Unitary skew-dilations of Hilbert space operators

Main Article Content

Vidal Agniel

Abstract

The aim of this paper is to study, for a given sequence (ρn )n≥1 of complex numbers, the class of Hilbert space operators possessing (ρn)-unitary dilations. This is the class of bounded linear operators T acting on a Hilbert space H, whose iterates Tn can be represented as Tn = ρnPHUn|H , n ≥ 1, for some unitary operator U acting on a larger Hilbert space, containing H as a closed subspace. Here PH is the projection from this larger space onto H. The case when all ρn ’s are equal to a positive real number ρ leads to the class Cρ introduced in the 1960s by Foias and Sz.-Nagy, while the case when all ρn ’s are positive real numbers has been previously considered by several authors. Some applications and examples of operators possessing (ρn)-unitary dilations, showing a behavior different from the classical case, are given in this paper.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Article Details

How to Cite
Agniel, V. (2020). Unitary skew-dilations of Hilbert space operators. Extracta Mathematicae, 35(2), 137-184. Retrieved from https://publicaciones.unex.es/index.php/EM/article/view/2605-5686.35.2.137
Section
Banach Spaces and Operator Theory

Funding data

References

[1] T. Ando, C.K. Li, Operator radii and unitary operators, Oper. Matrices 4 (2) (2010), 273 – 281.
[2] T. Ando, K. Nishio, Convexity properties of operator radii associated with unitary ρ-dilations, Michigan Math. J. 20 (1973), 303 – 307.
[3] T. Ando, K. Okubo,, Hölder-type inequalities associated with operator radii and Schur products, Linear and Multilinear Algebra 43 (1-3) (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 Hilbert-space 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 York-London-Melbourne, 1970.
[9] P. Găvruţa, On a problem of Bernard Chevreau concerning the ρ-contractions, 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 power-bounded 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 ρ-dilations, Michigan Math. J. 18 (1971), 149 – 159.
[13] N. Kalton, Quasi-Banach Spaces, in “Handbook of the Geometry of Banach Spaces, Vol. 2”, North-Holland, 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, Springer-Verlag, Berlin, 1996.
[17] A. Rácz, Unitary skew-dilations, (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 ρ-dilations 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 power-bounded 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.