## Abstract

Given a bounded operator Q on a Hilbert space H, a pair of bounded operators (T_{1},T_{2}) 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.

Original language | English |
---|---|

Article number | 131 |

Journal | Complex Analysis and Operator Theory |

Volume | 18 |

Issue number | 6 |

DOIs | |

State | Published - 1 Sep 2024 |

## Keywords

- 32A70
- 46E22
- 47A13
- 47A20
- 47A56
- 47B32
- 47B38
- Brehmer positivity
- Hardy space
- Isometric dilation
- q-Commuting contractions
- Q-commuting contractions
- Szegö positivity

## ASJC Scopus subject areas

- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics