Theory of Q-commuting contractions: Joint reducing subspaces and orthogonal decompositions

A family [Formula: see text] of operators acting on a Hilbert space [Formula: see text] is said be Q-commuting if there exists a commuting family [Formula: see text] of unitaries such that [Formula: see text] and [Formula: see text] for all [Formula: see text] in [Formula: see text] with [Formula: see text]. The family [Formula: see text] is said to be doubly Q-commuting if it is [Formula: see text]-commuting and [Formula: see text] for all [Formula: see text] in [Formula: see text] with [Formula: see text]. In this paper, we obtain the following main results. (i) It is well known that any number of doubly commuting isometries admit a Wold-type decomposition. Also, examples from the literature show that such decomposition does not hold in general for commuting isometries. We generalize this result to any doubly [Formula: see text]-commuting family of contractions and get a canonical decomposition for them. Then we obtain a Wold-type decomposition for any doubly [Formula: see text]-commuting family of isometries as a special case. (ii) Further, we show that a similar decomposition is possible for an arbitrary doubly [Formula: see text]-commuting family of c.n.u. contractions, where all members jointly orthogonally decompose into pure isometries (i.e. unilateral shifts) and completely non-isometry (c.n.i.) contractions. (iii) We present a decomposition result for [Formula: see text]-commuting tuples of contractions which is analogous to Burdak’s work on commuting contractions. (iv) Also, we provide decomposition results for a pair of contractions [Formula: see text] satisfying any of the following relations: [Formula: see text] where [Formula: see text] is any unitary that commutes with the product [Formula: see text].