Classification and dilation for q-commuting 2 × 2 scalar matrices

Abstract

A tuple T _ = ( T 1 , … , T k ) of operators on a Hilbert space H is said to be q-commuting with ‖ q ‖ = 1 or simply q - commuting if there is a family of scalars q = { q i j ∈ C : | q i j | = 1 , q i j = q j i − 1 , 1 ≤ i < j ≤ k } such that T i T j = q i j T j T i for 1 ≤ i < j ≤ k . Moreover, if each q i j = − 1 , then T _ is called an anti-commuting tuple . A well-known result due to Holbrook [5] 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 up to similarity. Then we prove that a q -commuting tuple which is unitarily equivalent to any of these three types, admits a q ˜ -unitary dilation, where q ˜ ⊆ q ∪ { 1 } . A special emphasis is given to the dilation of an anti-commuting tuple of 2 × 2 scalar matrix contractions.