Bibliographic record
Abstract
Abstract. InthisnotewepointouthowatheoremofGamelinandGarnettfromfunctiontheory can be used to establish a spectral mapping theorem for an arbitrary contractionand an associated class of H 1 -functions.1. IntroductionLet H be a separable, in nite dimensional, complex Hilbert space, and denote byL(H) the algebra of all bounded linear operators on H. For T in L(H), we write, as usual,˙(T) for the spectrum of T. Recall rst that if T satis es kTk 1, then T is calleda contraction (operator). If T is a contraction with the property that in its canonicaldecomposition T = T 1 T 2 ; where T 1 is completely nonunitary (c.n.u.) and T 2 is a unitaryoperator (cf. [11]), it happens that T 2 is an absolutely continuous unitary operator, then Tis called an absolutely continuous contraction (a.c.c.). We will denote by D the open unitdisc in the complex plane C, and by H 1 = H 1 (D) the algebra of all bounded analyticfunctions on D with the supremum norm k k 1 .If f is any function in H 1 and T is an a.c.c., then the operator f(T) is de ned bythe Nagy-Foias functional calculus (cf. [2, p.34]), and the relationship between ˙(f(T))and f(˙(T)) is sometimes quite complicated. This relationship has been studied fairlyextensively; for example, in [5], [8], [6], [7], and [1], where perhaps the best results are tobe found. On the other hand, if U is a singular unitary operator, then f(U) cannot even bede ned for an arbitrary f in H
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How this classification was reachedexpand
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 | 0.006 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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) | 0.003 | 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; both teacher heads agree on what is shown here.
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".