Hostname: page-component-5c6d5d7d68-xq9c7 Total loading time: 0 Render date: 2024-08-29T11:06:46.519Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Essential Spectra of Quasisimilar Operators

Published online by Cambridge University Press:  20 November 2018

Domingo A. Herrero
Domingo A. Herrero*
Affiliation:
Arizona State University, Temple, Arizona
Rights & Permissions [Opens in a new window]

Extract

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.

B. Sz.-Nagy and C. Foiaş gave the first examples of quasisimilar operators with different spectra [25]. Indeed, quasisimilar operators can even have different spectral radius [19] (see also [15]). Nevertheless, T. B. Hoover has shown in [19] that if T and S are quasisimilar, then σ(T)∩σ(S) ≠ ∅, where σ(R) denotes the spectrum of the (bounded linear) operator R. In [12], the author improved Hoover's result by showing that each component of σ(S) intersects σ(T), and viceversa. This, in turn, was further improved by L. A. Fialkow in [7]. (Actually, Fialkow's results were obtained independently of [12].)

L. A. Fialkow [7] and L. R. Williams [27] independently proved that σe(T)∩σe(S) ≠ ∅, where σe(R) denotes the essential spectrum of R. Several authors have raised the following natural question:

Is it also true that each component of σe(S) intersects σe(T), and viceversa? [7] [21] [26].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 6 , 01 December 1988 , pp. 1436 - 1457
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Apostol, C., Quasitriangularity in Hilbert space, Indiana Univ. Math. J. 22 (1973), 817–825.Google Scholar
2. Apostol, C., Operators quasisimilar to a normal operator, Proc. Amer. Math. Soc. 53 (1975), 104–106.Google Scholar
3. Apostol, C., Douglas, R. G. and Foiaş, C., Quasi-similarity models for nilpotent operators, Trans. Amer. Math. Soc. 224 (1976), 407–415.Google Scholar
4. Apostol, C., Fialkow, L. A., Herrero, D. A. and Voiculescu, D., Approximation of Hilbert space operators, Volume II, Research Notes in Math. 102 (Pitman, Boston-London-Melbourne, 1984).Google Scholar
5. Bercovici, H., Foiaş, C. and Pearcy, C. M., Dual algebras with applications to invariant subspaces and dilation theory, Regional Conference Series in Mathematics 56 (Amer. Math. Soc, Providence, R.I., 1985).CrossRefGoogle Scholar
6. Colojara, I. and Foiaş, C., Theory of generalized spectral operators (Gordon and Breach, New York, 1968).Google Scholar
7. Fialkow, L. A., A note on quasisimilarity of operators, Acta Sci. Math. (Szeged) 39 (1977), 67–85.Google Scholar
8. Fialkow, L. A., A note on quasisimilarity. II, Pac. J. Math. 70 (1977), 151–162.Google Scholar
9. Fialkow, L. A., Weighted shifts quasisimilar to quasinilpotent operators, Acta Sci. Math. (Szeged) 42 (1980), 71–79.Google Scholar
10. Fialkow, L. A., Quasisimilarity orbits and closures of similarity orbits of operators, J. Operator Theory 14 (1985), 215–238.Google Scholar
11. Gohberg, I. C. and Krein, M. G., The basic propositions on defect numbers, root numbers and indices of linear operators, Uspehi Mat. Nauk. SSSR 12 (1957), no. 2(74), 43–118 (Russian); English Transi. Amer. Math. Soc. Transi. (2)13 (1960), 185–264.Google Scholar
12. Herrero, D. A., On the spectra of the restrictions of an operator, Trans. Amer. Math. Soc. 233 (1977), 45–58.Google Scholar
13. Herrero, D. A., Quasisimilarity does not preserve the hyperlattice, Proc. Amer. Math. Soc. 65 (1978), 80–84.Google Scholar
14. Herrero, D. A., On multicyclic operators, Integral Equations and Operator Theory 1 (1978), 57–102.Google Scholar
15. Herrero, D. A., Quasisimilar operators with different spectra, Acta Sci. Math. (Szeged) 41 (1979), 101–118.Google Scholar
16. Herrero, D. A., Approximation of Hilbert space operators, Volume I, Research Notes in Math. 72, (Pitman, Boston-London-Melbourne, 1982).Google Scholar
17. Herrero, D. A., The Fredholm structure of an n-multicyclic operator, Indiana Univ. Math. J. 36 (1987), 549–566.Google Scholar
18. Herrero, D. A., Most quasitriangular operators are triangular, most biquasitriangular operators are bitriangular, J. Operator Theory 20 (1988), 251–267.Google Scholar
19. Hoover, T. B., Quasisimilarity of operators, Illinois J. Math. 16 (1972), 672–686.Google Scholar
20. Kato, T., Perturbation theory for linear operators (Springer-Verlag, New York, 1966).Google Scholar
21. Shapiro, J. H., Zeros of functions in weighted Bergman spaces, Michigan Math. J. 24 (1977), 243–256.Google Scholar
22. Shields, A. L., Weighted shift operators and analytic function theory, in Math. Surveys 13 (Amer. Math. Soc, Providence, R.I., 1974), 49–128.Google Scholar
23. Stampfli, J. G., Quasisimilarity of operators, Proc. R. Ir. Acad. 81 A (1981), 109–119.Google Scholar
24. Sz.-Nagy, B. and Foiaş, C., Propriétés des fonctions caractéristiques, modèles triangulaires et une classification des contractions, C.R. Acad. Sci. Paris 258 (1963), 3413–3415.Google Scholar
25. Sz.-Nagy, B. and Foiaş, C., Harmonie analysis of operators on Hilbert space (North-Holland, Amsterdam, 1970).Google Scholar
26. Sz.-Nagy, B. and Foiaş, C., Modèles de Jordan pour une class d'operateurs de classe Co, Acta Sci. Math (Szeged) 31 (1970), 287–296.Google Scholar
27. Williams, L. R., Quasisimilar operators have overlapping essential spectra, preprint (1976).Google Scholar
28. Williams, L. R., Equality of essential spectra of quasisimilar quasinormal operators, J. Operator Theory 3 (1980), 57–69.Google Scholar