Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-15T11:48:18.264Z Has data issue: false hasContentIssue false
  • English
  • Français

Proximity and similarity of operators II

Published online by Cambridge University Press:  18 May 2009

Charles Burnap  and
Alan Lambert
Charles Burnap
Affiliation:
Department of MathematicsThe University of North CarolinaAt Charlotte Charlotte, North Carolina 28223, U.S.A.
Alan Lambert
Affiliation:
Department of MathematicsThe University of North CarolinaAt Charlotte Charlotte, North Carolina 28223, U.S.A.
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.

In this paper we continue the examination of the question of similarity of operators A and B begun in reference [3]. In that article, a similarity result was obtained based on a measure of closeness, or proximity, of the uniformly continuous semigroups etA and etB, t>0. The operators considered were elements of ℬ(ℋ), the algebra of bounded operators on a Hilbert space ℬWe now wish to relax this requirement and replace ℬ(ℋ) by a complex Banach algebra ℬ with unit I. In Section 2 we give a necessary condition for the similarity of A, B ∈ ℋ. We then give a condition sufficient to guarantee A and B are approximately similar (as defined in reference [5]). In Section 3 we restrict our attention to the case where ℋ = ℋ(ℋ). There we give a condition which guarantees A, B ∈ ℋ(ℋ) are intertwined by a Fredholm operator. This leads naturally into a discussion of proximity-similarity in the Calkin algebra si. This is the subject of Section 4. Following reference [7] we define a metric p on N(ℋ), the normal elements of ℋ We show (N(ℋ), p) is a complete metric space and that the unitary orbit of ℋ (N(ℋ) p)is the p-connected component of a in N (ℋ).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 2 , May 1990 , pp. 205 - 213
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Apostol, C., Fialkow, L., Herrero, D., and Voiculescu, D., Approximation of Hilbert space operators, Volume II, Research Notes in Mathematics No. 102 (Pitman, 1984).Google Scholar
2.Brown, L. S., Douglas, R. G., and Fillmore, P. A., Unitary equivalence modulo the compact operators and extensions of compact operators, in Proceedings of a Conference on Operator Theory, Lecture Notes in Mathematics No. 345 (Springer-Verlag, 1973) 59–128.Google Scholar
3.Burnap, C. and Lambert, A., Proximity and similarity for operators, Proc. Royal Irish Acad., 87 Sect. A (1987), 95–102.Google Scholar
4.Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc, 17 (1966), 413–415.CrossRefGoogle Scholar
5.Hadwin, D. W., An asymptotic double commutant theorem for C*-algebras, Trans. Amer. Math. Soc, 244 (1978), 273–297.Google Scholar
6.Herrero, D., Approximation of Hilbert space operators, Volume I, Research notes in Mathematics No. 72 (Pitman, 1982).Google Scholar
7.Lambert, A., Equivalence for groups and semi-groups of operators, Indiana Univ. Math. J. 24 (1975), 879–885.CrossRefGoogle Scholar