2024/10/21 by Sourav Pal, Pal, Sourav, Prajakta Sahasrabuddhe +3
Engineering · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Random Matrices and Applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2410.16134
openalex publication_date 2024/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A tuple \underlineT=(T1, \dotsc, Tk) of operators on a Hilbert space \mathcal H is said to be q-commuting with ‖q‖=1 or simply q-commuting if there is a family of scalars q=\qij ∈ \mathbb C : |qij|=1, qij=qji-1, 1 ≤ i < j ≤ k \ such that Ti Tj =qijTj Ti for 1 ≤ i < j ≤ k. Moreover, if each qij=-1, then \underlineT is called an anti-commuting tuple. A well-known result due to Holbrook \citeHolbrook states that a commuting k-tuple consisting of 2 × 2 scalar matrix contractions always dilates to a commuting k-tuple of unitaries for any k≥ 1. To find a generalization of this result for a q-commuting k-tuple of 2× 2 scalar matrix contractions, we first classify such tuples into three types upto similarity. Then we prove that a q-commuting tuple which is unitarily equivalent to any of these three types, admits a \widetildeq-unitary dilation, where \widetilde q ⊆ q ∪ \1\. A special emphasis is given to the dilation of an anti-commuting tuple of 2 × 2 scalar matrix contractions.