The paper is devoted to pairs of commuting isometries. A unique decomposition of such pairs into a compatible and a completely non-compatible part is constructed. The compatible part is fully described.
The paper summarizes, clarifies and supplements results on decompositions and a model for pairs of commuting isometries. The geometrical model is given whenever it is possible. The remaining problem of describing the model for completely non-compatible pairs of isometries is reduced to completely non-compatible pairs of unilateral shifts. Moreover, subspaces generating incompatibility are determined.
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.
Mathematics Subject Classification (2000). Primary 47B20; Secondary 47A13.
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