Shift-Type Properties of Commuting, Completely Non Doubly Commuting Pairs of Isometries

Abstract

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 isometry with respect to its wandering vectors which strongly corresponds with Lebesgue decomposition. © 2014 The Author(s).