Continuity of transformations which leave invariant certain translation invariant subspaces

Barry E. Johnson

Vol. 20 (1967), No. 2, 223–230

DOI: 10.2140/pjm.1967.20.223

Abstract

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

Milestones

Received: 8 October 1965

Published: 1 February 1967