bounded linear operators operating within a Hilbert C∗-module over A. Furthermore, our work establishes a connection between these limits and recent research findings that provide accurate upper and lower bounds for the numerical radius of linear operators. The primary objective of this investigation is to explore various specific scenarios of interest and extend existing inequalities found in the literature to encompass the Euclidean radius of two operators in a Hilbert A-module. Additionally, our study presents conclusions that reveal relationships between the operator norm, the typical numerical radius of a composite operator, and the Euclidean operator radius. Furthermore, we introduce several new inequalities involving the Euclidean numerical radius and Euclidean operator norm of 2-tuple operators. These inequalities offer both lower and upper bounds for the Euclidean numerical radius of 2-tuple operators, as well as for the sum and product of 2-tuple operators. We also delve into the study of Euclidean numerical radius inequalities for 2×2 operator matrices whose entries consist of 2-tuple operators, leading to the derivation of some Euclidean operator radius inequalities. Additionally, we establish an inequality for the Euclidean operator norm of 2×2 operator matrices.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".