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 H, a pair of bounded operators (T₁,T₂) on H is said to be Q-commuting if one of the following holds: (Formula presented.) 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_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≥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.