PreprintCC BY 4.0arXiv
Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators
Abstract
We study the numerical radius ω(T) of a bounded linear operator T on a complex Hilbert space. Write T=A+iB, where A and B are the real and imaginary parts of T. A central result is the exact formula ω(T)=π/2sup|x|=1∫01 |⟨ (Acos πt+Bsin πt)x,x⟩| dt . For every n∈ℕ, we also prove the power form ω2n(T)=4n/binom2nn sup|x|=1∫01 ⟨ (Acos πt+Bsin πt)x,x⟩2n dt . This yields sharp upper bounds; for n=1, together with the matching integral lower bound, it recovers Kittaneh's classical two-sided inequality 1/4|T*T+TT*|≤ ω2(T)≤ 1/2|T*T+TT*|. Furthermore, we obtain exact numerical radius formulas for some block operator matrices, including a 3×3 nilpotent matrix and longer alternating paths. The factor π/2 is best possible.
Related
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BibTeX
@misc{audeh2026integral,
title = {{Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators}},
author = {Wasim Audeh and Fuad Kittaneh and Hamid Reza Moradi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15242},
eprint = {2610.15242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15242},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators
AU - Wasim Audeh
AU - Fuad Kittaneh
AU - Hamid Reza Moradi
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15242
UR - https://arxiv.org/abs/2610.15242
AB - We study the numerical radius ω(T) of a bounded linear operator T on a complex Hilbert space. Write T=A+iB, where A and B are the real and imaginary parts of T. A central result is the exact formula ω(T)=π/2sup_|x|=1∫_0^1 |⟨ (Acos πt+Bsin πt)x,x⟩| dt . For every n∈ℕ, we also prove the power form ω^2n(T)=4^n/binom2nn sup_|x|=1∫_0^1 ⟨ (Acos πt+Bsin πt)x,x⟩^2n dt . This yields sharp upper bounds; for n=1, together with the matching integral lower bound, it recovers Kittaneh's classical two-sided inequality 1/4|T^*T+TT^*|≤ ω^2(T)≤ 1/2|T^*T+TT^*|. Furthermore, we obtain exact numerical radius formulas for some block operator matrices, including a 3×3 nilpotent matrix and longer alternating paths. The factor π/2 is best possible.
ER -