Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T08:45:11.128Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetrisable operators :Part III Hilbert space operators Symmetrisable by bounded operators

Published online by Cambridge University Press:  09 April 2009

J. P. O. Silberstein
J. P. O. Silberstein
Affiliation:
Department of Mathematics, University of Western Australia.
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 fact that the most general symmetrisable operators in Hilbert Space do not possess a number of the desirable properties of such operators in unitary spaces makes it necessary to look for a more restricted class of operators. There are two reasons for our particular choice. In the first place many of the conditions introduced in the course of Part II concerned reltionships between the domain of the symmetrising operator H and the domain and range of the symmetrisable operator A. These conditions are now all automatically satisfied. The other reason is that the construction used in section 4 to relate symmetrisable operators to certain symmetric operators clearly required that either H or H−1 was bounded. The case of H−1 bounded has already been dealt with in section 9 and shown to be fairly simple. The case in which H is bounded is clearly of considerable complexity, since we have already seen (example in proof of Theorem 10.6.) that the con |H| the bound of H by

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 1 , February 1964 , pp. 31 - 48
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Dixmier, J., Bull. Soc. Math. France, 77 (1949) 11–101.Google Scholar
[2]Friedrichs, K. O., Math. Annalen 109 (1934) 465–487.Google Scholar
[3]von Neumann, J., Functional Operators Vol. II, Annals of Mathematics Studies 22, Princeton 1950.Google Scholar
[4]Silberstein, J. P. O., Dissertation, Cambridge, 1952.Google Scholar
[5]Silberstein, J. P. O., Symmetrisable Operators, This Journal 2 (1961–1962), 381–402.Google Scholar
[6]Silberstein, J. P. O., Symmetrisable Operators Part II, This Journal 3 (1963).Google Scholar
[7]Stone, M. H., Linear Transformations in Hilbert Space, Amer. Math. Soc. Colloquium Publications XV, New York 1932.Google Scholar