On generalized elements with respect to linear operators
Magnus Giertz · Pacific Journal of Mathematics · 1967
Let Xbe 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 ele-ments to which A has a natural extension. When A is a closed Hubert space transformation, its domain can in this way be extended to a weakly complete space (Theorem 1). For a self-adjoint operator T this extension X may be regarded as the dual of a perfect countably Hubert 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 ex-tension of functions of T to X, and a spectral theory for T conclude the paper. The most widely accepted and used method of defining distributions