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On the joint spectra of doubly commuting n-tuples of semi-normal operators

  • Muneo Chō (a1) and A. T. Dash (a2)

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Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that

.

A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such that

A point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ jn.

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References

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1.Berberian, S. K., Approximate proper vectors, Proc. Amer. Math. Soc. 13 (1962, 111114).
2.Choi, M. D. and Davis, C., The spectral mapping theorem for joint approximate point spectrum, Bull. Amer. Math. Soc. 80 (1974), 317321.
3.Curto, R. E., Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), 129159.
4.Dash, A. T., Joint spectra, Studia Math. 45 (1973), 225237.
5.Dash, A. T., A note on joint approximate point spectrum, Mimeographed lecture notes, University of Guelph and Indiana University, 1972.
6.Putnam, C. R., On the spectra of semi-normal operators, Trans. Amer. Math. Soc. 119 (1965), 509523.
7.Putnam, C. R., Commutation properties of Hilbert space operators and related topics (Springer, 1967).
8.Taylor, J. L., A joint spectrum for several commuting operators, J. Func. 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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