A generalization of Andô's theorem and Parrott's example
Author:
David Opela
Journal:
Proc. Amer. Math. Soc. 134 (2006), 27032710
MSC (2000):
Primary 47A20
Published electronically:
April 7, 2006
MathSciNet review:
2213750
Fulltext PDF Free Access
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References 
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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 tuple of contractions that commute according to a graph without a cycle can be dilated to an tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.
 [A63]
T.
Andô, On a pair of commutative contractions, Acta Sci.
Math. (Szeged) 24 (1963), 88–90. MR 0155193
(27 #5132)
 [AM02]
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
(2003b:47001)
 [B02]
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
(2003k:47014)
 [GR69]
D.
Gașpar and A.
Rácz, An extension of a theorem of T. Andô,
Michigan Math. J. 16 (1969), 377–380. MR 0251586
(40 #4813)
 [Par70]
Stephen
Parrott, Unitary dilations for commuting contractions, Pacific
J. Math. 34 (1970), 481–490. MR 0268710
(42 #3607)
 [Pau02]
Vern
Paulsen, Completely bounded maps and operator algebras,
Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge
University Press, Cambridge, 2002. MR 1976867
(2004c:46118)
 [Pi01]
Gilles
Pisier, Similarity problems and completely bounded maps,
Second, expanded edition, Lecture Notes in Mathematics, vol. 1618,
SpringerVerlag, Berlin, 2001. Includes the solution to “The Halmos
problem”. MR 1818047
(2001m:47002)
 [Pi03]
Gilles
Pisier, Introduction to operator space theory, London
Mathematical Society Lecture Note Series, vol. 294, Cambridge
University Press, Cambridge, 2003. MR 2006539
(2004k:46097)
 [Po86]
Gelu
Popescu, Isometric dilations of 𝒫commuting
contractions, Rev. Roumaine Math. Pures Appl. 31
(1986), no. 5, 383–393. MR 856214
(87k:47012)
 [SF70]
Béla
Sz.Nagy and Ciprian
Foiaș, Harmonic analysis of operators on Hilbert space,
Translated from the French and revised, NorthHolland Publishing Co.,
Amsterdam, 1970. MR 0275190
(43 #947)
 [A63]
 T. Andô;
On a Pair of Commuting Contractions; Acta Sci. Math. (Szeged) (24) (1963), 8890. MR 0155193 (27:5132)
 [AM02]
 J. Agler, J. E. McCarthy;
Pick Interpolation and Hilbert Function Spaces; Graduate Studies in Mathematics 44, AMS, 2002. MR 1882259 (2003b:47001)
 [B02]
 T. Bhattacharyya;
Dilation of Contractive Tuples: A Survey; Proc. Centre Math. Appl. Austral. Nat. Univ. 40 (2002), 89126. MR 1953481 (2003k:47014)
 [GR69]
 D. Gaspar, A. Rácz;
An Extension of a Theorem of T. Andô; Michigan Math. J. (16) (1969), 377380. MR 0251586 (40:4813)
 [Par70]
 S. Parrott;
Unitary Dilations For Commuting Contractions; Pacific J. of Math. (34) (1970), no.2, 481490. MR 0268710 (42:3607)
 [Pau02]
 V. Paulsen;
Completely Bounded Maps and Operator Algebras; Cambridge University Press, 2002. MR 1976867 (2004c:46118)
 [Pi01]
 G. Pisier;
Similarity Problems and Completely Bounded Maps; 2nd, expanded edition, LNM 1618 (2001), SpringerVerlag. MR 1818047 (2001m:47002)
 [Pi03]
 G. Pisier;
Introduction to Operator Space Theory; Cambridge University Press, 2003. MR 2006539 (2004k:46097)
 [Po86]
 G. Popescu;
Isometric Dilations of Commuting Contractions; Rev. Roumaine Math. Pures Appl. (31) (1986), 383393. MR 0856214 (87k:47012)
 [SF70]
 B. Sz.Nagy, C. Foias;
Harmonic Analysis of Operators on Hilbert Space; NorthHolland, 1970. MR 0275190 (43:947)
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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:
http://dx.doi.org/10.1090/S0002993906083031
PII:
S 00029939(06)083031
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.
