Characterizations of Disjointness preserving operators on vector-valued function spaces
Bibliographic record
Abstract
We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper C 0 left-parenthesis upper X comma upper E right-parenthesis right-arrow upper C 0 left-parenthesis upper Y comma upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T : C_0(X, E) \to C_0(Y, F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact (resp. completely continuous) if and only if <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper T f equals sigma-summation Underscript n Endscripts delta Subscript x Sub Subscript n Subscript Baseline circled-times h Subscript n Baseline left-parenthesis f right-parenthesis for all f element-of upper C 0 left-parenthesis upper X comma upper E right-parenthesis comma EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mi>n</mml:mi> </mml:munder> <mml:msub> <mml:mi> δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo> ⊗ </mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="1em"/> <mml:mtext>for all </mml:mtext> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{align*} Tf = \sum _n \delta _{x_n} \otimes h_n (f) \quad \text {for all } f \in C_0(X,E), \end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n Baseline colon upper Y right-arrow upper B left-parenthesis upper E comma upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h_n : Y \to B(E,F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n Baseline left-parenthesis y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h_n(y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact (resp. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is uniformly completely continuous).
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".