Summary
In this chapter, we prove (actually three times) von Neumann’s inequality and its extension (due to Ando) for two mutually commuting contractions. We discuss the case of n > 2 mutually commuting contractions. We introduce the notion of semi-invariance. Finally, we show that Hilbert spaces are the only Banach spaces satisfying von Neumann’s inequality.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Nelson E.: The distinguished boundary of the unit operator ball. Proc. Amer. Math. Soc. 12 (1961) 994–995.
Sz.-Nagy B. and Foias C. Harmonic analysis of operators on Hilbert space. Akademiai Kiadd, Budapest 1970.
Varopoulos N.: On an inequality of von Neumann and an application of the metric theory of tensor products to Operators Theory. J. Funct. Anal. 16 (1974) 83–100.
Dixon P. and Drury S.: Unitary dilations, polynomial identities and the von Neumann inequality. Math. Proc. Cambridge Phil. Soc. 99 (1986) 115–122.
Drury S.: Remarks on von Neumann’s inequality. Banach spaces, Harmonic analysis, and probability theory. Proceedings (edited by R. Blei and S. Sydney), Storrs 80/81. Springer Lecture Notes 995, 14–32.
Holbrook J.: Polynomials in a matrix and its commutant. Linear Alg. Appl. 48 (1982) 293–301.
Holbrook J.: Inequalities of von Neumann type for small matrices. in “Function Spaces” (edited by K. Jarosz ). Marcel Dekker, New-York, 1992.
Holbrook J.: Interpenetration of ellipsoids and the polynomial bound of a matrix. Preprint, Sept. 1993.
Tonge A.: The von Neumann inequality for polynomials in several Hilbert Schmidt operators. J. London Math. Soc. 18 (1978) 519–526.
Tonge A.: Polarisation and the 2 dimensional Grothendieck inequality. Proc. Cambridge Phil. Soc. 95 (1984) 313–318.
Losert V. Properties of the Fourier algebra that are equivalent to amenability. Proc. Amer. Math. Soc. 92 (1984) 347–354.
Boiejko M.: Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality. Studia Math. 95 (1989) 107–118.
Popescu G.: Von Neumann inequality for (B(H)“)1. Math. Scand. 68 (1991) 292–304.
Popescu G.: Non-commutative disc algebras and their representations. 1994, Preprint to appear.
Popescu G.: Positive-definite functions on free semigroups. Preprint, 1994, to appear.
Popescu G.: Multi-analytic operators on Fock space. Math. Ann. to appear.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pisier, G. (1996). Von Neumann’s inequality and Ando’s generalization. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21537-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-662-21537-1_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60322-1
Online ISBN: 978-3-662-21537-1
eBook Packages: Springer Book Archive