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Unitarily Invariant Operator Norms

Published online by Cambridge University Press:  20 November 2018

C.-K. Fong  and
J. A. R. Holbrook
C.-K. Fong
Affiliation:
University of Toronto, Toronto, Ontario
J. A. R. Holbrook
Affiliation:
University of Guelph, Guelph, Ontario
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1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius

1

of a Hilbert space operator T. Among the many interesting developments, we may mention:

(a) C. Berger's proof of the “power inequality”

2

(b) R. Bouldin's result that

3

for any isometry V commuting with T;

(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;

(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 2 , 01 April 1983 , pp. 274 - 299
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Ando, T., Structure of operators with numerical radius one, Acta Sci. Math. 34 (1973), 11–15.Google Scholar
2. Ando, T., Topics in dilation theory, Sapporo (1973).Google Scholar
3. Ando, T. and Nishio, K., Convexity properties of operator radii associated with unitaryp-dilations, Michigan Math. J. 20 (1973), 303–307.Google Scholar
4. Ando, T. and Okubo, K., Operator radii of commuting products, Proc. Amer. Math. Soc. 56 (1976), 203–210.Google Scholar
5. Berger, C. A., A strange dilation theorem, Notices Amer. Math. Soc. 12 (1965), 590.Google Scholar
6. Berger, C. A. and Stampfli, J. G., Norm relations and skew dilations, Acta Sci. Math. 28 (1967), 191–195.Google Scholar
7. Berger, C. A. and Stampfli, J. G., Mapping theorems for the numerical range, Amer. J. Math. 89 (1967), 1047–1055.Google Scholar
8. Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras (Cambridge U. P., 1971).Google Scholar
9. Bonsall, F. F. and Duncan, J., Numerical ranges II (Cambridge U. P., 1973).Google Scholar
10. Bouldin, R., The numerical range of a product II, J. Math. Anal. Appl. 33 (1971), 212–219.Google Scholar
11. Holbrook, J. A. R., Multiplicative properties of the numerical radius in operator theory, J. reine angew. Math. 237 (1969), 166–174.Google Scholar
12. Holbrook, J. A. R., On the power-bounded operators of Sz.-Nagy and Foias, Acta Sci. Math. 29 (1968), 299–310. Google Scholar
13. Katznelson, Y., An introduction to harmonic analysis (Wiley, 1968).Google Scholar
14. Palmer, T. W., Characterizations of C*-algebras II, Trans. Amer. Math. Soc. 148 (1970), 577–588.Google Scholar
15. Pearcy, C., An elementary proof of the power inequality for the numerical radius, Michigan Math. J. 13 (1966), 289–291.Google Scholar
16. Schatten, R., Norm ideals of completely continuous operators (Springer, 1960).Google Scholar
17. Sz.-Nagy, B. and Foias, C., Harmonic analysis of operators on Hilbert space (North-Holland, 1970).Google Scholar
18. Williams, J. P., Schwarz norms for operators, Pacific J. Math. 24 (1968), 181–188.Google Scholar