Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-17T14:44:15.959Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on d-symmetric operators

Published online by Cambridge University Press:  17 April 2009

B. C. Gupta  and
P. B. Ramanujan
B. C. Gupta
Affiliation:
Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388120, Gujarat, India
P. B. Ramanujan
Affiliation:
Department of Mathematics, Saurashtra University, Rajkot 360 005, Gujarat, India.
Rights & Permissions [Opens in a new window]

Abstract

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.

An operator T on a complex Hilbert space is d-symmetric if , where is the uniform closure of the range of the derivation operator δT(X)=TX−XT. It is shown that if the commutator ideal of the inclusion algebra for a d-symmetric operator is the ideal of all compact operators then T has countable spectrum and T is a quasidiagonal operator. It is also shown that if for a d-symmetric operator I(T) is the double commutant of T then T is diagonal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 3 , June 1981 , pp. 471 - 475
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Anderson, Joel, Bunce, John W., Deddens, James A., and Williams, J. P., “C *-algebras and derivation ranges”, Acta Sci. Math. (Szeged) 40 (1978), 211–227.Google Scholar
[2]Brown, L.G., Douglas, R.G. and Fillmore, P.A., “Unitary equivalence modulo the compact operators and extensions of C* -algebras”,Proceedings of Conference on Operator Theory,Halifax, Nova Scotia,1973, 58–128 (Lecture Motes in Mathematics, 345. Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[3]Fillmore, P.A., Stampfli, J.G., and Williams, J.P., “On the essential numerical range, the essential spectrum, and a problem of Halmos”, Acta Sci. Math. (Szeged) 33 (1972), 179–192.Google Scholar
[4]Rudin, Walter, Real and complex analysis (McGraw Hill, New York, London, Sydney, 1966).Google Scholar
[5]Stampfli, Joseph G., “On self-adjoint derivation ranges”, Pacific J. Math. 82 (1979), 257–277.CrossRefGoogle Scholar