Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract Every norm v on C n induces two norm numerical ranges on the algebra M n of all n × n complex matrices, the spatial numerical range where v D is the norm dual to v , and the algebra numerical range where is the set of states on the normed algebra M n under the operator norm induced by v . For a symmetric norm v , we identify all linear maps on M n that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e. , linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if v is not the ℓ 1 , ℓ 2 , or ℓ ∞ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e. , of the form A ⟼ Q * AQ , where Q is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ℓ 1 and the ℓ ∞ norms, the results are quite different.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it