A version of Lomonosov’s theorem for collections of positive operators
Bibliographic record
Abstract
It is known that for every Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and every proper <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W upper O upper T"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>O</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">WOT</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -closed subalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a compact operator, then it is not transitive; that is, there exist non-zero <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x\in X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper X Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in X^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle f comma upper T x mathematical right-angle equals 0"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mi>x</mml:mi> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle f,Tx\rangle =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T element-of script upper A"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T\in \mathcal A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the case of algebras of adjoint operators on a dual Banach space, V. Lomonosov extended this result as follows: without having a compact operator in the algebra, one has <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue mathematical left-angle f comma upper T x mathematical right-angle EndAbsoluteValue less-than-or-equal-to double-vertical-bar upper T Subscript asterisk Baseline double-vertical-bar Subscript e"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.2em" minsize="1.2em">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mi>x</mml:mi> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-CLOSE"> <mml:mo maxsize="1.2em" minsize="1.2em">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo> ≤ </mml:mo> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\bigl \lv
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".