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:

$$\begin{aligned} T_1T_2=QT_2T_1 \text { or }T_1T_2=T_2QT_1 \text { or }T_1T_2=T_2T_1Q. \end{aligned}$$

We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88–90, 1963) for pairs of commuting contractions. In particular, for \(Q=qI_{\mathcal {H}}\), 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 (Acta Sci Math (Szeged) 22:106–111, 1961), we construct a class of n-tuples of q-commuting contractions and find isometric dilations explicitly for the class.