Vol. 20, No. 2, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Continuity of transformations which leave invariant certain translation invariant subspaces

Barry E. Johnson

Vol. 20 (1967), No. 2, 223–230

It is shown that a linear operator T : L2(X) L2(X) (X a locally compact group), with the property that TE E for each norm closed right translation invariant subspace E of L2(X), is necessarily continuous. In §5 the author shows that this is also true for L1(X) when X contains an element a which does not lie in any compact subgroup. An example is constructed to show that, in l(−∞,+),T can be discontinuous and still leave invariant each σ(l,l1) closed translation invariant subspace of l. If however T : l(−∞,+) l(−∞,+) leaves invariant all norm closed translation invariant subspaces, then T must be continuous.

Mathematical Subject Classification
Primary: 46.80
Secondary: 47.25
Received: 8 October 1965
Published: 1 February 1967
Barry E. Johnson