Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T01:28:53.187Z Has data issue: false hasContentIssue false

On the spectra of prespectral operators

Published online by Cambridge University Press:  18 May 2009

B. Nagy
Affiliation:
Department of Mathematics, Technical University, Budapest, Hungary
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 spectrum of a prespectral operator was investigated by Dowson in [4]. The question was left open there whether, if a prespectral operator has closed range, the same is true for its scalar part. In this paper we answer this in the affirmative and point out some consequences concerning the essential spectra of prespectral operators. Also, following Taylor and Halberg [11], we present the state diagram of a prespectral operator, which will show, in a sense, the sharpness of the results of the spectral theory of such operators.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1978

References

REFERENCES

1.Berkson, E. and Dowson, H. R., Prespectral operators, Illinois J. Math., 13 (1969), 291315.Google Scholar
2.Dowson, H. R., Restrictions of prespectral operators, J. London Math. Soc., (2) 1 (1969), 633642.CrossRefGoogle Scholar
3.Dowson, H. R., A commutativity theorem for prespectral operators, Illinois J. Math., 17 (1973), 525532.Google Scholar
4.Dowson, H. R., Some properties of prespectral operators, Proc. Roy. Irish Acad., Sect. A 74 (1974), 207221.Google Scholar
5.Dunford, N. and Schwartz, J. T., Linear operators—Part III: Spectral operators (Wiley-Interscience, New York, 1971).Google Scholar
6.Gramsch, B. and Lay, D., Spectral mapping theorems for essential spectra, Math. Ann., 192 (1971), 1732.Google Scholar
7.Nagy, B., Essential spectra of spectral operators, to appear.Google Scholar
9.Pietsch, A., Zur Theorie der cr-Transformationen in lokalkonvexen Vektorräumen, Math. Nachr., 21 (1960), 347369.CrossRefGoogle Scholar
10.Taylor, A. E., Introduction to functional analysis, (Wiley, New York, 1958).Google Scholar
11.Taylor, A. E. and Halberg, C. J. A., General theorems about a linear operator and its conjugate, J. Reine Angew. Math., 198 (1957), 93111.Google Scholar