On dilation and commuting liftings of n-tuples of commuting Hilbert space contractions

Zbigniew Burdak; Wiesław Grygierzec

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica (2020)

  • Volume: 19, page 121-139
  • ISSN: 2300-133X

Abstract

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The n-tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n-tuple. A series of such liftings leads to an isometric dilation of the n-tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite n-tuple has an isometric dilation.

How to cite

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Zbigniew Burdak, and Wiesław Grygierzec. "On dilation and commuting liftings of n-tuples of commuting Hilbert space contractions." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 19 (2020): 121-139. <http://eudml.org/doc/296809>.

@article{ZbigniewBurdak2020,
abstract = {The n-tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n-tuple. A series of such liftings leads to an isometric dilation of the n-tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite n-tuple has an isometric dilation.},
author = {Zbigniew Burdak, Wiesław Grygierzec},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
language = {eng},
pages = {121-139},
title = {On dilation and commuting liftings of n-tuples of commuting Hilbert space contractions},
url = {http://eudml.org/doc/296809},
volume = {19},
year = {2020},
}

TY - JOUR
AU - Zbigniew Burdak
AU - Wiesław Grygierzec
TI - On dilation and commuting liftings of n-tuples of commuting Hilbert space contractions
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2020
VL - 19
SP - 121
EP - 139
AB - The n-tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n-tuple. A series of such liftings leads to an isometric dilation of the n-tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite n-tuple has an isometric dilation.
LA - eng
UR - http://eudml.org/doc/296809
ER -

References

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  1. Andô, Tsuyoshi. "On a pair of commutative contractions." Acta Sci. Math. (Szeged) 24 (1963): 88-90. 
  2. Arhancet, Cédric, and Stephan Fackler, and Christian Le Merdy. "Isometric dilations and H1 calculus for bounded analytic semigroups and Ritt operators." Trans. Amer. Math. Soc. 369, no. 10 (2017): 6899-6933. 
  3. Ball, Joseph A., and Haripada Sau. "Rational dilation of tetrablock contractions revisited." J. Funct. Anal. 278, no. 1 (2020): 108275, 14 pp. 
  4. Barik, Sibaprasad, et all. "Isometric dilations and von Neumann inequality for a class of tuples in the polydisc." Trans. Amer. Math. Soc. 372, no. 2 (2019): 1429-1450. 
  5. Choi, Man-Duen, and Kenneth R. Davidson. "A 3 × 3 dilation counterexample." Bull. Lond. Math. Soc. 45, no. 3 (2013): 511-519. 
  6. Das, B. Krishna, and Jaydeb Sarkar. "Andô dilations, von Neumann inequality, and distinguished varieties." J. Funct. Anal. 272, no. 5 (2017): 2114-2131. 
  7. Fackler, Stephan, and Glück, Jochen. "A toolkit for constructing dilations on Banach spaces." Proc. Lond. Math. Soc. (3) 118, no. 2, (2019): 416-440. 
  8. Foias, Ciprian, and Arthur E. Frazho. The commutant lifting approach to interpolation problems. Vol. 44 of Operator Theory: Advances and Applications. Basel: Birkhäuser Verlag, 1990. 
  9. Keshari, Dinesh Kumar, and Nirupama Mallick. "q-commuting dilation." Proc. Amer. Math. Soc. 147, no. 2 (2019): 655-669. 
  10. Müller, Vladimír. "Commutant lifting theorem for n-tuples of contractions." Acta Sci. Math. (Szeged) 59, no. 3-4 (1994): 465-474. 
  11. Parrott, Stephen. "Unitary dilations for commuting contractions." Pacific J. Math. 34 (1970): 481-490. 
  12. Paulsen, Vern. Completely bounded maps and operator algebras. Vol. 78 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2002. 
  13. Popescu, Gelu. "Andô dilations and inequalities on noncommutative varieties." J. Funct. Anal. 272, no. 9 (2017): 3669-3711. 
  14. Russo, Benjamin. "Lifting commuting 3-isometric tuples." Oper. Matrices 11, no. 2 (2017): 397-433. 
  15. Szokefalvi-Nagy, Béla. "Sur les contractions de l’espace de Hilbert." Acta Sci. Math. (Szeged) 15 (1953): 87-92. 
  16. Szokefalvi-Nagy, Béla and Ciprian Foias. Harmonic analysis of operators on Hilbert space. Amsterdam-London: North-Holland Publishing Co.; New York: American Elsevier Publishing Co., Inc.; Budapest: Akadémiai Kiadó, 1970. 
  17. Varopoulos, Nicholas Th. "On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory." J. Functional Analysis 16 (1974): 83-100. 

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