Vol. 23, No. 1, 1967

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On generalized elements with respect to linear operators

Magnus Giertz

Vol. 23 (1967), No. 1, 47–67

Let X be the domain of a linear transformation A. Certain subspaces of the second algebraic conjugate Xff, obtained by the application of a weak completion process to some suitable subspace of X, may be regarded as spaces of generalized elements to which A has a natural extension. When A is a closed Hilbert space transformation, its domain can in this way be extended to a weakly complete space (Theorem 1). For a selfadjoint operator T this extension X may be regarded as the dual of a perfect countably Hilbert space precisely if T has a compact inverse (Theorem 2). Any element in X is obtained by a repeated application of the extended transformation T to some element in X (Theorem 3). A discussion of the extension of functions of T to X, and a spectral theory for T conclude the paper.

Mathematical Subject Classification
Primary: 47.25
Received: 20 June 1966
Published: 1 October 1967
Magnus Giertz