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Pairs (V, V′) of commuting, completely non doubly commuting isometries are studied. We show, that the space of the minimal unitary extension of V (denoted by U) is a closed linear span of subspaces reducing U to bilateral shifts. Moreover, the restriction of V′ to the maximal subspace reducing V to a unitary operator is a unilateral shift. We also get a new hyperreducing decomposition of a single...
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