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On the numerical radius parallelism and the numerical radius Birkhoff orthogonality

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Jiaye Bi, Huayou Xie, Yongjin Li

Functional Analysis · Submitted 2 Dec 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Notably, our findings reveal that numerical radius parallelism lacks transitivity, and numerical radius Birkhoff orthogonality is neither left nor right additive.

how it is stated · asserted 1.00 read the record →

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arXiv ID2412.01735 (version 1)
Pith IDZMMEVL3H, Pith's permanent record of this paper and its review
CitationHuayou Xie, Jiaye Bi, Yongjin Li. "On the numerical radius parallelism and the numerical radius Birkhoff orthogonality." arXiv:2412.01735 (2024). https://pith.science/paper/ZMMEVL3H
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,160 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.60 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.26 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper, we generalize the notions of numerical radius parallelism and numerical radius Birkhoff orthogonality, originally formulated for operators on Hilbert spaces, to operators on normed spaces. We then proceed to demonstrate their fundamental properties. Notably, our findings reveal that numerical radius parallelism lacks transitivity, and numerical radius Birkhoff orthogonality is neither left nor right additive. Additionally, we offer characterizations for both concepts. Furthermore, we establish a connection between numerical radius parallelism and numerical radius Birkhoff orthogonality.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper, we generalize the notions of numerical radius parallelism and numerical radius Birkhoff orthogonality, originally formulated for operators on Hilbert spaces, to operators on normed spaces. contribution 1.00 0.88 0.07 asserted 0.99
s1 We then proceed to demonstrate their fundamental properties. contribution 0.79 0.86 0.03 asserted 0.83
s2 Notably, our findings reveal that numerical radius parallelism lacks transitivity, and numerical radius Birkhoff orthogonality is neither left nor right additive. result 1.00 0.95 0.61 asserted 1.00
s3 Additionally, we offer characterizations for both concepts. contribution 0.96 0.95 0.06 asserted 0.99
s4 Furthermore, we establish a connection between numerical radius parallelism and numerical radius Birkhoff orthogonality. result 0.52 0.95 0.18 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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  • full text
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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

In this paper, we generalize the notions of numerical radius parallelism and numerical radius Birkhoff orthogonality, originally formulated for operators on Hilbert spaces, to operators on normed spaces. We then proceed to demonstrate their fundamental properties. Notably, our findings reveal that numerical radius parallelism lacks transitivity, and numerical radius Birkhoff orthogonality is neither left nor right additive. Additionally, we offer characterizations for both concepts. Furthermore, we establish a connection between numerical radius parallelism and numerical radius Birkhoff orthogonality.

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