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A generalization of Andô's theorem and Parrott's example


Author: David Opela
Journal: Proc. Amer. Math. Soc. 134 (2006), 2703-2710
MSC (2000): Primary 47A20
DOI: https://doi.org/10.1090/S0002-9939-06-08303-1
Published electronically: April 7, 2006
MathSciNet review: 2213750
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Abstract: Andô's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting contractions. We generalize both of these results as follows. Any $ n$-tuple of contractions that commute according to a graph without a cycle can be dilated to an $ n$-tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.


References [Enhancements On Off] (What's this?)

  • [A-63] T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 0155193
  • [AM-02] Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002. MR 1882259
  • [B-02] Tirthankar Bhattacharyya, Dilation of contractive tuples: a survey, Surveys in analysis and operator theory (Canberra, 2001) Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 89–126. MR 1953481
  • [GR-69] D. Gaşpar and A. Rácz, An extension of a theorem of T. Andô, Michigan Math. J. 16 (1969), 377–380. MR 0251586
  • [Par-70] Stephen Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970), 481–490. MR 0268710
  • [Pau-02] Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
  • [Pi-01] Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047
  • [Pi-03] Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539
  • [Po-86] G. Popescu;
    Isometric Dilations of $ {\mathcal P}$-Commuting Contractions;
    Rev. Roumaine Math. Pures Appl. (31) (1986), 383-393. MR 0856214 (87k:47012)
  • [SF-70] Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR 0275190

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Additional Information

David Opela
Affiliation: Department of Mathematics, Campus Box 1146, Washington University in Saint Louis, Saint Louis, Missouri 63130
Email: opela@math.wustl.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08303-1
Keywords: Unitary dilations, commuting contractions, And\^o's theorem
Received by editor(s): January 2, 2005
Received by editor(s) in revised form: April 12, 2005
Published electronically: April 7, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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