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

Abstract

We call an $n$-tuple of $2\times 2$ matrices a ``\textit{$2\times 2$ matrix ${\Gamma }_n$-contraction}'' if its joint eigenvalues (joint spectrum for a commuting tuple of matrices) are contained in ${\Gamma }_n$. In this short note on ${\Gamma }_n$-contraction, we produce two families of $2\times 2$ matrix ${\Gamma }_n$-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 ${\Gamma }_n$-isometric dilation.