Unitary skewdilations of Hilbert space operators
Main Article Content
Abstract
The aim of this paper is to study, for a given sequence (ρ_{n} )_{n≥1} of complex numbers, the class of Hilbert space operators possessing (ρ_{n})unitary dilations. This is the class of bounded linear operators T acting on a Hilbert space H, whose iterates T^{n} can be represented as T^{n} = ρ_{n}P_{H}U^{n}_{H} , n ≥ 1, for some unitary operator U acting on a larger Hilbert space, containing H as a closed subspace. Here P_{H} is the projection from this larger space onto H. The case when all ρ_{n} ’s are equal to a positive real number ρ leads to the class C_{ρ} introduced in the 1960s by Foias and Sz.Nagy, while the case when all ρ_{n} ’s are positive real numbers has been previously considered by several authors. Some applications and examples of operators possessing (ρ_{n})unitary dilations, showing a behavior different from the classical case, are given in this paper.
Downloads
Metrics
Article Details
Funding data

Agence Nationale de la Recherche
Grant numbers ANR17CE400021 
Agence Nationale de la Recherche
Grant numbers ANR11LABX000701
References
[2] T. Ando, K. Nishio, Convexity properties of operator radii associated with unitary ρdilations, Michigan Math. J. 20 (1973), 303 – 307.
[3] T. Ando, K. Okubo,, Höldertype inequalities associated with operator radii and Schur products, Linear and Multilinear Algebra 43 (13) (1997), 53 – 61.
[4] C. Badea, Operators near completely polynomially dominated ones and similarity problems, J. Operator Theory 49 (1) (2003), 3 – 23.
[5] C. Badea, G. Cassier, Constrained von Neumann inequalities, Adv. Math. 166 (2) (2002), 260 – 297.
[6] C.A. Berger, J.G. Stampfli, Norm relations and skew dilations, Acta Sci. Math. (Szeged) 28 (1967), 191 – 195.
[7] C. Davis, The shell of a Hilbertspace operator, II, Acta Sci. Math. (Szeged) 31 (1970), 301 – 318.
[8] P.A. Fillmore, “Notes on Operator Theory”, Van Nostrand Reinhold Mathematical Studies, No. 30, Van Nostrand Reinhold Co., New YorkLondonMelbourne, 1970.
[9] P. Găvruţa, On a problem of Bernard Chevreau concerning the ρcontractions, Proc. Amer. Math. Soc. 136 (9) (2008), 3155 – 3158.
[10] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887 – 933.
[11] J.A.R. Holbrook, On the powerbounded operators of Sz.Nagy and Foiaş, Acta Sci. Math. (Szeged) 29 (1968), 299 – 310.
[12] J.A.R. Holbrook, Inequalities governing the operator radii associated with unitary ρdilations, Michigan Math. J. 18 (1971), 149 – 159.
[13] N. Kalton, QuasiBanach Spaces, in “Handbook of the Geometry of Banach Spaces, Vol. 2”, NorthHolland, Amsterdam, 2003, 1099 – 1130.
[14] K. Okubo, T. Ando, Operator radii of commuting products, Proc. Amer. Math. Soc. 56 (1976), 203 – 210.
[15] K. Okubo, T. Ando, Constants related to operators of class Cρ, Manuscripta Math. 16 (4) (1975), 385 – 394.
[16] G. Pisier, “Similarity Problems and Completely Bounded Maps”, Lecture Notes in Mathematics, 1618, SpringerVerlag, Berlin, 1996.
[17] A. Rácz, Unitary skewdilations, (Romanian, with English summary), Stud. Cerc. Mat. 26 (1974), 545 – 621.
[18] A. Salhi, H. Zerouali, On a ρn dilation of operator in Hilbert spaces, Extracta Math. 31 (1) (2016), 11 – 23.
[19] N.P. Stamatiades, “Unitary ρdilations and the Holbrook Radius for Bounded Operators on Hilbert Space”, Ph.D. Thesis, University of London, Royal Holloway College, United Kingdom, 1982.
[20] C.Y. Suen, WA contractions, Positivity 2 (4) (1998), 301 – 310.
[21] C.Y. Suen, Wρ contractions, Soochow J. Math. 24 (1) (1998), 1 – 8.
[22] B. Sz.Nagy, C. Foiaş,, On certain classes of powerbounded operators in Hilbert space, Acta Sci. Math. (Szeged) 27 (1966), 17 – 25.
[23] B. Sz.Nagy, C. Foias, H. Bercovici, L. Kérchy, “Harmonic Analysis of Operators on Hilbert Space”, Second edition, Universitext, Springer, New York, 2010.
[24] J.P. Williams, Schwarz norms for operators, Pacific J. Math. 24 (1968), 181 – 188.