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In the literature, we have several results associated with canonicaldecomposition of commuting contractions, dilations and Brehmer's positivity. Inthis paper, we generalize a few of these results to q-commuting contractions.Here we mainly deal with q-commuting and doubly q-commuting contractions when$|q|=1$ or when $q$ is a unitary operator.

Theory of q-commuting contractions : decompositions, dilations and Brehmer's positivity