A generalization of Ando's dilation, and isometric dilations for a class of tuples of $q$-commuting contractions

Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1,T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: $$T_1{T}_2=Q{T}_2{T}_1\text{ or }{T}_1{T}_2=T_{2}Q{T}_1\text{ or }{T}_1{T}_2=T_2{T}_{1}Q\mathrm{.}$$ We give an explicit construction of isometric dilations for pairs of $Q$-commuting contractions for unitary $Q$, which generalizes the isometric dilation of Ando [2] for pairs of commuting contractions. In particular, for $Q=q{I}_{\mathcal{H}}$, where $q$ is a complex number of modulus $1$, this gives, as a corollary, an explicit construction of isometric dilations for pairs of $q$-commuting contractions which are well studied. There is an extended notion of $q$-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an $n$-tuple of $q$-commuting contractions, where $n\ge 3$. Generalizing the class of commuting contractions considered by Brehmer [8], we construct a class of $n$-tuples of $q$-commuting contractions and find isometric dilations explicitly for the class.