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A note on the joint operator norm of hermitian operators on Banach spaces

  • Muneo Chō (a1) and Tadasi Huruya (a2)

Extract

Let X be a complex Banach space and H be a hermitian operator on X. Then in [7] Sinclair proved that r(H) = ¶H¶, where r(H) and ¶H¶ are the spectral radius and the operator norm of H, respectively.

For a commuting n-tuple T = (T1,…, Tn) of operators on X, we denote the (Taylor) joint spectrum of T by σ(T) (see [9]) and define the joint operator norm ¶T¶ and the joint spectral radius r(T) by

and

respectively.

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Copyright

References

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1.Bonsall, F. F. and Duncan, J., Numerical ranges II (Cambridge Univ. Press, 1973).
2.Chō, M., Joint spectra of commuting normal operators on Banach spaces, Glasgow Math. J. 30 (1988), 339345.
3.Chō, M. and Takaguchi, M., Some classes of commuting n-tuple of operators, Studia Math. 80 (1984), 245259.
4.Chō, M. and Yamaguchi, H., A simple example of a normal operator Tsuch that r(T)<¶T¶, Proc. Amer. Math. Soc. 108 (1990), 143.
5.Rosenblum, M., On the operator equation BX – XB =Q, Duke Math. J. 23 (1956), 263269.
6.Shaw, S.-Y., On numerical ranges of generalized derivations and related properties, J. Australian Math. Soc.(A) 36 (1984), 134142.
7.Sinclair, A. M., The norm of a hermitian element in a Banach algebra, Proc. Amer. Math. Soc. 28 (1971), 446450.
8.Stampfli, J. G., The norm of a derivation, Pacific J. Math. 33 (1970), 737747.
9.Taylor, J. L., A joint spectrum for several commuting operators, J. Functional Anal. 6 (1970), 172191.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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