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The Families of Γn-Contractions Having Non-commutative Fundamental Operators
Abstract
We call an n-tuple of 2 × 2 matrices a ``\textit{2 × 2 matrix Γn-contraction}'' if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in Γn. In this short note on Γn-contraction, we produce two families of 2 × 2 matrix Γn-contractions whose fundamental operators do not satisfy the commutativity conditions and FiF*n−j − FjF*n−i = F*n−jFi − F*n−iFj, 1 ≤ i, j ≤ n−1, which shows that the commutativity as well as the conditions on fundamental operators mentioned above are sufficient, but not necessary, for the existence of Γn-isometric dilation.
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BibTeX
@misc{paul2026families,
title = {{The Families of $Γ_n$-Contractions Having Non-commutative Fundamental Operators}},
author = {Bhaskar Paul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39711},
eprint = {2609.39711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39711},
}
RIS
TY - UNPB
TI - The Families of Γ_n-Contractions Having Non-commutative Fundamental Operators
AU - Bhaskar Paul
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39711
UR - https://arxiv.org/abs/2609.39711
AB - We call an n-tuple of 2 × 2 matrices a ``\textit{2 × 2 matrix Γ_n-contraction}'' if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in Γ_n. In this short note on Γ_n-contraction, we produce two families of 2 × 2 matrix Γ_n-contractions whose fundamental operators do not satisfy the commutativity conditions and F_iF^*_n−j − F_jF^*_n−i = F^*_n−jF_i − F^*_n−iF_j, 1 ≤ i, j ≤ n−1, which shows that the commutativity as well as the conditions on fundamental operators mentioned above are sufficient, but not necessary, for the existence of Γ_n-isometric dilation.
ER -