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Weyl Spectral Identity and Biquasitriangularity

Published online by Cambridge University Press:  29 October 2015

C. S. Kubrusly  and
B. P. Duggal
C. S. Kubrusly
Affiliation:
Catholic University of Rio de Janeiro, 22453-900, Rio de Janeiro, Brazil ([email protected])
B. P. Duggal
Affiliation:
University of Niš, Department of Mathematics, PO Box 224, Niš, Serbia ([email protected])
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Abstract

Let A and B be operators acting on infinite-dimensional complex Banach spaces. We say that the Weyl spectral identity holds for the tensor product A⊗B if σw(A⊗B) = σw(A)·σ(B)∪σ(A)·σw(B), where σ(·) and σw(·) stand for the spectrum and the Weyl spectrum, respectively. Conditions on A and B for which the Weyl spectral identity holds are investigated. Especially, it is shown that if A and B are biquasitriangular (in particular, if the spectra of A and B have empty interior), then the Weyl spectral identity holds. It is also proved that if A and B are biquasitriangular, then the tensor product A ⊗ B is biquasitriangular.

Keywords

tensor productWeyl spectral identitybiquasitriangular operatorsPrimary 47A80Secondary 47A53
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 363 - 375
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Aiena, P., Fredholm and local spectral theory, with applications to multipliers (Kluwer Academic, Dordrecht, 2004).Google Scholar
2. Apostol, C., Foiaş, C. and Voiculescu, D., Some results on non-quasitriangular operators IV, Rev. Romaine Math. Pure Appl. 18 (1973), 487–514.Google Scholar
3. Apostol, C., Foiaş, C. and Voiculescu, D., Some results on non-quasitriangular operators VI, Rev. Romaine Math. Pure Appl. 18 (1973), 1473–1494.Google Scholar
4. Brown, A. and Pearcy, C., Spectra of tensor products of operators, Proc. Am. Math. Soc. 17 (1966), 162–166.Google Scholar
5. Brown, A. and Pearcy, C., Introduction to operator theory I: elements of functional analysis, Graduate Texts in Mathematics, Volume 55 (Springer, 1977).Google Scholar
6. Duggal, B. P. and Kubrusly, C. S., Biquasitriangularity and derivations, Funct. Analysis Approx. Computat. 6 (2014), 41–48.Google Scholar
7. Duggal, B. P., Djordjević, S. V. and Kubrusly, C. S., On the a-Browder and a-Weyl spectra of tensor products, Rend. Circ. Mat. Palermo 59 (2010), 473–481.CrossRefGoogle Scholar
8. Duggal, B., Harte, R. and Kim, A.-H., Weyl’s theorem, tensor products and multiplication operators II, Glasgow Math. J. 52 (2010), 705–709.Google Scholar
9. Eschmeier, J., Tensor products and elementary operators, J. Reine Angew. Math. 390 (1988), 47–66.Google Scholar
10. Halmos, P. R., Quasitriangular operators, Acta Sci. Math. (Szeged) 29 (1968), 283–293.Google Scholar
11. Ichinose, T., On the spectra of tensor products of linear operators in Banach spaces, J. Reine Angew. Math. 244 (1970), 119–153.Google Scholar
12. Ichinose, T., Spectral properties of tensor products of linear operators I, Trans. Am. Math. Soc. 235 (1978), 75–113.Google Scholar
13. Ichinose, T., Spectral properties of tensor products of linear operators II: the approximate point spectrum and Kato essential spectrum, Trans. Am. Math. Soc. 237 (1978), 223–254.Google Scholar
14. Kitson, D., Harte, R. and Hernandez, C., Weyl’s theorem and tensor products: a counterexample, J. Math. Analysis Applic. 378 (2011), 128–132.Google Scholar
15. Kubrusly, C. S., A concise introduction to tensor product, Far East J. Math. Sci. 22 (2006), 137–174.Google Scholar
16. Kubrusly, C. S., The elements of operator theory (Birkhäuser, 2011).Google Scholar
17. Kubrusly, C. S., Spectral theory of operators on Hilbert spaces (Birkhäuser, 2012).CrossRefGoogle Scholar
18. Kubrusly, C. S., Contractions T for which A is a projection, Acta Sci. Math. (Szeged) 80 (2014), 603–624.Google Scholar
19. Kubrusly, C. S. and Duggal, B. P., On Weyl and Browder spectra of tensor products, Glasgow Math. J. 50 (2008), 289–302.CrossRefGoogle Scholar
20. Kubrusly, C. S. and Duggal, B. P., On Weyl’s theorem and tensor products, Glasgow Math. J. 55 (2013), 139–144.Google Scholar
21. Pearcy, C. M., Some recent developments in operator theory, CBMS Regional Conference Series in Mathematics, Volume 36 (American Mathematical Society/CBMS, 1978).CrossRefGoogle Scholar
22. Ryan, R., Introduction to tensor products of Banach spaces (Springer, 2002).Google Scholar
23. Schechter, M., On the spectra of operators on tensor products, J. Funct. Analysis 4 (1969), 95–99.Google Scholar
24. Song, Y.-M. and Kim, A.-H., Weyl’s theorem for tensor products, Glasgow Math. J. 46 (2004), 301–304.CrossRefGoogle Scholar
25. Weidmann, J., Linear operators in Hilbert spaces (Springer, 1980).CrossRefGoogle Scholar