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Numerical Information of Hilbert Space Operators
Abstract
For a Hilbert space operator A∈ B(H), the numerical range map of A, denoted ϕA is the map sending unit vectors x in H to ⟨ Ax,x⟩. From this map we define the Numerical Information of A, denoted ΣA, which is the sigma algebra induced on the unit sphere by the numerical range map. One of the main results of the paper are that if A,B∈ B(H), and ϕB=f∘ ϕA, then f is affine. Our other main result is showing that ΣB⊂ΣA if and only if B=αRe(A)+βIm(A)+γI for some complex numbers α,β,γ.
Related
- The L2-Cartesian Numerical Radii of Hilbert Space Operators
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- Some Generalized Numerical Ranges of Real-linear Operators on Hilbert Spaces
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- Rhaly and generalized Hilbert operators between spaces of analytic functions
BibTeX
@misc{johnson2026numerical,
title = {{Numerical Information of Hilbert Space Operators}},
author = {Dylan Johnson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29826},
eprint = {2610.29826},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29826},
}
RIS
TY - UNPB
TI - Numerical Information of Hilbert Space Operators
AU - Dylan Johnson
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29826
UR - https://arxiv.org/abs/2610.29826
AB - For a Hilbert space operator A∈ B(H), the numerical range map of A, denoted ϕ_A is the map sending unit vectors x in H to ⟨ Ax,x⟩. From this map we define the Numerical Information of A, denoted Σ_A, which is the sigma algebra induced on the unit sphere by the numerical range map. One of the main results of the paper are that if A,B∈ B(H), and ϕ_B=f∘ ϕ_A, then f is affine. Our other main result is showing that Σ_B⊂Σ_A if and only if B=αRe(A)+βIm(A)+γI for some complex numbers α,β,γ.
ER -