Unitary skew-dilations of Hilbert space operators
DOI:
https://doi.org/10.17398/2605-5686.35.2.137Palabras clave:
Hilbert space operators, Dilations, Compressions of linear operators, Functional calculi, Numerical radius, ρ-radii, ρ-classes, (ρn )-classesResumen
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.
Descargas
Referencias
T. Ando, C.K. Li, Operator radii and unitary operators, Oper. Matrices 4 (2) (2010), 273 – 281.
T. Ando, K. Nishio, Convexity properties of operator radii associated with unitary ρ-dilations, Michigan Math. J. 20 (1973), 303 – 307.
T. Ando, K. Okubo,, Hölder-type inequalities associated with operator radii and Schur products, Linear and Multilinear Algebra 43 (1-3) (1997), 53 – 61.
C. Badea, Operators near completely polynomially dominated ones and similarity problems, J. Operator Theory 49 (1) (2003), 3 – 23.
C. Badea, G. Cassier, Constrained von Neumann inequalities, Adv. Math. 166 (2) (2002), 260 – 297.
C.A. Berger, J.G. Stampfli, Norm relations and skew dilations, Acta Sci. Math. (Szeged) 28 (1967), 191 – 195.
C. Davis, The shell of a Hilbert-space operator, II, Acta Sci. Math. (Szeged) 31 (1970), 301 – 318.
P.A. Fillmore, “Notes on Operator Theory”, Van Nostrand Reinhold Mathematical Studies, No. 30, Van Nostrand Reinhold Co., New York-London-Melbourne, 1970.
P. Găvruţa, On a problem of Bernard Chevreau concerning the ρ-contractions, Proc. Amer. Math. Soc. 136 (9) (2008), 3155 – 3158.
P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887 – 933.
J.A.R. Holbrook, On the power-bounded operators of Sz.-Nagy and Foiaş, Acta Sci. Math. (Szeged) 29 (1968), 299 – 310.
J.A.R. Holbrook, Inequalities governing the operator radii associated with unitary ρ-dilations, Michigan Math. J. 18 (1971), 149 – 159.
N. Kalton, Quasi-Banach Spaces, in “Handbook of the Geometry of Banach Spaces, Vol. 2”, North-Holland, Amsterdam, 2003, 1099 – 1130.
K. Okubo, T. Ando, Operator radii of commuting products, Proc. Amer. Math. Soc. 56 (1976), 203 – 210.
K. Okubo, T. Ando, Constants related to operators of class Cρ, Manuscripta Math. 16 (4) (1975), 385 – 394.
G. Pisier, “Similarity Problems and Completely Bounded Maps”, Lecture Notes in Mathematics, 1618, Springer-Verlag, Berlin, 1996.
A. Rácz, Unitary skew-dilations, (Romanian, with English summary), Stud. Cerc. Mat. 26 (1974), 545 – 621.
A. Salhi, H. Zerouali, On a ρn -dilation of operator in Hilbert spaces, Extracta Math. 31 (1) (2016), 11 – 23.
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.
C.-Y. Suen, WA contractions, Positivity 2 (4) (1998), 301 – 310.
C.-Y. Suen, Wρ -contractions, Soochow J. Math. 24 (1) (1998), 1 – 8.
B. Sz.-Nagy, C. Foiaş,, On certain classes of power-bounded operators in Hilbert space, Acta Sci. Math. (Szeged) 27 (1966), 17 – 25.
B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, “Harmonic Analysis of Operators on Hilbert Space”, Second edition, Universitext, Springer, New York, 2010.
J.P. Williams, Schwarz norms for operators, Pacific J. Math. 24 (1968), 181 – 188.
