Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-20T12:36:44.828Z Has data issue: false hasContentIssue false

Joint spectra for commuting operators

Published online by Cambridge University Press:  20 January 2009

A. KällstrÖm
Affiliation:
Uppsala UniversityDepartment of Mathematics, Thunbergsvagen 3, S-752 38 UppsalaSweden
B.D. Sleeman
Affiliation:
Department of Mathematical SciencesThe UniversityDundee Dd1 4HnScotland, U.K.
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.

The theory of joint spectra for commuting operators in a Hilbert space has recently been studied by several authors (Vasilescu [11,12], Curto [4,5], and Cho-Takaguchi[2,3]). In this paper we willuse the definition by Taylor [10] of the joint spectrum to show that thejoint spectrum is determined by the action of certain "Laplacians"(cf. Curto [4,5]) of a chain-complex of Hilbert spaces.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1.Arens, R. and Calderon, P., Analytic functions of several Banach algebra elements, Ann. Math. 62 (1955), 204216.Google Scholar
2.Cho, M. and Takaguchi, M., Identity of Taylor's joint spectrum and Dash's joint spectrum, Stud. Math. 70 (1982), 225229.CrossRefGoogle Scholar
3.Cho, M. and Takaguchi, M., Boundary of Taylor's joint spectrum for two commuting operators, Rev. Roum. Math. Pures et Appl. 27 (1982), 863866.Google Scholar
4.Curto, R. E., Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), 129159.Google Scholar
5.Curto, R. E., On the connectedness of invertible n-tuples, Ind. Univ. Math. J. 29 (1980), 393406.Google Scholar
6.Dash, A. T., Joint spectra, Stud. Math. 45 (1973), 225237.CrossRefGoogle Scholar
7.Dash, A. T., On a conjecture concerning joint spectra, J. Func. Anal. 6 (1970), 165171.Google Scholar
8.Kato, T., Perturbation Theory for Linear Operators (Springer-Verlag (2nd ed.), 1976).Google Scholar
9.Putnam, C. R., Commutation Properties of Hilbert Space Operators and Related Topics. (Springer-Verlag, 1967).CrossRefGoogle Scholar
10.Taylor, J. L., A joint spectrum for several commuting operators, J. Func. Anal. 6 (1970), 172191.Google Scholar
11.Vasilescu, F. H., On pairs of commuting operators, Stud. Math. 62 (1978), 203207.CrossRefGoogle Scholar
12.Vasilescu, F. H., A characterization of the joint spectrum in Hilbert spaces, Rev. Roum. Math. Pures et Appl. 22 (1977), 10031009.Google Scholar