On the ρ-numerical radius of rank-one operators and parametrized Buzano-type inequalities
Abstract
Let ℋ be a complex Hilbert space and ρ>0. We establish the explicit formula ωρ(a⊗ b)=1/ρ|a| |b|+|1−1/ρ||⟨ a,b⟩|, a,b∈ℋ, for the ρ-numerical radius of rank-one operators, a unified expression interpolating between the operator norm, the numerical radius, and the spectral radius, which correspond to ρ=1,2, and ρ→∞, respectively. As an application, we derive a parametrized family of Buzano-type inequalities for four vectors. We extract from it an explicit closed-form bound and, under a reality condition which is automatically satisfied in real Hilbert spaces, we determine the best bound in the family in closed form. Examples show that the resulting bounds can be strictly sharper than both the Cauchy--Schwarz and the Buzano inequality, and the classical Buzano inequality is recovered as a boundary case, which yields an alternative proof of it.
Related
- Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators
- Octonionic Kittaneh Numerical Radius Inequalities
- Quaternionic Kittaneh Numerical Radius Inequalities
- Spectral and numerical radius preservers on pairs of unitarily similar operators
- Reversed and Refined Berezin Number Inequalities via Buzano’s Inequality and Polar Decomposition
BibTeX
@misc{stankovic2026ρnumerical,
title = {{On the $ρ$-numerical radius of rank-one operators and parametrized Buzano-type inequalities}},
author = {Hranislav Stanković},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00037},
eprint = {2609.00037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00037},
}
RIS
TY - UNPB
TI - On the ρ-numerical radius of rank-one operators and parametrized Buzano-type inequalities
AU - Hranislav Stanković
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00037
UR - https://arxiv.org/abs/2609.00037
AB - Let ℋ be a complex Hilbert space and ρ>0. We establish the explicit formula ω_ρ(a⊗ b)=1/ρ|a| |b|+|1−1/ρ||⟨ a,b⟩|, a,b∈ℋ, for the ρ-numerical radius of rank-one operators, a unified expression interpolating between the operator norm, the numerical radius, and the spectral radius, which correspond to ρ=1,2, and ρ→∞, respectively. As an application, we derive a parametrized family of Buzano-type inequalities for four vectors. We extract from it an explicit closed-form bound and, under a reality condition which is automatically satisfied in real Hilbert spaces, we determine the best bound in the family in closed form. Examples show that the resulting bounds can be strictly sharper than both the Cauchy--Schwarz and the Buzano inequality, and the classical Buzano inequality is recovered as a boundary case, which yields an alternative proof of it.
ER -