Von Neumann Type Inequalities for NC Operator Balls
Abstract
We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let Ω=𝔻Q be the NC operator ball of a linear pencil Q with linearly independent coefficients. If Ω(1) admits a transitive subgroup of automorphisms with NC lifts and Ω(1) has the bounded H∞-Gleason property at the origin, then |f(T)|≤ CΩ,n|f|H∞(Ω(1)) for every commuting tuple T∈Ω(n), with CΩ,n=(2ρΩKΩ)n−1; here KΩ is the Gleason constant and ρΩ is the sum of the suprema of the coordinate functions on Ω(1). The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where ρΩ=d, yielding an unconditional estimate (2dKΩ)n−1. For a general type I Cartan domain Rp,q, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded H∞-Gleason property at the origin, which we do not establish when min(p,q)≥ 2.
Related
- Bohr-type inequalities for certain integral transforms of operator-valued analytic functions on shifted disc
- Pólya's conjecture for higher-dimensional Neumann balls
- Full-radius dimension-free maximal inequalities for discrete Euclidean balls
- A Brezis--Merle inequality for the biharmonic operator with Dirichlet boundary conditions in four-dimensional balls
- On the spectrum of the biharmonic Steklov operator with Neumann boundary conditions acting on forms of the unit Euclidean ball
BibTeX
@misc{kiri2026neumann,
title = {{Von Neumann Type Inequalities for NC Operator Balls}},
author = {Ran Kiri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20355},
eprint = {2610.20355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20355},
}
RIS
TY - UNPB
TI - Von Neumann Type Inequalities for NC Operator Balls
AU - Ran Kiri
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20355
UR - https://arxiv.org/abs/2610.20355
AB - We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let Ω=𝔻_Q be the NC operator ball of a linear pencil Q with linearly independent coefficients. If Ω(1) admits a transitive subgroup of automorphisms with NC lifts and Ω(1) has the bounded H^∞-Gleason property at the origin, then |f(T)|≤ C_Ω,n|f|_H^∞(Ω(1)) for every commuting tuple T∈Ω(n), with C_Ω,n=(2ρ_ΩK_Ω)^n−1; here K_Ω is the Gleason constant and ρ_Ω is the sum of the suprema of the coordinate functions on Ω(1). The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where ρ_Ω=d, yielding an unconditional estimate (2dK_Ω)^n−1. For a general type I Cartan domain R_p,q, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded H^∞-Gleason property at the origin, which we do not establish when min(p,q)≥ 2.
ER -