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A generalization of the Powers-Størmer-Ogata inequality
Abstract
We show that for any positive operator monotone function f the inequality ∫[0,∞) f(t) d lVert EΔφ,ψ(t) ξψrVert2 + f'∞ φ(1−s(ψ)) ≥ f(1)/2 ( φ(1) + ψ(1) − lVert φ− ψrVert ) holds, where φ, ψ∈ ℳ*,+ are two normal positive linear functionals on a von Neumann algebra ℳ, and Δφ,ψ is the associated relative modular operator. Choosing f(t) = ts with s ∈ [0,1], the Powers-Størmer-Ogata inequality is recovered.
Related
BibTeX
@misc{frob2026generalization,
title = {{A generalization of the Powers-Størmer-Ogata inequality}},
author = {Markus B. Fröb},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16184},
eprint = {2609.16184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16184},
}
RIS
TY - UNPB
TI - A generalization of the Powers-Størmer-Ogata inequality
AU - Markus B. Fröb
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16184
UR - https://arxiv.org/abs/2609.16184
AB - We show that for any positive operator monotone function f the inequality ∫_[0,∞) f(t) d lVert E^Δ_φ,ψ(t) ξ_ψrVert^2 + f'_∞ φ(1−s(ψ)) ≥ f(1)/2 ( φ(1) + ψ(1) − lVert φ− ψrVert ) holds, where φ, ψ∈ ℳ_*,+ are two normal positive linear functionals on a von Neumann algebra ℳ, and Δ_φ,ψ is the associated relative modular operator. Choosing f(t) = t^s with s ∈ [0,1], the Powers-Størmer-Ogata inequality is recovered.
ER -