A Wold-type decomposition for commuting isometric pairs
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Abstract:
We obtain a Wold-type decomposition theorem for an arbitrary pair of commuting isometries $V$ on a Hilbert space. More precisely, $V$ can be uniquely decomposed into the orthogonal sum between a bi-unitary, a shift-unitary, a unitary-shift and a weak bi-shift part, that is, a part $S=(S_1,S_2)$ that can be characterized by the condition that $S_1S_2,\ S_1|_{\bigcap _{n\ge 0}\ker S_2^*S_1^n}$ and $S_2|_{\bigcap _{n\ge 0}\ker S_1^*S_2^n}$ are shifts. Moreover, $S$ contains bi-shift and modified bi-shift maximal parts.References
- C. A. Berger, L. A. Coburn, and A. Lebow, Representation and index theory for $C^*$-algebras generated by commuting isometries, J. Functional Analysis 27 (1978), no. 1, 51–99. MR 0467392, DOI 10.1016/0022-1236(78)90019-8
- D. Gaşpar and P. Gaşpar, Wold decompositions and the unitary model for bi-isometries, to appear.
- D. Gaşpar and N. Suciu, Intertwining properties of isometric semigroups and Wold type decompositions, Operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985) Oper. Theory Adv. Appl., vol. 24, Birkhäuser, Basel, 1987, pp. 183–193. MR 903071
- Dumitru Gaşpar and Nicolae Suciu, On the Wold decomposition of isometric semigroups, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 99–108. MR 820514
- Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15. MR 0107177
- M. Kosiek, Functional calculus and common invariant subspaces, Uniwersytet Jagiellonski Wydanie I, Krakow, 2001.
- H. Langer, Ein Zerspaltungssatz für Operatoren im Hilbertraum, Acta Math. Acad. Sci. Hungar. 12 (1961), 441–445 (German, with Russian summary). MR 139954, DOI 10.1007/BF02023926
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- Béla Sz.-Nagy and Ciprian Foiaş, Sur les contractions de l’espace de Hilbert. IV, Acta Sci. Math. (Szeged) 21 (1960), 251–259 (French). MR 126149
- Dan Popovici, On the structure of c.n.u. bi-isometries. II, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 329–347. MR 1916584
- Marek Słociński, On the Wold-type decomposition of a pair of commuting isometries, Ann. Polon. Math. 37 (1980), no. 3, 255–262. MR 587496, DOI 10.4064/ap-37-3-255-262
- I. Suciu, On the semi-groups of isometries, Studia Math. 30 (1968), 101–110. MR 229093, DOI 10.4064/sm-30-1-101-110
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Additional Information
- Dan Popovici
- Affiliation: Department of Mathematics, University of the West Timişoara, RO-300223 Timişoara, Bd. Vasile Pârvan nr. 4, Romania
- Email: popovici@math.uvt.ro
- Received by editor(s): September 13, 2002
- Received by editor(s) in revised form: April 21, 2003
- Published electronically: February 26, 2004
- Additional Notes: This work was supported by the EEC Research Training Network: “Analysis and Operators”, contract no. HPRN-CT-2000-00116
- Communicated by: Joseph A. Ball
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2303-2314
- MSC (2000): Primary 47A13, 47A45
- DOI: https://doi.org/10.1090/S0002-9939-04-07331-9
- MathSciNet review: 2052406