The Families of ${\mathrm{\Gamma }}_n$-Contractions Having Non-commutative Fundamental Operators

We call an $n$-tuple of $2\times 2$ matrices a ``$2\times 2\mathit{\text{matrix}}{\mathrm{\Gamma }}_n\mathit{\text{-contraction}}$‘’ if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in ${\mathrm{\Gamma }}_n$. In this short note on ${\mathrm{\Gamma }}_n$-contraction, we produce two families of $2\times 2$ matrix ${\mathrm{\Gamma }}_n$-contractions whose fundamental operators do not satisfy the commutativity conditions and \begin{equation*} \begin{aligned} F_iF^_{n-j} - F_jF^{n-i} = F^*{n-j}F_i - F^_{n-i}F_j, \quad 1 \le i, j \le n-1, \end{aligned} \end{equation} which shows that the commutativity as well as the conditions on fundamental operators mentioned above are sufficient, but not necessary, for the existence of ${\mathrm{\Gamma }}_n$-isometric dilation.