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

A family T = {Ti : i ∈ J} of operators acting on a Hilbert space H is said be Qcommuting if there exists a commuting family Q = {Qij ∈ B(H ) : Qij = Q*ji, i ≠ j ∈ J} of unitaries such that TiTj = QijTjTi and TkQij = QijTk for all i, j, k in J with i ≠ j. The family T is said to be doubly Q-commuting if it is Q-commuting and TiT*j = Q*ijT*j Ti for all i, j in J with i ≠ j. In this paper, we obtain the following main results. (i) It is well known that any number of doubly commuting isometries admit a Woldtype 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 Q-commuting family of contractions and get a canonical decomposition for them. Then we obtain a Wold-type decomposition for any doubly Q-commuting family of isometries as a special case. (ii) Further, we show that a similar decomposition is possible for an arbitrary doubly Q-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 Q-commuting tuples of contractions which is analogous to Burdaks work on commuting contractions. (iv) Also, we provide decomposition results for a pair of contractions (T1, T2) satisfying any of the following relations: T1T2 = QT2T1, T1T2 = T2QT1, T1T2 = T2T1Q, where Q is any unitary that commutes with the product T1T2.