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Functional model for commuting isometries. (English) Zbl 0693.47013
An isometry V on a complex, separable Hilbert space induces a commutative semigroup of isometries \(\{V^ n\}_{n\geq 0}\). The well-known Wold decomposition for V then yields a corresponding decomposition for this semigroup into semigroups of unitary operators and unilateral shifts. This paper generalizes this result to the setting of the so-called (commutative) compatible semigroup of isometries \(\{T_ s\}_{s\in S}\). Here “compatible” means that the orthogonal projections onto the ranges of \(T_ s\) commute. Such isometries lie between doubly commuting isometries and commuting ones. The authors develop a canonical functional model for any finitely generated compatible semigroup of isometries, which generalizes the Wold decomposition of two doubly commuting isometries obtained before by M. Slociński and others.
Reviewer: Wu Pei Yuan

47A45 Canonical models for contractions and nonselfadjoint linear operators
Full Text: EuDML
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