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.
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Bercovici H., Douglas R.G., Foias C.: On the classification of multi-isometries. Acta Sci. Math (Szeged) 72, 639–661 (2006)
Bercovici H., Douglas R.G., Foias C.: Canonical models for bi-isometries. Oper. Theory Adv. Appl. 218, 177–205 (2012)
Berger C.A., Coburn L.A., Lebow A.: Representation and index theory for C*-algebras generated by commuting isometries. J. Funct. Anal. 27, 51–99 (1978)
Burdak Z., Kosiek M., Słociński M.: The canonical Wold decomposition of commuting isometries with finite dimensional wandering spaces. Bull. des Sci. Math. 137, 653–658 (2013)
Burdak Z.: On decomposition of pairs of commuting isometries. Ann. Polon. Math. 84, 121–135 (2004)
Burdak Z.: On a decomposition for pairs of commuting contractions. Studia Math. 181(1), 33–45 (2007)
Catepillán X., Ptak M., Szymański W.: Multiple Canonical decompositions of families of operators and a model of quasinormal families. Proc. Am. Math. Soc. 121, 1165–1172 (1994)
Conway J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York, Inc. (1990)
Gaspar D., Gaspar P.: Wold decompositions and the unitary model for bi-isometries. Integral Equ. Oper. Theory 49, 419–433 (2004)
Gaspar D., Suciu N.: Wold decompositions for commutative families of isometries. An. Univ. Timisoara Ser. Stint. Mat. 27, 31–38 (1989)
Hoffman K.: Banach Spaces of Analytic Functions. Prentice Hall, Inc., Englewood Cliffs (1962)
Kosiek M.: Fuglede-type decompositions of representations. Studia Math. 151, 87–98 (2002)
Mlak W.: Intertwinning operators. Studia Math. 43, 219–233 (1972)
Popovici D.: A Wold-type decomposition for commuting isometric pairs. Proc. Am. Math. Soc. 132, 2303–2314 (2004)
Ptak M.: On the reflexivity of the pairs of isometries and of tensor products of some operator algebras. Studia Math. 83, 47–53 (1986)
Słociński M.: On the Wold type decomposition of a pair of commuting isometries. Ann. Polon. Math. 37, 255–262 (1980)
Suciu I.: On the semigroups of isometries. Studia Math. 30, 101–110 (1968)
Wold H.: A Study in the Analysis of Stationary Time Series. Almkvist and Wiksell, Stockholm (1954)
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The research of the second and the fourth author was supported by the MNiSzW (Ministry of Science and Higher Education) grant NN201 546438 (2010–2013).
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Burdak, Z., Kosiek, M., Pagacz, P. et al. Shift-Type Properties of Commuting, Completely Non Doubly Commuting Pairs of Isometries. Integr. Equ. Oper. Theory 79, 107–122 (2014). https://doi.org/10.1007/s00020-014-2135-z
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DOI: https://doi.org/10.1007/s00020-014-2135-z