Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-31T16:57:59.307Z Has data issue: false hasContentIssue false
  • English
  • Français

A structure theorem for operators with closed range

Published online by Cambridge University Press:  17 April 2009

James Guyker
James Guyker
Affiliation:
Department of Mathematics, State University of New York, College at Buffalo, Buffalo, New York, USA.
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.

A characterization has previously been given for linear transformations in Hilbert space whose first N + 1 powers are partial isometries. An analogous characterization is now obtained for transformations whose first N+ 1 powers have closed ranges. A hypothesis (that transformations have no isometric part) is found to be unnecessary in previous work.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 2 , April 1978 , pp. 169 - 186
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Ben-Israel, Adi, Greville, Thomas N.E., Generalized inverses: theory and applications (John Wiley & Sons, New York, London, Sydney, Toronto, 1974).Google Scholar
[2]Branges, Louis de, Rovnyak, James, “Canonical models in quantum scattering theory”,Perturbation theory and its applications in quantum mechanics, 295–392 (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Instit., University of Wisconsin, Madison, 16. John Wiley & Sons, New York, London, Sydney, 1966).Google Scholar
[3]Erdelyi, I van, “Partial isometries closed under multiplication on Hilbert spaces”, J. Math. Anal. Appl. 22 (1968), 546–551.CrossRefGoogle Scholar
[4]Fishel, B., “Partial isometries which are sums of shifts”, Math. Proc. Cambridge Philos. Soc. 78 (1975), 107–110.CrossRefGoogle Scholar
[5]Guyker, James, “On partial isometries with no isometric part”, Pacific J. Math. 62 (1976), 419–433.CrossRefGoogle Scholar
[6]Guyker, James, “Reducing subspaces of contractions with no isometric part”, Proc. Amer. Math. Soc. 45 (1974), 411–413.CrossRefGoogle Scholar
[7]Halmos, Paul R., A Hilbert space problem book (Van Nostrand, Princeton, New Jersey; Toronto; London; 1967)Google Scholar
[8]Halmos, P.R. & Wallen, L.J., “Powers of partial isometries”, J. Math. Mech. 19 (1970), 657–663.Google Scholar
[9]Lambert, Alan, “Unitary equivalence and reducibility of invertibly weighted shifts”, Bull. Austral. Math. Soc. 5 (1971), 157–173.CrossRefGoogle Scholar
[10]HиHoльсчий, H.H. [Nikol'skiĭ, N.K.], “0б иHβapиaHTHbix ⊓oд⊓pocTpaHCTβax βзβeiщeHHbix o⊓epatopoβ CдβиΓa” [Invariant subspaces of weighted shift operators],Mat. Sb. (N.S.) 74 (116) (1967), 171–190. English Transl.: Math. USSR Sb. 3 (1967), 159–176.Google Scholar