Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T08:10:22.988Z Has data issue: false hasContentIssue false
  • English
  • Français

A Dilation and Norm in Several Variable Operator Theory

Published online by Cambridge University Press:  20 November 2018

Paul Binding ,
D. R. Farenick  and
Chi-Kwong Li
Paul Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N1N4
D. R. Farenick
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For every m-tuple of operators acting on a Hilbert space, it is shown that there exists a common dilation of these operators to mcommuting normal operators on some larger Hilbert space. We then introduce a norm on the m-fold cartesian product of ℬ(ℋ) that is defined to be, for a given w-tuple, the infimum of the joint spectral radii of all joint normal dilations of the m operators. This norm has several good features, one of which is that it is invariant under the passage to adjoints.

Keywords

15A6047A2047A30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 3 , 01 June 1995 , pp. 449 - 461
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Ando, T., On a pair of commuting contractions, Acta Sci. Math. 24(1963), 88–90.Google Scholar
2. Atkinson, F.V., Multiparameter Eigenvalue Problems, Volume I, Academic Press, New York, 1972.Google Scholar
3. Binding, P., Simultaneous diagonalisation of several Hermitian matrices, SI AM J. Matrix Anal. Appl. 11(1990), 531–536.Google Scholar
4. Binding, P. and Li, C.-K., Joint ranges of Hermitian matrices and simultaneous diagonalization, Linear Algebra Appl. 151(1991), 157–168.Google Scholar
5. Brown, L.G., Douglas, R.G. and Fillmore, P.A., Unitary equivalence modulo the compact operators and extensions of C*-algebras, Springer-Verlag, Lecture Notes in Math. 345, 1973. 58–128.Google Scholar
6. Bunce, J.W., Models for n-tuples of noncommuting operators, J. Funct. Anal. 57(1984), 21–30.Google Scholar
7. Cho, M. and Takaguchi, M., Some classes of commuting m-tuples of operators, Studia Math. 80(1984), 245–259.Google Scholar
8. Cho, M. and Zelazko, W., On geometric spectral radius of commuting m-tuples of operators, Hokkaido Math. J. 21(1992), 251–258.Google Scholar
9. Curto, R.E., Applications of several complex variables to multiparameter spectral theory, Surveys of Recent Results in Operator Theory, Volume II, Pitman Research Notes in Math. 192, Longman,Inc, 1988. 25–90.Google Scholar
10. Douglas, R.G., Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.Google Scholar
11. Hadwin, D.W., Completely positive maps and approximate equivalence, Indiana Univ. Math. J. 36(1987), 211–228.Google Scholar
12. Halmos, P.R., Normal dilations and extensions of operators, Summa Brasil Math. 2(1950), 125–134.Google Scholar
13. Halmos, P.R., Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970), 887–933.Google Scholar
14. Isaev, H.A., On multiparameter spectral theory, Dokl. Akad. Nauk SSSR 229(1976), 284–287.Google Scholar
15. Kallstrom, A. and Sleeman, B.D., Solvability of a linear operator system, J. Math. Anal. Appl. 55(1976), 785–793.Google Scholar
16. Kosir, T., Commuting matrices and mulitparameter eigenvalue problems, Ph.D. Thesis, University of Calgary, 1993.Google Scholar
17. Mirman, B.A., Numerical range and norm of a linear operator, Voronež. Gos. Univ., Trudy Sem. Funk. Anal. 10(1968), 51–55.Google Scholar
18. Muller, V. and Vasilescu, F.-H., Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117(1993), 979–990.Google Scholar
19. Nakamura, Y., Numerical range and norm, Math. Japon. 27(1982), 149–150.Google Scholar
20. Parrott, S.K., Unitary dilations for commuting contractions, Pacific J. Math. 34(1970), 481–490.Google Scholar
21. Pryde, A., A Bauer-Fike theorem for the joint spectrum of commuting matrices, Linear Algebra Appl. 173(1992), 221–230.Google Scholar
22. Rynne, B.P., Multiparameter spectral theory and Taylor s spectrum in Hilbert space, Proc. Edinburgh Math. Soc. 31(1988), 127–144.Google Scholar
23. Sleeman, B.D. Multiparameter spectral theory in Hilbert space, Research Notes in Math. 22, Pitman, 1978.Google Scholar
24. Wintner, A., Zur Theorie der beschrànkten Bilinearformen, Math. Z. 39(1929), 228–282.Google Scholar