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A characterization of spectral operators on Hilbert spaces

Published online by Cambridge University Press:  18 May 2009

Ernst Albrecht
Ernst Albrecht
Affiliation:
Fachbereich Mathematik, Universität des Saarlandes, D-6600 Saarbrücken
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Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following property

Conversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 23 , Issue 1 , January 1982 , pp. 91 - 95
Copyright
Copyright © Glasgow Mathematical Journal Trust 1982

References

REFERENCES

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