1. Some of the results of the present article can be obtained on the basis of theorems derived by E. Cartan in hisLecons sur la Géométric Projective Complexe(Gauthier-Villars, Paris, 1931).
2. I am much indebted to Professor S. Bochner for drawing my attention to the very profound investigations contained in this treatise, which deals with general linear and antilinear transformations. However, the direct derivations, given in the text of the present paper, are hardly longer than the reinterpretation and amplification of Cartan’s results (see particularly pp. 124–137) would have been.
3. The two vectors υωk and υω∗k form a plane in our Hilbert space. The line which corresponds to this plane in Cartan’s projective space is the invariant line of the passage cited in footnote reference 1.