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

Zbigniew Burdak, Wiesław Grygierzec

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.

Keywords


dilation; lifting; von Neumann inequality

Mathematics Subject Classification (2010)


Primary 47A20; Secondary 47A50

References


Andô, Tsuyoshi. "On a pair of commutative contractions." Acta Sci. Math. (Szeged) 24 (1963): 88-90.

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.

Ball, Joseph A., and Haripada Sau. "Rational dilation of tetrablock contractions revisited." J. Funct. Anal. 278, no. 1 (2020): 108275, 14 pp.

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.

Choi, Man-Duen, and Kenneth R. Davidson. "A 3 × 3 dilation counterexample." Bull. Lond. Math. Soc. 45, no. 3 (2013): 511-519.

Das, B. Krishna, and Jaydeb Sarkar. "Andô dilations, von Neumann inequality, and distinguished varieties." J. Funct. Anal. 272, no. 5 (2017): 2114-2131.

Fackler, Stephan, and Glück, Jochen. "A toolkit for constructing dilations on Banach spaces." Proc. Lond. Math. Soc. (3) 118, no. 2, (2019): 416-440.

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.

Keshari, Dinesh Kumar, and Nirupama Mallick. "q-commuting dilation." Proc. Amer. Math. Soc. 147, no. 2 (2019): 655-669.

Müller, Vladimír. "Commutant lifting theorem for n-tuples of contractions." Acta Sci. Math. (Szeged) 59, no. 3-4 (1994): 465-474.

Parrott, Stephen. "Unitary dilations for commuting contractions." Pacific J. Math. 34 (1970): 481-490.

Paulsen, Vern. Completely bounded maps and operator algebras. Vol. 78 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2002.

Popescu, Gelu. "Andô dilations and inequalities on noncommutative varieties." J. Funct. Anal. 272, no. 9 (2017): 3669-3711.

Russo, Benjamin. "Lifting commuting 3-isometric tuples." Oper. Matrices 11, no. 2 (2017): 397-433.

Szokefalvi-Nagy, Béla. "Sur les contractions de l’espace de Hilbert." Acta Sci. Math. (Szeged) 15 (1953): 87-92.

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.

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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