Vol. 101, No. 1, 1982

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Dilations on locally convex spaces

James E. Simpson

Vol. 101 (1982), No. 1, 185–192

The simultaneous dilation of a group of contractions on a Hilbert space by a group of isometries is generalized here to operators on locally convex spaces. The basic construction, using a quotient of a large direct sum, is found in the Hilbert space treatment. The particular difficulties to be overcome and innovations introduced here relate to the definition of contraction and isometry for operators on a locally convex space, and to the handling of various topologies on the operators under scrutiny. With these definitions the traditional dilation of a contraction by an isometry is recovered. Finally we have a variation of the basic dilation theorem particularly suited to semi-groups of operators on locally convex spaces and to spectral operators.

Mathematical Subject Classification 2000
Primary: 47A20
Secondary: 47D05, 46A05
Received: 17 March 1980
Revised: 20 May 1981
Published: 1 July 1982
James E. Simpson