Hyponormality of Generalized Cesàro Matrices of Every Positive Integer Order
Abstract
For a positive integer m and a real parameter α>−1, consider the generalized Cesàro matrix on ℓ2(ℕ0) with entries m(n−j+1)m−1/(n+α+1)m for j≤ n. We give a computer-assisted proof that this operator is hyponormal if and only if α≥0. The main algebraic ingredient is an explicit decomposition of an auxiliary defect operator into a positive diagonal operator and at most m rank-one operators. Discrete Gram polynomials determine the signs and coefficients of these terms. For nonnegative parameters, weighted estimates reduce positivity to scalar inequalities. A uniform analytic estimate covers every m≥32768, while finite rational certificates cover the remaining orders and entire parameter intervals, without parameter sampling or truncating an infinite operator. The proof also gives a uniform weighted lower bound for the self-commutator. For −1<α<0, finite-support vectors give negative quadratic forms for every order.
Related
- Simultaneous Unimodularity of Positive Integer Matrices and Their Entrywise Cubes
- Banach-Cesaro positive operator-valued measure
- Complete mapping criteria for generalized Cesàro operators between Hardy spaces
- Integer symplectic matrices in expanding boxes
- D(N)-quadruples in upper-triangular 2×2 integer matrices
BibTeX
@misc{ma2026hyponormality,
title = {{Hyponormality of Generalized Cesàro Matrices of Every Positive Integer Order}},
author = {Yicen Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00193},
eprint = {2610.00193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00193},
}
RIS
TY - UNPB
TI - Hyponormality of Generalized Cesàro Matrices of Every Positive Integer Order
AU - Yicen Ma
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00193
UR - https://arxiv.org/abs/2610.00193
AB - For a positive integer m and a real parameter α>−1, consider the generalized Cesàro matrix on ℓ^2(ℕ_0) with entries m(n−j+1)_m−1/(n+α+1)_m for j≤ n. We give a computer-assisted proof that this operator is hyponormal if and only if α≥0. The main algebraic ingredient is an explicit decomposition of an auxiliary defect operator into a positive diagonal operator and at most m rank-one operators. Discrete Gram polynomials determine the signs and coefficients of these terms. For nonnegative parameters, weighted estimates reduce positivity to scalar inequalities. A uniform analytic estimate covers every m≥32768, while finite rational certificates cover the remaining orders and entire parameter intervals, without parameter sampling or truncating an infinite operator. The proof also gives a uniform weighted lower bound for the self-commutator. For −1<α<0, finite-support vectors give negative quadratic forms for every order.
ER -