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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

MSC:
47A45 Canonical models for contractions and nonselfadjoint linear operators
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References:
[1] D. Gaspar N. Suciu: On the structure of isometric semigroups. Operator Theory: Adv. and Appl. 14, Birkhäuser Verlag Basel, 1984, 125-139.
[2] D. Gaspar N. Suciu: Intertwinings of isometric semigroups and Wold type decompositions. Sem. de operatori liniari si analiză armonică, No. 3, 1985.
[3] P. R. Halmos: Introduction to Hilbert space and the theory of spectral multiplicity. Chelsea Publishing Comp. New York, 1951. · Zbl 0045.05702
[4] P. R. Halmos: Measure Theory. New York, 1950. · Zbl 0040.16802
[5] H. Helson D. Lowdenslager: Prediction theory and Fourier series in several variables II. Acta Math. 106 (1961), 175-213. · Zbl 0102.34802
[6] K. Horák V. Müller: On the structure of commuting isometries. CMUC 28 (1987), 165-171. · Zbl 0619.47014
[7] G. Kalliampur V. Mandrekar: Nondeterministic random fields and Wold and Halmos decompositions for commuting isometries. Prediction Theory and Harmonic Analysis, North Holland Publ. Comp., 1983, 165-190.
[8] M. Slociński: On the Wold-type decomposition of a pair of commuting isometries. Annales Polonici Math. 37 (1980), 255-262.
[9] M. Slociński: Models of commuting contractions and isometries. Report of the 11th Conference on Operator Theory Bucharest 1986
[10] I. Suciu: On the semigroups of isometries. Studia Math. 30. · Zbl 0184.15906
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