Vol. 20, No. 2, 1967

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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